A123862 - OEIS

A123862

Expansion of f(q)*f(q^7)/(f(-q)*f(-q^7)) in powers of q where f() is a Ramanujan theta function.

1, 2, 2, 4, 6, 8, 12, 18, 26, 34, 48, 64, 84, 112, 146, 192, 246, 316, 402, 508, 640, 804, 1008, 1248, 1548, 1910, 2344, 2872, 3510, 4276, 5184, 6280, 7578, 9120, 10956, 13128, 15702, 18724, 22292, 26480, 31392, 37148, 43884, 51760, 60912, 71592

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OFFSET

0,2

COMMENTS

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Euler transform of period 28 sequence [ 2, -1, 2, 0, 2, -1, 4, 0, 2, -1, 2, 0, 2, -2, 2, 0, 2, -1, 2, 0, 4, -1, 2, 0, 2, -1, 2, 0, ...].

G.f. A(x) satisfies 0=f(A(x), A(x^2)) where f(u, v)=(u-1)^2 -2*u*v*(v-1).

a(n) ~ exp(2*Pi*sqrt(n/7)) / (4 * 7^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jun 03 2018

MATHEMATICA

QP := QPochhammer; a[n_]:= SeriesCoefficient[QP[-q]*QP[-q^7]/( QP[q]* QP[q^7]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Jan 04 2018 *)

PROG

(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x^2+A)*eta(x^14+A))^3/ (eta(x+A)*eta(x^7+A))^2/ (eta(x^4+A)*eta(x^28+A)), n))}

CROSSREFS

Cf. A123648(n)=a(n)/2 if n>0.

Sequence in context: A018129 A361298 A091915 * A089647 A274152 A274155

Adjacent sequences: A123859 A123860 A123861 * A123863 A123864 A123865

KEYWORD

nonn

AUTHOR

Michael Somos, Oct 14 2006

STATUS

approved