A131961 - OEIS

%I #11 Mar 12 2021 22:24:44

%S 1,1,3,2,2,1,0,3,2,4,2,0,1,2,2,3,0,2,2,2,4,0,1,4,2,2,1,0,2,0,4,0,2,4,

%T 4,1,0,4,0,2,3,0,2,2,4,0,0,2,2,0,2,3,2,4,2,2,0,3,4,4,0,0,2,0,0,4,0,2,

%U 0,2,1,0,8,2,2,2,2,3,2,4,0,0,0,2,2,4,0

%N Expansion of f(x, x^2) * f(x^2, x^2) in powers of x where f(, ) is Ramanujan's general theta function.

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

%H G. C. Greubel, <a href="/A131961/b131961.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 Expansion of phi(x^2) * phi(-x^3) / chi(-x) in powers of x where phi(), chi() are Ramanujan theta functions.

%F Expansion of q^(-1/24) * eta(q^3)^2 * eta(q^4)^5 / (eta(q) * eta(q^2) * eta(q^6) * eta(q^8)^2) in powers of q.

%F Euler transform of period 24 sequence [ 1, 2, -1, -3, 1, 1, 1, -1, -1, 2, 1, -4, 1, 2, -1, -1, 1, 1, 1, -3, -1, 2, 1, -2, ...].

%F a(25*n + 1) = a(n). a(25*n + 6) = a(25*n + 11) = a(25*n + 16) = a(25*n + 21) = 0.

%F a(n) = A123484(24*n + 1).

%F Expansion of phi(-x^3) * f(x^2)^2 / psi(-x) in powers of x where phi(), psi(), f() are Ramanujan theta functions. - _Michael Somos_, Nov 06 2015

%e G.f. = 1 + x + 3*x^2 + 2*x^3 + 2*x^4 + x^5 + 3*x^7 + 2*x^8 + 4*x^9 + 2*x^10 + ...

%e G.f. = q + q^25 + 3*q^49 + 2*q^73 + 2*q^97 + q^121 + 3*q^169 + 2*q^193 + 4*q^217 + ...

%t a[ n_] := If[ n < 0, 0, With[ {m = 24 n + 1}, DivisorSum[ m, KroneckerSymbol[ -12, #] Mod[m/#, 2] &]]]; (* _Michael Somos_, Nov 06 2015 *)

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^3] EllipticTheta[ 3, 0, x^2] QPochhammer[ -x, x], {x, 0, n}]; (* _Michael Somos_, Nov 06 2015 *)

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^2] QPochhammer[ -x, x^3] QPochhammer[ -x^2, x^3] QPochhammer[ x^3], {x, 0, n}]; (* _Michael Somos_, Nov 06 2015 *)

%o (PARI) {a(n) = if( n<0, 0, n = 24*n + 1; sumdiv(n, d, kronecker( -12, d) * (n/d %2)))};

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^2 * eta(x^4 + A)^5 / (eta(x + A) * eta(x^2 + A) * eta(x^6 + A) * eta(x^8 + A)^2), n))};

%Y Cf. A123484.

%K nonn

%O 0,3

%A _Michael Somos_, Aug 02 2007