A322187 a(n) is the coefficient of x^n*y^n/n in log( Product_{n>=1} 1/(1 - x^(2*n-1) - y^(2*n-1)) ), for n >= 1.

1, 3, 13, 35, 131, 471, 1723, 6435, 24349, 92393, 352727, 1352183, 5200313, 20058321, 77559203, 300540195, 1166803127, 4537569063, 17672631919, 68923264585, 269128942459, 1052049481893, 4116715363823, 16123801860855, 63205303219531, 247959266474091, 973469712897103, 3824345300380465, 15033633249770549, 59132290782710201, 232714176627630575, 916312070471295267, 3609714217009191161, 14226520737620288421

Offset: 1

Author

Paul D. Hanna, Dec 07 2018

Keywords

• nonn

Examples

			L.g.f.: L(x) = x + 3*x^2/2 + 13*x^3/3 + 35*x^4/4 + 131*x^5/5 + 471*x^6/6 + 1723*x^7/7 + 6435*x^8/8 + 24349*x^9/9 + 92393*x^10/10 + 352727*x^11/11 + 1352183*x^12/12 + ...
RELATED SERIES.
exp(L(x)) = 1 + x + 2*x^2 + 6*x^3 + 15*x^4 + 45*x^5 + 140*x^6 + 448*x^7 + 1483*x^8 + 5027*x^9 + 17311*x^10 + 60469*x^11 + 213678*x^12 + ... + A322188(n)*x^n + ...
The table of coefficients of x^n*y^k/(n+k) in
log( Product_{n>=1} 1/(1 - x^(2*n-1) - y^(2*n-1)) ) = (1*x + 1*y) + (1*x^2 + 2*x*y + 1*y^2)/2 + (4*x^3 + 3*x^2*y + 3*x*y^2 + 4*y^3)/3 + (1*x^4 + 4*x^3*y + 6*x^2*y^2 + 4*x*y^3 + 1*y^4)/4 + (6*x^5 + 5*x^4*y + 10*x^3*y^2 + 10*x^2*y^3 + 5*x*y^4 + 6*y^5)/5 + (4*x^6 + 6*x^5*y + 15*x^4*y^2 + 26*x^3*y^3 + 15*x^2*y^4 + 6*x*y^5 + 4*y^6)/6 + (8*x^7 + 7*x^6*y + 21*x^5*y^2 + 35*x^4*y^3 + 35*x^3*y^4 + 21*x^2*y^5 + 7*x*y^6 + 8*y^7)/7 + (1*x^8 + 8*x^7*y + 28*x^6*y^2 + 56*x^5*y^3 + 70*x^4*y^4 + 56*x^3*y^5 + 28*x^2*y^6 + 8*x*y^7 + 1*y^8)/8 + ...
begins
n=0: [0, 1, 1, 4, 1, 6, 4, 8, 1, 13, 6, ..., A000593(k), ...];
n=1: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...];
n=2: [1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...];
n=3: [4, 4, 10, 26, 35, 56, 93, 120, 165, 232, ...];
n=4: [1, 5, 15, 35, 70, 126, 210, 330, 495, 715, ...];
n=5: [6, 6, 21, 56, 126, 262, 462, 792, 1287, 2002, ...];
n=6: [4, 7, 28, 93, 210, 462, 942, 1716, 3003, 5035, ...];
n=7: [8, 8, 36, 120, 330, 792, 1716, 3446, 6435, 11440, ...];
n=8: [1, 9, 45, 165, 495, 1287, 3003, 6435, 12870, 24310, ...];
n=9: [13, 10, 55, 232, 715, 2002, 5035, 11440, 24310, 48698, ...];
n=10: [6, 11, 66, 286, 1001, 3018, 8008, 19448, 43758, 92378, ...]; ...
in which the diagonal of coefficients of x^n*y^n/(2*n) equals
[0, 2, 6, 26, 70, 262, 942, 3446, 12870, 48698, ..., 2*a(n), ...],
which is twice this sequence.
The related infinite product may be written as the following series expansion
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) * (1 - x^11 - y^11) * ...) = 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) + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..512

Crossrefs

Cf. A322188, A322198, A322203, A000593.

Programs

PARI

N=35;
{L = sum(n=1, N+1, -log(1 - x^(2*n-1) - y^(2*n-1) +x*O(x^N) +y*O(y^N)) ); }
{a(n) = polcoeff( n*polcoeff( L, n, x), n, y)}
for(n=1, N, print1( a(n), ", ") )

Formula

a(n) ~ 2^(2*n-1) / sqrt(Pi*n). - Vaclav Kotesovec, Jun 18 2019