cp's OEIS Frontend

A207648 Expansion of e.g.f. Sum_{n>=0} 1/(n+1)! * Product_{k=1..n} ((1+x)^(n+k) - 1).

%I A207648 #16 Mar 15 2023 14:48:24
%S A207648 1,1,5,60,1192,34790,1378380,70445130,4478636736,344722776048,
%T A207648 31454679473280,3345722335272240,409180573835161920,
%U A207648 56883771843543627840,8902319140111902785280,1555438839901675382253600,301239031844599064651635200,64260075520580099615272097280
%N A207648 Expansion of e.g.f. Sum_{n>=0} 1/(n+1)! * Product_{k=1..n} ((1+x)^(n+k) - 1).
%C A207648 Compare e.g.f. to: Sum_{n>=0} 1/(n+1)! * Product_{k=1..n} (n+k)*x, which is a g.f. of Catalan numbers (A000108).
%H A207648 Vaclav Kotesovec, <a href="/A207648/b207648.txt">Table of n, a(n) for n = 0..275</a>
%F A207648 E.g.f.: Sum_{n>=0} 1/(n+1)! * Product_{k=1..n} ((1+x)^(n+k) - 1).
%e A207648 E.g.f.: A(x) = 1 + x + 5*x^2/2! + 60*x^3/3! + 1192*x^4/4! + 34790*x^5/5! +...
%e A207648 such that, by definition,
%e A207648 A(x) = 1 + ((1+x)^2-1)/2! + ((1+x)^3-1)*((1+x)^4-1)/3! + ((1+x)^4-1)*((1+x)^5-1)*((1+x)^6-1)/4! + ((1+x)^5-1)*((1+x)^6-1)*((1+x)^7-1)*((1+x)^8-1)/5! +...
%o A207648 (PARI) {a(n)=n!*polcoeff(sum(m=0,n,1/(m+1)!*prod(k=1,m,(1+x)^(m+k)-1 +x*O(x^n)) ),n)}
%o A207648 for(n=0,25,print1(a(n),", "))
%Y A207648 Cf. A207649.
%K A207648 nonn
%O A207648 0,3
%A A207648 _Paul D. Hanna_, Feb 19 2012