OFFSET
0,2
COMMENTS
Formula (2.6) given in the Borwein link implies the identity
(1 + 9*x*P(x)^3/P(x^9)^3)^3 = 1 + 27*x*P(x)^12/P(x^3)^12
for the partition function P(x) = 1/Product_{n>=1} (1 - x^n) (A000041).
For fixed integer m, the g.f. G(x) that satisfies (1 + 9*x*G(x)^3/G(x^9)^3)^3 = 1 + 27*x*G(x)^m/G(x^3)^m forms an integer power series in x only when m = 12 (the partition function P(x)), m = 6 (the g.f. A(x) of this sequence), or m = 0.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..730
J. M. Borwein, P. B. Borwein and F. Garvan, Some Cubic Modular Identities of Ramanujan, Trans. Amer. Math. Soc. 343 (1994), p. 39, formula (2.6).
EXAMPLE
G.f.: A(x) = 1 + 3*x + 27*x^2 + 321*x^3 + 4257*x^4 + 61155*x^5 + 924543*x^6 + 14492790*x^7 + 233421669*x^8 + 3839329521*x^9 + ...
RELATED SERIES.
(1 + 9*x*A(x)^3/A(x^9)^3)^3 = 1 + 27*x + 486*x^2 + 8019*x^3 + 131706*x^4 + 2182626*x^5 + 36571014*x^6 + 619550127*x^7 + ...
which also equals 1 + 27*x*A(x)^6/A(x^3)^6.
A(x)^3/A(x^9)^3 = 1 + 9*x + 108*x^2 + 1476*x^3 + 21465*x^4 + 327321*x^5 + 5163804*x^6 + 83575719*x^7 + 1379999187*x^8 + ...
A(x)^6/A(x^3)^6 = 1 + 18*x + 297*x^2 + 4878*x^3 + 80838*x^4 + 1354482*x^5 + 22946301*x^6 + 392705496*x^7 + 6782271912*x^8 + ...
PROG
(PARI) {a(n) = my(V=[1], A); for(m=0, n, V = concat(V, 0); A = Ser(V);
V[#V] = polcoef( (1 + 9*x*Ser(A)^3/subst(Ser(A), x, x^9)^3 )^3 - 1 - 27*x*Ser(A)^6/subst(Ser(A), x, x^3)^6, #V)/81 ); V[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 16 2024
STATUS
approved