A022577 - OEIS

%I #51 Sep 08 2022 08:44:46

%S 1,12,78,376,1509,5316,16966,50088,138738,364284,913824,2203368,

%T 5130999,11585208,25444278,54504160,114133296,234091152,471062830,

%U 931388232,1811754522,3471186596,6556994502,12222818640,22502406793,40944396120,73680871326,131211105208,231355524048,404110659732

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

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

%H Seiichi Manyama, <a href="/A022577/b022577.txt">Table of n, a(n) for n = 0..1000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

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

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

%F Expansion of k^2 / (16 * q * k') in powers of q^2. - _Michael Somos_, Jul 22 2011

%F Expansion of q^(-1/2) * (k/4) / (1 - k^2) in powers of q. - _Michael Somos_, Jul 16 2005

%F Expansion of q^(-1/2) * (eta(q^2) / eta(q))^12 in powers of q. - _Michael Somos_, Jul 16 2005

%F Euler transform of period 2 sequence [12, 0, ...]. - _Michael Somos_, Jul 16 2005

%F 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

%F 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

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

%F a(n) ~ exp(2 * Pi * sqrt(n)) / (128 * n^(3/4)). - _Vaclav Kotesovec_, Mar 05 2015

%F a(0) = 1, a(n) = (12/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Apr 03 2017

%e G.f. = 1 + 12*x + 78*x^2 + 376*x^3 + 1509*x^4 + 5316*x^5 + 16966*x^6 + ...

%e G.f. = q + 12*q^3 + 78*q^5 + 376*q^7 + 1509*q^9 + 5316*q^11 + 16966*q^13 + ...

%p N:= 50:

%p G:= mul(1+x^m,m=1..N+1)^12:

%p S:= series(G,x,N+1):

%p seq(coeff(S,x,n),n=0..N); # _Robert Israel_, Feb 26 2018

%t a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (m / 16 / q)^(1/2) / (1 - m), {q, 0, n}]]; (* _Michael Somos_, Jul 22 2011 *)

%t a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (m / 16 / q) /(1-m)^(1/2), {q, 0, 2 n}]]; (* _Michael Somos_, Jul 22 2011 *)

%t CoefficientList[QPochhammer[-1, q]^12/4096+O[q]^30, q] (* _Jean-François Alcover_, Nov 27 2015 *)

%t With[{nmax=50}, CoefficientList[Series[Product[(1+q^k)^12, {k,1,nmax}], {q, 0, nmax}],q]] (* _G. C. Greubel_, Feb 25 2018 *)

%o (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 */

%o (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 */

%o (PARI) m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+q^n)^12)) \\ _G. C. Greubel_, Feb 25 2018

%o (Magma) Coefficients(&*[(1+x^m)^12:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // _G. C. Greubel_, Feb 25 2018

%Y Column k=12 of A286335.

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

%K nonn

%O 0,2

%A _N. J. A. Sloane_

%E More terms added by _G. C. Greubel_, Feb 25 2018