A135828 - OEIS

A135828

Expansion of psi(x^2)^8 * (psi(x)^8 + psi(-x)^8) / 2 in powers of x^2 where psi() is a Ramanujan theta function.

1, 36, 378, 2200, 8955, 28836, 78558, 188568, 410805, 828080, 1564686, 2804976, 4809370, 7927380, 12643560, 19594632, 29568204, 43626708, 63094550, 89501040, 124916931, 171803652, 232822908, 311683680, 412601490, 539849556, 699657642, 898801400, 1143680535

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OFFSET

0,2

COMMENTS

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

Expansion of q^(-3) * ( eta(q^2)^24 + eta(q)^16 * eta(q^4)^8 ) / ( 2 * eta(q)^8 * eta(q^2)^16 / eta(q^4)^16 ) in powers of q^2.

7680 * a(n) = A008774(2*n + 3).

Convolution of A007331 and A045823.

EXAMPLE

G.f. = 1 + 36*x + 378*x^2 + 2200*x^3 + 8955*x^4 + 28836*x^6 + 78558*x^7 + ...

G.f. = q^3 + 36*q^5 + 378*q^7 + 2200*q^9 + 8955*q^11 + 28836*q^13 + 78558*q^15 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x]^8 (EllipticTheta[ 2, 0, x^(1/2)]^8 + EllipticTheta[ 2, Pi/4, x^(1/2)]^8 16) / 131072, {x, 0, 2 n + 3}]; (* Michael Somos, Oct 15 2015 *)

PROG

(PARI) {a(n) = my(A); if( n<0, 0, n *= 2; A = x * O(x^n); polcoeff( ( eta(x^2 + A)^24 + eta(x + A)^16 * eta(x^4 + A)^8 ) / ( 2 * eta(x + A)^8 * eta(x^2 + A)^16 / eta(x^4 + A)^16 ), n))};

(Magma) Basis( ModularForms( Gamma1(4), 8), 60)[4]; /* Michael Somos, Oct 15 2015 */

CROSSREFS

Cf. A007331, A008774, A045823.

Sequence in context: A222492 A203282 A071232 * A250805 A254644 A034686

Adjacent sequences: A135825 A135826 A135827 * A135829 A135830 A135831

KEYWORD

nonn

AUTHOR

Michael Somos, Nov 29 2007

STATUS

approved