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%I M0266 #145 Aug 03 2020 01:35:44
%S 1,1,1,1,2,2,3,3,4,5,6,7,9,10,12,14,17,19,23,26,31,35,41,46,54,61,70,
%T 79,91,102,117,131,149,167,189,211,239,266,299,333,374,415,465,515,
%U 575,637,709,783,871,961,1065,1174,1299,1429,1579,1735,1913,2100,2311,2533,2785
%N Number of partitions of n into parts 5k+1 or 5k+4.
%C Expansion of Rogers-Ramanujan function G(x) in powers of x.
%C Same as number of partitions into distinct parts where the difference between successive parts is >= 2.
%C As a formal power series, the limit of polynomials S(n,x): S(n,x)=sum(T(i,x),0<=i<=n); T(i,x)=S(i-2,x).x^i; T(0,x)=1,T(1,x)=x; S(n,1)=A000045(n+1), the Fibonacci sequence. - Claude Lenormand (claude.lenormand(AT)free.fr), Feb 04 2001
%C The Rogers-Ramanujan identity is 1 + Sum_{n >= 1} t^(n^2)/((1-t)*(1-t^2)*...*(1-t^n)) = Product_{n >= 1} 1/((1-t^(5*n-1))*(1-t^(5*n-4))).
%C Coefficients in expansion of permanent of infinite tridiagonal matrix:
%C 1 1 0 0 0 0 0 0 ...
%C x 1 1 0 0 0 0 0 ...
%C 0 x^2 1 1 0 0 0 ...
%C 0 0 x^3 1 1 0 0 ...
%C 0 0 0 x^4 1 1 0 ...
%C ................... - _Vladeta Jovovic_, Jul 17 2004
%C Also number of partitions of n such that the smallest part is greater than or equal to number of parts. - _Vladeta Jovovic_, Jul 17 2004
%C Also number of partitions of n such that if k is the largest part, then each of {1, 2, ..., k-1} occur at least twice. Example: a(9)=5 because we have [3, 2, 2, 1, 1], [2, 2, 2, 1, 1, 1], [2, 2, 1, 1, 1, 1, 1], [2, 1, 1, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1]. - _Emeric Deutsch_, Feb 27 2006
%C Also number of partitions of n such that if k is the largest part, then k occurs at least k times. Example: a(9)=5 because we have [3, 3, 3], [2, 2, 2, 2, 1], [2, 2, 2, 1, 1, 1], [2, 2, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1]. - _Emeric Deutsch_, Apr 16 2006
%C a(n) = number of NW partitions of n, for n >= 1; see A237981.
%C For more about the generalized Rogers-Ramanujan series G[i](x) see the Andrews-Baxter and Lepowsky-Zhu papers. The present series is G[1](x). - _N. J. A. Sloane_, Nov 22 2015
%C Convolution of A109700 and A109697. - _Vaclav Kotesovec_, Jan 21 2017
%D G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109, 238.
%D G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1999; Exercise 6(e), p. 591.
%D Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 669.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 107.
%D G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, pp. 90-92.
%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, pp. 290-291.
%D H. P. Robinson, Letter to N. J. A. Sloane, Jan 04 1974.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vaclav Kotesovec, <a href="/A003114/b003114.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from T. D. Noe)
%H G. E. Andrews, <a href="http://www.mat.univie.ac.at/~slc/opapers/s25andrews.html">Three aspects of partitions</a>, Séminaire Lotharingien de Combinatoire, B25f (1990), 1 p.
%H G. E. Andrews, <a href="http://dx.doi.org/10.1090/S0273-0979-07-01180-9">Euler's "De Partitio Numerorum"</a>, Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573.
%H George E. Andrews; R. J. Baxter, <a href="https://doi.org/10.1080/00029890.1989.11972207">A motivated proof of the Rogers-Ramanujan identities</a>, Amer. Math. Monthly 96 (1989), no. 5, 401-409.
%H R. K. Guy, <a href="/A005169/a005169_6.pdf">Letter to N. J. A. Sloane</a>, Sep 25 1986.
