cp's OEIS Frontend

A262050 Expansion of f(-x)^2 * f(-x^10) / phi(-x)^3 in powers of x where phi(), f() are Ramanujan theta functions.

%I A262050 #10 Feb 16 2025 08:33:27
%S A262050 1,4,11,28,63,132,264,504,928,1660,2892,4924,8221,13480,21750,34592,
%T A262050 54288,84168,129048,195816,294282,438324,647413,948748,1380107,
%U A262050 1993632,2860984,4080172,5784560,8154900,11435142,15953124,22147824,30604868,42102636,57672312
%N A262050 Expansion of f(-x)^2 * f(-x^10) / phi(-x)^3 in powers of x where phi(), f() are Ramanujan theta functions.
%C A262050 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A262050 G. C. Greubel, <a href="/A262050/b262050.txt">Table of n, a(n) for n = 0..1000</a>
%H A262050 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A262050 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A262050 Expansion of q^(-1/2) * eta(q^2)^3 * eta(q^10) / eta(q)^4 in powers of q.
%F A262050 Euler transform of period 10 sequence [ 4, 1, 4, 1, 4, 1, 4, 1, 4, 0, ...].
%F A262050 2 * a(n) = A138526(2*n + 1) = - A261968(2*n + 1).
%e A262050 G.f. = 1 + 4*x + 11*x^2 + 28*x^3 + 63*x^4 + 132*x^5 + 264*x^6 + 504*x^7 + ...
%e A262050 G.f. = q + 4*q^3 + 11*q^5 + 28*q^7 + 63*q^9 + 132*q^11 + 264*q^13 + ...
%t A262050 a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2 QPochhammer[ x^10] / EllipticTheta[ 4, 0, x]^3, {x, 0, n}];
%o A262050 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^10 + A) / eta(x + A)^4, n))};
%Y A262050 Cf. A138526, A261968.
%K A262050 nonn
%O A262050 0,2
%A A262050 _Michael Somos_, Sep 09 2015