cp's OEIS Frontend

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

%I A222033 #8 Sep 24 2021 16:19:28
%S A222033 1,3,12,64,429,3459,32578,350928,4254819,57339343,850210608,
%T A222033 13755324192,241123857361,4552433489355,92097902228022,
%U A222033 1987543508858416,45576279808372215,1106640757105043895,28364428977533987380,765303225207132783360,21681823874743612308981
%N A222033 G.f.: Sum_{n>=0} Product_{k=1..n} (1 - 1/(1+k*x)^3).
%C A222033 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.
%e A222033 G.f.: A(x) = 1 + 3*x + 12*x^2 + 64*x^3 + 429*x^4 + 3459*x^5 + 32578*x^6 +...
%e A222033 where
%e A222033 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) +...
%t A222033 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 *)
%o A222033 (PARI) {a(n)=polcoeff(sum(m=0, n, prod(k=1, m,1-1/(1+k*x +x*O(x^n))^3)),n)}
%o A222033 for(n=0, 25, print1(a(n), ", "))
%Y A222033 Cf. A136127.
%K A222033 nonn
%O A222033 0,2
%A A222033 _Paul D. Hanna_, Feb 06 2013