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%I M0566 N0204 #171 Aug 10 2023 10:30:16
%S 1,1,2,3,4,6,9,12,16,22,29,38,50,64,82,105,132,166,208,258,320,395,
%T 484,592,722,876,1060,1280,1539,1846,2210,2636,3138,3728,4416,5222,
%U 6163,7256,8528,10006,11716,13696,15986,18624,21666,25169,29190,33808,39104,45164
%N Number of partitions with no even part repeated; partitions of n in which no parts are multiples of 4.
%C Also number of partitions of n where no part appears more than three times.
%C a(n) satisfies Euler's pentagonal number (A001318) theorem, unless n is in A062717 (see Fink et al.).
%C Also number of partitions of n in which the least part and the differences between consecutive parts is at most 3. Example: a(5)=6 because we have [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1] and [1,1,1,1,1]. - _Emeric Deutsch_, Apr 19 2006
%C Equals A000009 convolved with its aerated variant, = polcoeff A000009 * A000041 * A010054 (with alternate signs). - _Gary W. Adamson_, Mar 16 2010
%C Equals left border of triangle A174715. - _Gary W. Adamson_, Mar 27 2010
%C The Cayley reference is actually to A083365. - _Michael Somos_, Feb 24 2011
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Convolution of A000009 and A035457. - _Vaclav Kotesovec_, Aug 23 2015
%C Convolution inverse is A082303. - _Michael Somos_, Sep 30 2017
%D A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.
%D I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.5.2).
%D M. D. Hirschhorn, The Power of q, Springer, 2017. See ped page 303ff.
%D R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 241.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Seiichi Manyama, <a href="/A001935/b001935.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from T. D. Noe)
%H George 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. (See Th. 9.)
%H George E. Andrews, <a href="https://hal.archives-ouvertes.fr/hal-03498190/">Partition Identities for Two-Color Partitions</a>, Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 2021, Special Commemorative volume in honour of Srinivasa Ramanujan, 2021, 44, pp.74-80. hal-03498190. See Theorem 1.4 p. 75.
%H Riccardo Aragona, Roberto Civino, and Norberto Gavioli, <a href="https://arxiv.org/abs/2301.06347">A modular idealizer chain and unrefinability of partitions with repeated parts</a>, arXiv:2301.06347 [math.RA], 2023.
%H Cristina Ballantine and Mircea Merca, <a href="https://doi.org/10.1007/s11139-016-9845-6">Parity of sums of partition numbers and squares in arithmetic progressions</a>, The Ramanujan Journal, 2016.
%H A. Cayley, <a href="/A001934/a001934.pdf">A memoir on the transformation of elliptic functions</a>, Philosophical Transactions of the Royal Society of London (1874): 397-456; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, included in Vol. 9. [Annotated scan of pages 126-129.]
%H S.-C. Chen, <a href="http://dx.doi.org/10.1016/j.disc.2011.02.025">On the number of partitions with distinct even parts</a>, Discrete Math., 311 (2011), 940-943.
%H A. Fink, R. K. Guy and M. Krusemeyer, <a href="https://doi.org/10.11575/cdm.v3i2.61940">Partitions with parts occurring at most thrice</a>, Contributions to Discrete Mathematics, Vol 3, No 2 (2008).
%H M. D. Hirschhorn and J. A. Sellers, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Sellers/sellers32.html">A Congruence Modulo 3 for Partitions into Distinct Non-Multiples of Four</a>, Article 14.9.6, Journal of Integer Sequences, Vol. 17 (2014).
%H Joro, <a href="http://mathoverflow.net/questions/59192/">Is "OEIS A001935 Number of partitions with no even part repeated" efficiently computable mod 4?</a>
%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 15.
%H Mircea Merca, <a href="https://dx.doi.org/10.1016/j.jnt.2016.12.015">New relations for the number of partitions with distinct even parts</a>, Journal of Number Theory 176 (July 2017), 1-12.
%H Alexander Patkowski, <a href="http://demmath.mini.pw.edu.pl/archive/dm42_2/4.pdf">On some partitions where even parts do not repeat</a>, Demonstratio Mathematica Volume 42, Issue 2 (Jun 2009), pp. 259-263.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PartitionFunctionb.html">Partition Function b_k</a> and <a href="http://mathworld.wolfram.com/PartitionFunctionP.html">Partition Function P</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Glaisher%27s_theorem">Glaisher's Theorem</a>.
%F Euler transform of period 4 sequence [ 1, 1, 1, 0, ...].
%F Expansion of q^(-1/8) * eta(q^4) / eta(q) in powers of q. - _Michael Somos_, Mar 19 2004
%F Expansion of psi(-x) / phi(-x) = psi(x) / phi(-x^2) = psi(x^2) / psi(-x) = chi(x) / chi(-x^2)^2 = 1 / (chi(x) * chi(-x)^2) = 1 / (chi(-x) * chi(-x^2)) = f(-x^4) / f(-x) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions. - _Michael Somos_, Jul 08 2011
%F G.f.: Product(j>=1, 1 + x^j + x^(2*j) + x^(3*j)). - _Jon Perry_, Mar 30 2004
%F G.f.: Product_{k>=1} (1+x^k)^(2-k%2). - _Jon Perry_, May 05 2005
%F G.f.: Product_{k>0} (1 + x^(2*k)) / (1 - x^(2*k-1)) = 1 + Sum_{k>0}(Product_{i=1..k} (x^i + 1) / (x^-i - 1)).
