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A260158 Expansion of psi(x)^4 * psi(-x^3) / f(x) in powers of x where psi, f() are Ramanujan theta functions. 4
1, 3, 4, 6, 7, 6, 10, 12, 13, 15, 14, 18, 18, 21, 22, 18, 25, 27, 28, 24, 26, 33, 34, 42, 37, 30, 36, 42, 43, 45, 38, 48, 49, 42, 54, 42, 56, 57, 58, 60, 43, 63, 64, 66, 67, 63, 70, 60, 73, 84, 62, 78, 79, 72, 72, 66, 90, 87, 88, 90, 74, 78, 98, 96, 97, 78 (list; graph; refs; listen; history; text; internal format)
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..10000

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of q^(-5/6) * eta(q^2)^5 * eta(q^3) * eta(q^4) * eta(q^12) / (eta(q)^3 * eta(q^6)) in powers of q.

Euler transform of period 12 sequence [ 3, -2, 2, -3, 3, -2, 3, -3, 2, -2, 3, -4, ...].

4 * a(n) = A260109(3*n + 2) = A124815(6*n + 5).

a(2*n + 1) = 3 * A260295(n).

EXAMPLE

G.f. = 1 + 3*x + 4*x^2 + 6*x^3 + 7*x^4 + 6*x^5 + 10*x^6 + 12*x^7 + 13*x^8 + ...

G.f. = q^7 + 3*q^23 + 4*q^39 + 6*q^55 + 7*q^71 + 6*q^87 + 10*q^103 + 12*q^119 + ...

MATHEMATICA

a[ n_] := If[ n < 0, 0, With[ {m = 6 n + 5}, DivisorSum[ m, m/# KroneckerSymbol[ 12, #]&] / 4]];

a[ n_] := SeriesCoefficient[ 2^(-9/2) x^(-7/8) EllipticTheta[ 2, 0, x^(1/2)]^4 EllipticTheta[ 2, Pi/4, x^(3/2)] / QPochhammer[ -x], {x, 0, n}];

PROG

(PARI) {a(n) = my(m); if( n<0, 0, m = 6*n + 5; sumdiv( m, d, m/d * kronecker( 12, d)) / 4)};

(PARI) {a(n) = if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A) / (eta(x + A)^3 * eta(x^6 + A)), n))};

CROSSREFS

Cf. A124815, A260109, A260295.

Sequence in context: A283740 A167161 A129000 * A317093 A181590 A243293

Adjacent sequences:  A260155 A260156 A260157 * A260159 A260160 A260161

KEYWORD

nonn,changed

AUTHOR

Michael Somos, Nov 09 2015

STATUS

approved

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Last modified July 17 23:21 EDT 2019. Contains 325109 sequences. (Running on oeis4.)