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%I #71 Apr 19 2024 11:14:36
%S 1,1,2,3,5,2,6,5,7,5,12,6,12,6,10,11,16,7,20,15,12,12,22,10,25,12,20,
%T 18,30,10,32,21,24,16,30,21,36,20,24,25,42,12,42,36,35,22,46,22,43,25,
%U 32,36,52,20,60,30,40,30,60,30,62,32,42,43,60,24,66,48,44,30,72,35,72
%N Number of 5-core partitions of n.
%C Number 11 of the 74 eta-quotients listed in Table I of Martin (1996).
%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988; see p. 54 (1.52).
%H Seiichi Manyama, <a href="/A053723/b053723.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from T. D. Noe)
%H Subhajit Bandyopadhyay and Nayandeep Deka Baruah, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL27/Baruah/bar6.html">Arithmetic Identities for Some Analogs of the 5-Core Partition Function</a>, J. Int. Seq. (2024) Vol. 27, Art. No. 24.4.5.
%H Bruce C. Berndt, <a href="http://www.math.uiuc.edu/~berndt/articles/rrcf.pdf">The Rogers-Ramanujan continued fraction</a>.
%H Bruce C. Berndt et al., <a href="http://doi.org/10.1016/S0377-0427(99)00033-3">The Rogers-Ramanujan continued fraction</a>, Journal of Computational and Applied Mathematics, Vol. 105, No. 1-2 (1999), 9-24.
%H Freeman J. Dyson, <a href="https://doi.org/10.1090/S0002-9904-1972-12971-9">Missed opportunities</a>, Bull. Amer. Math. Soc. 78 (1972), 635-652. see pages 636-637.
%H Frank Garvan, Dongsu Kim, and Dennis Stanton, <a href="https://www-users.cse.umn.edu/~stant001/PAPERS/cra2.pdf">Cranks and t-cores</a>.
%H Frank Garvan, Dongsu Kim, and Dennis Stanton, <a href="https://gdz.sub.uni-goettingen.de/id/PPN356556735_0101?tify=%7B%22pages%22%3A%5B7%5D%2C%22view%22%3A%22toc%22%7D">Cranks and t-cores</a>, Inventiones Math. 101 (1990) 1-17.
%H Yves Martin, <a href="http://dx.doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
%H Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>, 2016.
%F Given g.f. A(x), then B(q) = q * A(q) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = v^3 + 2 * u*v*w + 4 * u*w^2 - u^2*w. - _Michael Somos_, May 02 2005
%F G.f.: (1/x) * (Sum_{k>0} Kronecker(k, 5) * x^k / (1 - x^k)^2). - _Michael Somos_, Sep 02 2005
%F G.f.: Product_{k>0} (1 - x^(5*k))^5 / (1 - x^k) = 1/x * (Sum_{k>0} k * x^k * (1 - x^k) * (1 - x^(2*k)) / (1 - x^(5*k))). - _Michael Somos_, Jun 17 2005
%F G.f.: (1/x) * Sum_{a, b, c, d, e in Z^5} x^((a^2 + b^2 + c^2 + d^2 + e^2) / 10) where a + b + c + d + e = 0, (a, b, c, d, e) == (0, 1, 2, 3, 4) (mod 5). - [Dyson 1972] _Michael Somos_, Aug 08 2007
%F Euler transform of period 5 sequence [ 1, 1, 1, 1, -4, ...].
%F Expansion of q^(-1) * eta(q^5)^5 / eta(q) in powers of q.
%F a(n) = b(n + 1) where b() is multiplicative with b(5^e) = 5^e, b(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1, 4 (mod 5), b(p^e) = (p^(e+1) + (-1)^e) / (p + 1) if p == 2, 3 (mod 5).
%F Convolution inverse of A109063. a(n) = (-1)^n * A138512(n+1).
%F Convolution of A227216 and A229802. - _Michael Somos_, Jun 10 2014
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (5 t)) = (1/5)^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A109064. - _Michael Somos_, May 17 2015
%F Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A328717. - _Amiram Eldar_, Nov 23 2023
%e G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 2*x^5 + 6*x^6 + 5*x^7 + 7*x^8 + ...
%e G.f. = q + q^2 + 2*q^3 + 3*q^4 + 5*q^5 + 2*q^6 + 6*q^7 + 5*q^8 + 7*q^9 + ...
%t a[n_]:=Total[KroneckerSymbol[#, 5]*n/# & /@ Divisors[n]]; Table[a[n], {n, 1, 73}] (* _Jean-François Alcover_, Jul 26 2011, after PARI prog. *)
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^5]^5 / QPochhammer[ x], {x, 0, n}]; (* _Michael Somos_, Jul 13 2012 *)
%t a[ n_] := With[{m = n + 1}, If[ m < 1, 0, DivisorSum[ m, m/# KroneckerSymbol[ 5, #] &]]]; (* _Michael Somos_, Jul 13 2012 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^5 + A)^5 / eta(x + A), n))};
%o (PARI) {a(n) = if( n<0, 0, n++; sumdiv( n, d, kronecker( d, 5) * n/d))};
%o (PARI) {a(n) = if( n<0, 0, n++; direuler( p=2, n, 1 / ((1 - p*X) * (1 - kronecker( p, 5) * X)))[n])};
%Y Column t=5 of A175595.
%Y Cf. A053724, A109063, A109064, A138512.
%Y Cf. A227216, A229802, A328717.
%K easy,nonn,mult
%O 0,3
%A _James A. Sellers_, Feb 11 2000
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