A322198 a(n) is the coefficient of x^n*y^n in Product_{n>=1} 1/(1 - x^(2*n-1) - y^(2*n-1)).

1, 2, 6, 24, 84, 312, 1174, 4420, 16772, 64014, 245212, 942668, 3634914, 14051530, 54440336, 211331906, 821779372, 3200447054, 12481364146, 48736064248, 190513382908, 745492958862, 2919891150694, 11446207136530, 44905452622268, 176300343498632, 692629144937724, 2722834581642342, 10710164125130394, 42151077430686344, 165975440541202824, 653864689092828458

Offset

0,2

Links

Paul D. Hanna, Table of n, a(n) for n = 0..512

Formula

a(n) ~ c * 4^n / sqrt(n), where c = 1/(sqrt(Pi) * QPochhammer(1/4)) = 0.819402796697705077405540985476846791094716961849197... - Vaclav Kotesovec, Jun 18 2019, updated Mar 17 2024

Example

G.f.: A(x) = 1 + 2*x + 6*x^2 + 24*x^3 + 84*x^4 + 312*x^5 + 1174*x^6 + 4420*x^7 + 16772*x^8 + 64014*x^9 + 245212*x^10 + 942668*x^11 + 3634914*x^12 + ...
RELATED SERIES.
The product P(x,y) = Product_{n>=1} 1/(1 - x^(2*n-1) - y^(2*n-1)) = 1/( (1 - x - y) * (1 - x^3 - y^3) * (1 - x^5 - y^5) * (1 - x^7 - y^7) * (1 - x^9 - y^9) * ...)
may be expressed as the series expansion
P(x,y) = 1 + (1*x + 1*y) + (1*x^2 + 2*x*y + 1*y^2) + (2*x^3 + 3*x^2*y + 3*x*y^2 + 2*y^3) + (2*x^4 + 5*x^3*y + 6*x^2*y^2 + 5*x*y^3 + 2*y^4) + (3*x^5 + 7*x^4*y + 11*x^3*y^2 + 11*x^2*y^3 + 7*x*y^4 + 3*y^5) + (4*x^6 + 10*x^5*y + 18*x^4*y^2 + 24*x^3*y^3 + 18*x^2*y^4 + 10*x*y^5 + 4*y^6) + (5*x^7 + 14*x^6*y + 28*x^5*y^2 + 42*x^4*y^3 + 42*x^3*y^4 + 28*x^2*y^5 + 14*x*y^6 + 5*y^7) + (6*x^8 + 19*x^7*y + 42*x^6*y^2 + 71*x^5*y^3 + 84*x^4*y^4 + 71*x^3*y^5 + 42*x^2*y^6 + 19*x*y^7 + 6*y^8) + ...
in which this sequence equals the coefficients of x^n*y^n for n >= 0.

Prog

(PARI) N=35;
{P = 1/prod(n=1, N+1, (1 - x^(2*n-1) - y^(2*n-1) +x^2*O(x^N) +y^2*O(y^N)) ); }
{a(n) = polcoeff( polcoeff( P, n, x), n, y)}
for(n=0, N, print1( a(n), ", ") )

Crossrefs

Cf. A322187, A322211.

Sequence in context: A189568 A002742 A048120 * A003759 A378168 A217527

Adjacent sequences: A322195 A322196 A322197 * A322199 A322200 A322201

Keyword

nonn

Author

Paul D. Hanna, Dec 07 2018

Status

approved