%H R. K. Guy, <a href="/A005728/a005728.pdf">Letter to N. J. A. Sloane, 1987</a>
%H R. K. Guy, <a href="/A000081/a000081.pdf">Letter to N. J. A. Sloane, 1988-04-12</a> (annotated scanned copy)
%H R. K. Guy, <a href="http://www.jstor.org/stable/2322249">The strong law of small numbers</a>. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
%H R. K. Guy, <a href="/A005165/a005165.pdf">The strong law of small numbers</a>. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
%H P. Jacob and P. Mathieu, <a href="http://arXiv.org/abs/hep-th/0505097">Parafermionic derivation of Andrews-type multiple-sums</a>, arXiv:hep-th/0505097, 2005.
%H James Lepowsky and Minxian Zhu, <a href="http://arxiv.org/abs/1205.6570">A motivated proof of Gordon's identities</a>, The Ramanujan Journal 29.1-3 (2012): 199-211.
%H I. Martinjak, D. Svrtan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Martinjak/mart3.html">New Identities for the Polarized Partitions and Partitions with d-Distant Parts</a>, J. Int. Seq. 17 (2014) # 14.11.4.
%H Herman P. Robinson, <a href="/A003105/a003105.pdf">Letter to N. J. A. Sloane, Jan 1974</a>.
%H A. V. Sills, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v10i1r13">Finite Rogers-Ramanujan type identities</a>, Electron. J. Combin. 10 (2003), Research Paper 13, 122 pp.
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Rogers-RamanujanIdentities.html">Rogers-Ramanujan Identities.</a>
%H Mingjia Yang, Doron Zeilberger, <a href="https://arxiv.org/abs/1910.08989">Systematic Counting of Restricted Partitions</a>, arXiv:1910.08989 [math.CO], 2019.
%F G.f.: Sum_{k>=0} x^(k^2)/(Product_{i=1..k} 1-x^i).
%F The g.f. above is the special case D=2 of sum(n>=0, x^(D*n*(n+1)/2 - (D-1)*n) / prod(k=1..n, 1-x^k) ), the g.f. for partitions into distinct part where the difference between successive parts is >= D. - _Joerg Arndt_, Mar 31 2014
%F G.f.: 1 + sum(i=1, oo, x^(5i+1)/prod(j=1 or 4 mod 5 and j<=5i+1, 1-x^j) + x^(5i+4)/prod(j=1 or 4 mod 5 and j<=5i+4, 1-x^j)). - _Jon Perry_, Jul 06 2004
%F G.f.: (Product_{k>0} 1 + x^(2*k)) * (Sum_{k>=0} x^(k^2) / (Product_{i=1..k} 1 - x^(4*i))). - _Michael Somos_, Oct 19 2006
%F Euler transform of period 5 sequence [ 1, 0, 0, 1, 0, ...]. - _Michael Somos_, Oct 15 2008
%F Expansion of f(-x^5) / f(-x^1, -x^4) in powers of x where f(,) is the Ramanujan general theta function. - _Michael Somos_, May 17 2015
%F Expansion of f(-x^2, -x^3) / f(-x) in powers of x where f(,) is the Ramanujan general theta function. - _Michael Somos_, Jun 13 2015
%F a(n) ~ phi^(1/2) * exp(2*Pi*sqrt(n/15)) / (2 * 3^(1/4) * 5^(1/2) * n^(3/4)) * (1 - (3*sqrt(15)/(16*Pi) + Pi/(60*sqrt(15))) / sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Aug 23 2015, extended Jan 24 2017
%F a(n) = (1/n)*Sum_{k=1..n} A284150(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, Mar 21 2017
%e G.f. = 1 + x + x^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 4*x^8 + 5*x^9 + ...
%e G.f. = 1/q + q^59 + q^119 + q^179 + 2*q^239 + 2*q^299 + 3*q^359 + 3*q^419 + ...
