A222033 G.f.: Sum_{n>=0} Product_{k=1..n} (1 - 1/(1+k*x)^3).

1, 3, 12, 64, 429, 3459, 32578, 350928, 4254819, 57339343, 850210608, 13755324192, 241123857361, 4552433489355, 92097902228022, 1987543508858416, 45576279808372215, 1106640757105043895, 28364428977533987380, 765303225207132783360, 21681823874743612308981

Offset: 0

Author

Paul D. Hanna, Feb 06 2013

Keywords

• nonn

Comments

Compare to the g.f. of A136127: Sum_{n>=0} Product_{k=1..n} (1 - 1/(1+k*x)^2), where A136127(n) equals the number of permutations of {1,2,...,n} having excedance set {1,2,...,k} for some k=0...n-1.

Examples

			G.f.: A(x) = 1 + 3*x + 12*x^2 + 64*x^3 + 429*x^4 + 3459*x^5 + 32578*x^6 +...
where
A(x) = 1 + (1 - 1/(1+x)^3) + (1 - 1/(1+x)^3)*(1 - 1/(1+2*x)^3) + (1 - 1/(1+x)^3)*(1 - 1/(1+2*x)^3)*(1 - 1/(1+3*x)^3) + (1 - 1/(1+x)^3)*(1 - 1/(1+2*x)^3)*(1 - 1/(1+3*x)^3)*(1 - 1/(1+4*x)^3) +...
		

Crossrefs

Cf. A136127.

Programs

Mathematica

CoefficientList[Series[Sum[Product[1-1/(1+k x)^3,{k,n}],{n,0,20}],{x,0,20}],x] (* Harvey P. Dale, Sep 24 2021 *)

PARI

{a(n)=polcoeff(sum(m=0, n, prod(k=1, m,1-1/(1+k*x +x*O(x^n))^3)),n)}
for(n=0, 25, print1(a(n), ", "))