cp's OEIS Frontend

A231274 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (k + x) / (1 - k*x - x^2).

%I A231274 #23 Jul 14 2025 00:15:12
%S A231274 1,1,4,18,104,736,6232,61632,698144,8917120,126807520,1987075872,
%T A231274 34018221728,631698903712,12645901972000,271482140140704,
%U A231274 6221487421328672,151587364647728032,3912949321334320672,106670353381399285920,3062317963564624162592,92345208262957730327968
%N A231274 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (k + x) / (1 - k*x - x^2).
%C A231274 Compare to the identity: Sum_{n>=0} x^n*Product_{k=1..n} -(k + x)/(1 - k*x - x^2) = 1 - x.
%C A231274 Compare also to the identity: Sum_{n>=0} x^n*Product_{k=1..n} (k + x)/(1 + k*x + x^2) = (1+x^2)/(1-x).
%H A231274 Vaclav Kotesovec, <a href="/A231274/b231274.txt">Table of n, a(n) for n = 0..268</a>
%F A231274 a(n) ~ n! / (2 * (log(2))^(n+1)). - _Vaclav Kotesovec_, Oct 31 2014
%F A231274 G.f. (conjecture): 1/2 + (1/2)*Sum_{n >= 0} (2*x)^n * Product_{k = 1..n} (k + x)/(1 + k*x). - _Peter Bala_, Jul 06 2025
%e A231274 G.f.: A(x) = 1 + x + 4*x^2 + 18*x^3 + 104*x^4 + 736*x^5 + 6232*x^6 +...
%e A231274 where
%e A231274 A(x) = 1 + x*(1+x)/(1-x-x^2) + x^2*(1+x)*(2+x)/((1-x-x^2)*(1-2*x-x^2)) + x^3*(1+x)*(2+x)*(3+x)/((1-x-x^2)*(1-2*x-x^2)*(1-3*x-x^2)) + x^4*(1+x)*(2+x)*(3+x)*(4+x)/((1-x-x^2)*(1-2*x-x^2)*(1-3*x-x^2)*(1-4*x-x^2)) +...
%o A231274 (PARI) {a(n)=polcoeff( sum(m=0, n, x^m*prod(k=1, m, (k+x)/(1-k*x-x^2 +x*O(x^n))) ), n)}
%o A231274 for(n=0, 30, print1(a(n), ", "))
%Y A231274 Cf. A231352, A231291.
%K A231274 nonn
%O A231274 0,3
%A A231274 _Paul D. Hanna_, Nov 06 2013