A053723 Number of 5-core partitions of n.

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, 18, 30, 10, 32, 21, 24, 16, 30, 21, 36, 20, 24, 25, 42, 12, 42, 36, 35, 22, 46, 22, 43, 25, 32, 36, 52, 20, 60, 30, 40, 30, 60, 30, 62, 32, 42, 43, 60, 24, 66, 48, 44, 30, 72, 35, 72

Offset

0,3

Comments

Number 11 of the 74 eta-quotients listed in Table I of Martin (1996).

References

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988; see p. 54 (1.52).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)

Subhajit Bandyopadhyay and Nayandeep Deka Baruah, Arithmetic Identities for Some Analogs of the 5-Core Partition Function, J. Int. Seq. (2024) Vol. 27, Art. No. 24.4.5.

Bruce C. Berndt, The Rogers-Ramanujan continued fraction.

Bruce C. Berndt et al., The Rogers-Ramanujan continued fraction, Journal of Computational and Applied Mathematics, Vol. 105, No. 1-2 (1999), 9-24.

Freeman J. Dyson, Missed opportunities, Bull. Amer. Math. Soc. 78 (1972), 635-652. see pages 636-637.

Frank Garvan, Dongsu Kim, and Dennis Stanton, Cranks and t-cores.

Frank Garvan, Dongsu Kim, and Dennis Stanton, Cranks and t-cores, Inventiones Math. 101 (1990) 1-17.

Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.

Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers, 2016.

Formula

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

G.f.: (1/x) * (Sum_{k>0} Kronecker(k, 5) * x^k / (1 - x^k)^2). - Michael Somos, Sep 02 2005

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

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

Euler transform of period 5 sequence [ 1, 1, 1, 1, -4, ...].

Expansion of q^(-1) * eta(q^5)^5 / eta(q) in powers of q.

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).

Convolution inverse of A109063. a(n) = (-1)^n * A138512(n+1).

Convolution of A227216 and A229802. - Michael Somos, Jun 10 2014

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

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A328717. - Amiram Eldar, Nov 23 2023

Example

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 + ...
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 + ...

Mathematica

a[n_]:=Total[KroneckerSymbol[#, 5]*n/# & /@ Divisors[n]]; Table[a[n], {n, 1, 73}] (* Jean-François Alcover, Jul 26 2011, after PARI prog. *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^5]^5 / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Jul 13 2012 *)
a[ n_] := With[{m = n + 1}, If[ m < 1, 0, DivisorSum[ m, m/# KroneckerSymbol[ 5, #] &]]]; (* Michael Somos, Jul 13 2012 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^5 + A)^5 / eta(x + A), n))};
(PARI) {a(n) = if( n<0, 0, n++; sumdiv( n, d, kronecker( d, 5) * n/d))};
(PARI) {a(n) = if( n<0, 0, n++; direuler( p=2, n, 1 / ((1 - p*X) * (1 - kronecker( p, 5) * X)))[n])};

Crossrefs

Column t=5 of A175595.

Cf. A053724, A109063, A109064, A138512.

Cf. A227216, A229802, A328717.

Sequence in context: A239692 A126833 A138512 * A201652 A359609 A272636

Adjacent sequences: A053720 A053721 A053722 * A053724 A053725 A053726

Keyword

easy,nonn,mult,changed

Author

James A. Sellers, Feb 11 2000

Status

approved