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A129402 Expansion of phi(x^3) * psi(x^4) + x * phi(x) * psi(x^12) in powers of x where phi(), psi() are Ramanujan theta functions. 12
1, 1, 2, 2, 1, 2, 0, 2, 0, 0, 2, 0, 3, 1, 2, 2, 2, 4, 0, 0, 0, 0, 2, 0, 3, 0, 2, 4, 0, 2, 0, 2, 0, 0, 0, 0, 2, 3, 4, 2, 1, 2, 0, 2, 0, 0, 2, 0, 2, 2, 2, 2, 4, 2, 0, 0, 0, 0, 0, 0, 3, 0, 4, 2, 0, 2, 0, 2, 0, 0, 0, 0, 4, 3, 2, 2, 0, 4, 0, 2, 0, 0, 4, 0, 1, 0, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

REFERENCES

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 83, Eq. (32.57).

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..10000

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of f(x^2) * f(-x^3) / (chi(-x) * chi(x^6)) in powers of x where f(), chi() are Ramanujan theta functions.

Expansion of q^(-1/2) * eta(q^3) * eta(x^4)^3 * eta(q^6) * eta(q^24) / (eta(q) * eta(q^8) * eta(q^12)^12) in powers of q.

Euler transform of period 24 sequence [ 1, 1, 0, -2, 1, -1, 1, -1, 0, 1, 1, -2, 1, 1, 0, -1, 1, -1, 1, -2, 0, 1, 1, -2, ...].

a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = 1, b(p^e) = e+1 if p == 1, 5, 7, 11 (mod 24), b(p^e) = (1 + (-1)^e)/2 if p == 13, 17, 19, 23 (mod 24).

G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 24^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A190611.

a(12*n + 6) = a(12*n + 8) = a(12*n + 9) = a(12*n + 11) = 0. a(3*n + 1) = a(n).

a(n) = A000377(2*n + 1). a(3*n + 2) = 2 * A128582(n). a(12*n) = A113780(n).

a(n) = (-1)^n * A190615(n) = (-1)^floor( (n+1) / 2) * A128580(n). - Michael Somos, Nov 11 2015

a(2*n) = A261118(n). a(2*n + 1) = A261119(n). a(3*n) = A261115(n). - Michael Somos, Nov 11 2015

a(4*n) = A260308(n). a(4*n + 1) = A257920(n). a(4*n + 2) = 2 * A259895(n). - Michael Somos, Nov 11 2015

a(n) = - A261122(4*n + 2). - Michael Somos, Nov 11 2015

EXAMPLE

G.f = 1 + x + 2*x^2 + 2*x^3 + x^4 + 2*x^5 + 2*x^7 + 2*x^10 + 3*x^12 + x^13 + 2*x^14 + ...

G.f. = q + q^3 + 2*q^5 + 2*q^7 + q^9 + 2*q^11 + 2*q^15 + 2*q^21 + 3*q^25 + q^27 + ...

MATHEMATICA

a[ n_] := If[ n < 0, 0, DivisorSum[ 2 n + 1, KroneckerSymbol[ -6, #] &]]; (* Michael Somos, Nov 11 2015 *)

a[ n_] := SeriesCoefficient[ QPochhammer[ -x^2] QPochhammer[ x^3] QPochhammer[ -x, x] QPochhammer[ x^6, -x^6], {x, 0, n}]; (* Michael Somos, Nov 11 2015 *)

PROG

(PARI) {a(n) = if( n<0, 0, n = 2*n+1; sumdiv( n, d, kronecker( -6, d)))};

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A) * eta(x^4 + A)^3 * eta(x^6 + A) * eta(x^24 + A) / (eta(x + A) * eta(x^8 + A) * eta(x^12 + A)^2), n))};

CROSSREFS

Cf. A000377, A128580, A128582, A113780, A190615, A257920, A259895, A260308, A261115, A261118, A261119, A261122.

Sequence in context: A094022 A134177 A190615 * A128580 A104405 A156381

Adjacent sequences:  A129399 A129400 A129401 * A129403 A129404 A129405

KEYWORD

nonn

AUTHOR

Michael Somos, Apr 13 2007

STATUS

approved

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Last modified January 23 04:16 EST 2020. Contains 331168 sequences. (Running on oeis4.)