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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988; see p. 54 (1.52).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.
Subhajit Bandyopadhyay and Nayandeep Deka Baruah, Arithmetic Identities for Some Analogs of 5-core Partition Function, arXiv:2409.02034 [math.NT], 2024.
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).
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
KEYWORD
easy,nonn,mult
AUTHOR
James A. Sellers, Feb 11 2000
STATUS
approved