A182507 G.f.: Sum_{n>=0} n! * 2^(n*(n-1)/2) * x^n / Product_{k=1..n} (1 + k*2^k*x).

1, 1, 2, 12, 232, 12848, 1858464, 663242944, 562426769024, 1103780804371200, 4916976475489286656, 48986367134323580374016, 1078808700869188981508990976, 52024935094126934151475827453952, 5451309776848243787358722272838524928

Offset: 0

Author

Paul D. Hanna, May 03 2012

Keywords

• nonn

Comments

Compare the g.f. to the identities:

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

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

First differs from A309615 at a(5) = 12848, A309615(5) = 19230. - Gus Wiseman, Aug 11 2019

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 12*x^3 + 232*x^4 + 12848*x^5 + 1858464*x^6 +...
such that
A(x) = 1 + x/(1+2*x) + 2!*2^1*x^2/((1+1*2*x)*(1+2*4*x)) + 3!*2^3*x^3/((1+1*2*x)*(1+2*4*x)*(1+3*8*x)) + 4!*2^6*x^4/((1+1*2*x)*(1+2*4*x)*(1+3*8*x)*(1+4*16*x)) +...
		

Links

• Hsien-Kuei Hwang, Emma Yu Jin, and Michael J. Schlosser, Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow, arXiv:2012.13570 [math.CO], 2020.

Crossrefs

Cf. A005329, A182489, A309615.

Programs

PARI

{a(n)=polcoeff(sum(m=0,n,m!*2^(m*(m-1)/2)*x^m/prod(k=1,m,1+k*2^k*x +x*O(x^n))),n)}
for(n=0,20,print1(a(n),", "))