A022577 Expansion of Product_{m>=1} (1+x^m)^12.

1, 12, 78, 376, 1509, 5316, 16966, 50088, 138738, 364284, 913824, 2203368, 5130999, 11585208, 25444278, 54504160, 114133296, 234091152, 471062830, 931388232, 1811754522, 3471186596, 6556994502, 12222818640, 22502406793, 40944396120, 73680871326, 131211105208, 231355524048, 404110659732

Offset

0,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

G.f.: Product_{k>=1} ( 1 + x^k )^12.

Expansion of chi(-x)^-12 in powers of x where chi() is a Ramanujan theta function.

Expansion of k^2 / (16 * q * k') in powers of q^2. - Michael Somos, Jul 22 2011

Expansion of q^(-1/2) * (k/4) / (1 - k^2) in powers of q. - Michael Somos, Jul 16 2005

Expansion of q^(-1/2) * (eta(q^2) / eta(q))^12 in powers of q. - Michael Somos, Jul 16 2005

Euler transform of period 2 sequence [12, 0, ...]. - Michael Somos, Jul 16 2005

Given g.f. A(x), then B(q) = (q * A(q^2))^2 satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (4096*u*v + 48*u + 1)*v - u^2 . - Michael Somos, Jul 16 2005

G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 1/64 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A007249. - Michael Somos, Jul 22 2011

A124863(n) = (-1)^n * a(n). A007096(4*n + 2) = 8 * a(n). Convolution inverse of A007249.

a(n) ~ exp(2 * Pi * sqrt(n)) / (128 * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015

a(0) = 1, a(n) = (12/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 03 2017

Example

G.f. = 1 + 12*x + 78*x^2 + 376*x^3 + 1509*x^4 + 5316*x^5 + 16966*x^6 + ...
G.f. = q + 12*q^3 + 78*q^5 + 376*q^7 + 1509*q^9 + 5316*q^11 + 16966*q^13 + ...

Maple

N:= 50:
G:= mul(1+x^m, m=1..N+1)^12:
S:= series(G, x, N+1):
seq(coeff(S, x, n), n=0..N); # Robert Israel, Feb 26 2018

Mathematica

a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (m / 16 / q)^(1/2) / (1 - m), {q, 0, n}]]; (* Michael Somos, Jul 22 2011 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (m / 16 / q) /(1-m)^(1/2), {q, 0, 2 n}]]; (* Michael Somos, Jul 22 2011 *)
CoefficientList[QPochhammer[-1, q]^12/4096+O[q]^30, q] (* Jean-François Alcover, Nov 27 2015 *)
With[{nmax=50}, CoefficientList[Series[Product[(1+q^k)^12, {k, 1, nmax}], {q, 0, nmax}], q]] (* G. C. Greubel, Feb 25 2018 *)

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, 1 + x^k, 1 + x * O(x^n))^12, n))}; /* Michael Somos, Jul 16 2005 */
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) / eta(x + A))^12, n))}; /* Michael Somos, Jul 16 2005 */
(PARI) m=50; q='q+O('q^m); Vec(prod(n=1, m, (1+q^n)^12)) \\ G. C. Greubel, Feb 25 2018
(Magma) Coefficients(&*[(1+x^m)^12:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 25 2018

Crossrefs

Column k=12 of A286335.

Cf. A000122, A000700, A007096, A007249, A010054, A121373, A124863.

Sequence in context: A121665 A341225 A124863 * A321949 A227707 A189493

Adjacent sequences: A022574 A022575 A022576 * A022578 A022579 A022580

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms added by G. C. Greubel, Feb 25 2018

Status

approved