%e From _Joerg Arndt_, Dec 27 2012: (Start)
%e The a(16)=17 partitions of 16 where all parts are 1 or 4 (mod 5) are
%e [ 1] [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ]
%e [ 2] [ 4 1 1 1 1 1 1 1 1 1 1 1 1 ]
%e [ 3] [ 4 4 1 1 1 1 1 1 1 1 ]
%e [ 4] [ 4 4 4 1 1 1 1 ]
%e [ 5] [ 4 4 4 4 ]
%e [ 6] [ 6 1 1 1 1 1 1 1 1 1 1 ]
%e [ 7] [ 6 4 1 1 1 1 1 1 ]
%e [ 8] [ 6 4 4 1 1 ]
%e [ 9] [ 6 6 1 1 1 1 ]
%e [10] [ 6 6 4 ]
%e [11] [ 9 1 1 1 1 1 1 1 ]
%e [12] [ 9 4 1 1 1 ]
%e [13] [ 9 6 1 ]
%e [14] [ 11 1 1 1 1 1 ]
%e [15] [ 11 4 1 ]
%e [16] [ 14 1 1 ]
%e [17] [ 16 ]
%e The a(16)=17 partitions of 16 where successive parts differ by at least 2 are
%e [ 1] [ 7 5 3 1 ]
%e [ 2] [ 8 5 3 ]
%e [ 3] [ 8 6 2 ]
%e [ 4] [ 9 5 2 ]
%e [ 5] [ 9 6 1 ]
%e [ 6] [ 9 7 ]
%e [ 7] [ 10 4 2 ]
%e [ 8] [ 10 5 1 ]
%e [ 9] [ 10 6 ]
%e [10] [ 11 4 1 ]
%e [11] [ 11 5 ]
%e [12] [ 12 3 1 ]
%e [13] [ 12 4 ]
%e [14] [ 13 3 ]
%e [15] [ 14 2 ]
%e [16] [ 15 1 ]
%e [17] [ 16 ]
%e (End)
%p g:=sum(x^(k^2)/product(1-x^j,j=1..k),k=0..10): gser:=series(g,x=0,65): seq(coeff(gser,x,n),n=0..60); # _Emeric Deutsch_, Feb 27 2006
%t CoefficientList[ Series[Sum[x^k^2/Product[1 - x^j, {j, 1, k}], {k, 0, 10}], {x, 0, 65}], x][[1 ;; 61]] (* _Jean-François Alcover_, Apr 08 2011, after _Emeric Deutsch_ *)
%t Table[Count[IntegerPartitions[n], p_ /; Min[p] >= Length[p]], {n, 24}] (* _Clark Kimberling, Feb 13 2014 *)
%t a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ x^1, x^5] QPochhammer[ x^4, x^5]), {x, 0, n}]; (* _Michael Somos_, May 17 2015 *)
%t a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{-1, 0, 0, -1, 0}[[Mod[k, 5, 1]]], {k, n}], {x, 0, n}]; (* _Michael Somos_, May 17 2015 *)
%t nmax = 60; kmax = nmax/5;
%t s = Flatten[{Range[0, kmax]*5 + 1}~Join~{Range[0, kmax]*5 + 4}];
%t Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* _Robert Price_, Aug 02 2020 *)
%o (PARI) {a(n) = my(t); if( n<0, 0, t = 1 + x * O(x^n); polcoeff( sum(k=1, sqrtint(n), t *= x^(2*k - 1) / (1 - x^k) * (1 + x * O(x^(n - k^2))), 1), n))}; /* _Michael Somos_, Oct 15 2008 */
%o (Haskell)
%o a003114 = p a047209_list where
%o p _ 0 = 1
%o p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
%o -- _Reinhard Zumkeller_, Jan 05 2011
%o (Haskell)
%o a003114 = p 1 where
%o p _ 0 = 1
%o p k m = if k > m then 0 else p (k + 2) (m - k) + p (k + 1) m
%o -- _Reinhard Zumkeller_, Feb 19 2013
%Y Cf. A003106, A003116, A127836, A003113, A006141, A039899, A039900.
%Y Cf. A188216 (least part k occurs at least k times).
%Y Cf. A047209, A203776, A237981.
%Y For the generalized Rogers-Ramanujan series G[1], G[2], G[3], G[4], G[5], G[6], G[7], G[8] see A003114, A003106, A006141, A264591, A264592, A264593, A264594, A264595. G[0] = G[1]+G[2] is given by A003113.
%Y Row sums of A268187.
%K easy,nonn,nice
%O 0,5
%A _N. J. A. Sloane_, _Herman P. Robinson_
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