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A131961
Expansion of f(x, x^2) * f(x^2, x^2) in powers of x where f(, ) is Ramanujan's general theta function.
12
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, 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, 0, 2, 1, 0, 8, 2, 2, 2, 2, 3, 2, 4, 0, 0, 0, 2, 2, 4, 0
OFFSET
0,3
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of phi(x^2) * phi(-x^3) / chi(-x) in powers of x where phi(), chi() are Ramanujan theta functions.
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.
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, ...].
a(25*n + 1) = a(n). a(25*n + 6) = a(25*n + 11) = a(25*n + 16) = a(25*n + 21) = 0.
a(n) = A123484(24*n + 1).
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
EXAMPLE
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 + ...
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 + ...
MATHEMATICA
a[ n_] := If[ n < 0, 0, With[ {m = 24 n + 1}, DivisorSum[ m, KroneckerSymbol[ -12, #] Mod[m/#, 2] &]]]; (* Michael Somos, Nov 06 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^3] EllipticTheta[ 3, 0, x^2] QPochhammer[ -x, x], {x, 0, n}]; (* Michael Somos, Nov 06 2015 *)
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 *)
PROG
(PARI) {a(n) = if( n<0, 0, n = 24*n + 1; sumdiv(n, d, kronecker( -12, d) * (n/d %2)))};
(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))};
CROSSREFS
Cf. A123484.
Sequence in context: A376990 A287823 A143378 * A276426 A317872 A049340
KEYWORD
nonn
AUTHOR
Michael Somos, Aug 02 2007
STATUS
approved