A366104 G.f. ( Chi(sqrt(x))^4 + Chi(-sqrt(x))^4 )/2, where Chi(x) = Product_{k >= 0} 1 + x^(2*k+1) is the g.f. of A000700.

1, 6, 17, 38, 84, 172, 325, 594, 1049, 1796, 3005, 4912, 7877, 12430, 19309, 29580, 44766, 66978, 99150, 145374, 211242, 304382, 435194, 617674, 870651, 1219352, 1697283, 2348888, 3232919, 4426546, 6030872, 8177986, 11039633, 14838518, 19862613, 26482878, 35175989, 46552818, 61393694

Offset

0,2

Comments

Compare with A224916 with g.f. ( Chi(sqrt(x))^4 - Chi(-sqrt(x))^4 )/(8*sqrt(x)),

A069910 with g.f. ( Chi(sqrt(x)) + Chi(-sqrt(x)) )/2,

A069911 with g.f. ( Chi(sqrt(x)) - Chi(-sqrt(x)) )/2,

A226622 with g.f. ( Chi(sqrt(x))^2 + Chi(-sqrt(x))^2 )/2 and

A226635 with g.f. ( Chi(sqrt(x))^2 - Chi(-sqrt(x))^2 )/(4*sqrt(x)),

Jacobi's "aequatio identica satis abstrusa" is the identity ( Chi(sqrt(x))^8 - Chi(-sqrt(x))^8 )/(16*sqrt(x)) = Product_{k >= 1} (1 + x^k)^8.

Links

Table of n, a(n) for n=0..38.

Formula

G.f.: Product_{k >= 1} (1 + x^(2*k))^2*(1 + x^(2*k-1))^6.

G.f.: x^(1/12) * eta(x^2)^10 * eta(x^4)^2 / ( eta(x) * eta(x^4) )^6.

Maple

with(QDifferenceEquations):
 seq(coeff((1/2)*expand(QPochhammer(-q, q^2, 40)^4 + QPochhammer(q, q^2, 40)^4), q, 2*n), n = 0..40);
#alternative program
seq(coeff(expand(QPochhammer(-q^2, q^2, 20)^2 * QPochhammer(-q, q^2, 20)^6), q, n), n = 0..40);

Crossrefs

Cf. A000700, A014705, A069910, A069911, A224916, A226622, A226635.

Sequence in context: A192756 A004799 A085278 * A080275 A061349 A213780

Adjacent sequences: A366101 A366102 A366103 * A366105 A366106 A366107

Keyword

nonn,easy

Author

Peter Bala, Sep 29 2023

Status

approved