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A131962
Expansion of psi(x) * phi(-x^12) / chi(-x^4) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.
10
1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 0, 2, 1, 0, 0, 1, 1, 1, 2, 0, 2, 0, 1, 1, 0, 2, 2, 1, 1, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 2, 3, 0, 1, 0, 1, 1, 1, 2, 0, 1, 1, 1, 1, 0, 3, 1, 1, 2, 0, 0, 1, 2, 0, 0, 1, 1, 2, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 2
OFFSET
0,11
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 q^(-7/24) * eta(q^2)^2 * eta(q^8) * eta(q^12)^2/( eta(q) * eta(q^4) * eta(q^24)) in powers of q.
Euler transform of period 24 sequence [ 1, -1, 1, 0, 1, -1, 1, -1, 1, -1, 1, -2, 1, -1, 1, -1, 1, -1, 1, 0, 1, -1, 1, -2, ...].
a(25*n + 7) = a(n). a(25*n + 2) = a(25*n + 12) = a(25*n + 17) = a(25*n + 22) = 0.
2 * a(n) = A123484(24*n + 7).
Expansion of chi(x) * f(-x^8) * phi(-x^12) in powers of x where phi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Nov 06 2015
EXAMPLE
G.f. = 1 + x + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + 2*x^10 + x^11 + ...
G.f. = q^7 + q^31 + q^79 + q^103 + q^127 + q^151 + q^175 + q^199 + q^223 + ...
MATHEMATICA
a[ n_] := If[ n < 0, 0, With[ {m = 24 n + 7}, DivisorSum[ m, KroneckerSymbol[ -12, #] Mod[m/#, 2] &] / 2]]; (* Michael Somos, Nov 06 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^8] EllipticTheta[ 4, 0, x^12] QPochhammer[ -x, x^2], {x, 0, n}]; (* Michael Somos, Nov 06 2015 *)
a[ n_] := SeriesCoefficient[ (1/2) x^(-1/8) EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 4, 0, x^12] QPochhammer[ -x^4, x^4], {x, 0, n}]; (* Michael Somos, Nov 06 2015 *)
PROG
(PARI) {a(n) = if( n<0, 0, n = 24*n + 7; sumdiv(n, d, kronecker( -12, d) * (n/d %2)) / 2)};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^8 + A) * eta(x^12 + A)^2 / (eta(x + A) * eta(x^4 + A) * eta(x^24 + A)), n))};
CROSSREFS
Cf. A123484.
Sequence in context: A154326 A330460 A027186 * A302236 A262929 A226862
KEYWORD
nonn
AUTHOR
Michael Somos, Aug 02 2007
STATUS
approved