A207557 G.f.: Sum_{n>=0} 1/(1+x)^(n^2-n) * Product_{k=1..n} ((1+x)^(2*k-1) - 1).

1, 1, 3, 12, 64, 420, 3276, 29581, 303389, 3483053, 44245695, 616103046, 9330961666, 152700926414, 2685132170466, 50488787588936, 1010864433071206, 21470488933116138, 482176661100286182, 11415700804801064258, 284169548252819022230, 7419733139418740010570

Offset

0,3

Comments

Compare g.f. to: Sum_{n>=0} 1/(1+x)^(n^2) * Product_{k=1..n} ((1+x)^(2*k-1) - 1), which is the g.f. of A179525.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..135

Hsien-Kuei Hwang, and Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], p. 36, 2019.

Formula

Given A(x) is the g.f. of this sequence, note that:

1 + x*A(x) = Sum_{n>=0} 1/(1+x)^(n^2+n) * Product_{k=1..n} ((1+x)^(2*k-1) - 1).

a(n) ~ 2*sqrt(6) * 12^(n+1) * n^(n+1) / (exp(n+Pi^2/24) * Pi^(2*n+3)). - Vaclav Kotesovec, May 07 2014

Example

G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 64*x^4 + 420*x^5 + 3276*x^6 +...
such that, by definition,
A(x) = 1 + ((1+x)-1) + ((1+x)-1)*((1+x)^3-1)/(1+x)^2 + ((1+x)-1)*((1+x)^3-1)*((1+x)^5-1)/(1+x)^6 + ((1+x)-1)*((1+x)^3-1)*((1+x)^5-1)*((1+x)^7-1)/(1+x)^20 +...

Prog

(PARI) {a(n)=polcoeff(sum(m=0, n, prod(k=1, m, (1+x)^(2*k-1)-1)/(1+x+x*O(x^n))^(m^2-m) ), n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A179525, A207556.

Sequence in context: A307724 A029851 A201720 * A235129 A222033 A341769

Adjacent sequences: A207554 A207555 A207556 * A207558 A207559 A207560

Keyword

nonn

Author

Paul D. Hanna, Feb 18 2012

Status

approved