%F G.f.: Sum_{n>=0} ( x^(n*(n+1)/2) * Product_{k=1..n} (1+x^k)/(1-x^k) ). - _Joerg Arndt_, Apr 07 2011
%F G.f.: P(x^4)/P(x) where P(x) = Product_{k>=1} 1-x^k. - _Joerg Arndt_, Jun 21 2011
%F A083365(n) = (-1)^n a(n). Convolution square is A001936. a(n) = A098491(n) + A098492(n). a(2*n) = A081055(n). a(2*n + 1) = A081056(n).
%F G.f.: (1+ 1/G(0))/2, where G(k) = 1 - x^(2*k+1) - x^(2*k+1)/(1 + x^(2*k+2) + x^(2*k+2)/G(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Jul 03 2013
%F G.f.: exp( Sum_{n>=1} (x^n/n) / (1 + (-x)^n) ). - _Paul D. Hanna_, Jul 24 2013
%F a(n) ~ Pi * BesselI(1, sqrt(8*n + 1)*Pi/4) / (2*sqrt(8*n + 1)) ~ exp(Pi*sqrt(n/2)) / (4 * (2*n)^(3/4)) * (1 + (Pi/(16*sqrt(2)) - 3/(4*Pi*sqrt(2))) / sqrt(n) + (Pi^2/1024 - 15/(64*Pi^2) - 15/128) / n). - _Vaclav Kotesovec_, Aug 23 2015, extended Jan 14 2017
%F a(n) = (1/n)*Sum_{k=1..n} A046897(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, Mar 25 2017
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (256 t)) = 1/2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A082303. - _Michael Somos_, Sep 30 2017
%e G.f. = 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 6*x^5 + 9*x^6 + 12*x^7 + 16*x^8 + 22*x^9 + ...
%e G.f. = q + q^9 + 2*q^17 + 3*q^25 + 4*q^33 + 6*q^41 + 9*q^49 + 12*q^57 + 16*q^65 + 22*q^73 + ...
%e a(5)=6 because we have [5], [4,1], [3,2], [3,1,1], [2,1,1,1] and [1,1,1,1,1].
%p g:=product((1+x^j)*(1+x^(2*j)),j=1..50): gser:=series(g,x=0,55): seq(coeff(gser,x,n),n=0..48); # _Emeric Deutsch_, Apr 19 2006
%p # second Maple program:
%p with(numtheory):
%p a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
%p `if`(irem(d, 4)=0, 0, d), d=divisors(j)), j=1..n)/n)
%p end:
%p seq(a(n), n=0..50); # _Alois P. Heinz_, Nov 24 2015
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q] / EllipticTheta[ 2, Pi/4, q^(1/2)] / (16 q)^(1/8), {q, 0, n}]; (* _Michael Somos_, Jul 11 2011 *)
%t a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, 4, n, 4}] / Product[ 1 - x^k, {k, n}], {x, 0, n}]; (* _Michael Somos_, Jul 08 2011 *)
%t CoefficientList[Series[Product[1+x^j+x^(2j)+x^(3j), {j,1,48}], {x,0,48}],x] (* _Jean-François Alcover_, May 26 2011, after _Jon Perry_ *)
%t QP = QPochhammer; CoefficientList[QP[q^4]/QP[q] + O[q]^50, q] (* _Jean-François Alcover_, Nov 24 2015 *)
%t a[0] = 1; a[n_] := a[n] = Sum[a[n-j] DivisorSum[j, If[Divisible[#, 4], 0, #]&], {j, 1, n}]/n; Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Feb 19 2016, after _Alois P. Heinz_ *)
%t Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 4], 0, 2] ], {n, 0, 49}] (* _Robert Price_, Jul 28 2020 *)
%o (PARI) {a(n) = if( n<0, 0, polcoeff( eta(x^4 + x * O(x^n)) / eta(x + x * O(x^n)), n))};
%o (PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint( 8*n + 1) - 1)\2, prod(i=1, k, (1 + x^i) / (x^-i - 1), 1 + x * O(x^n))), n))}; /* _Michael Somos_, Jun 01 2004 */
%o (PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/(1+(-x)^m+x*O(x^n))/m)),n)} \\ _Paul D. Hanna_, Jul 24 2013
%o (Haskell)
%o a001935 = p a042968_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_, Sep 02 2012
%Y Cf. A000009, A000726, A001936, A035959, A035985, A042968, A061198, A061199, A070048, A081055, A081056, A083365, A098491, A098492, A219601.
%Y Cf. A000041, A010054. - _Gary W. Adamson_, Mar 16 2010
%Y Cf. A174715. - _Gary W. Adamson_, Mar 27 2010
%Y Cf. A082303.
%Y Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_, _Simon Plouffe_, _Robert G. Wilson v_
%E More terms from _James A. Sellers_
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