cp's OEIS Frontend

A167138 G.f.: Sum_{n>=0} A167137(n)^2 * log(1+x)^n/n! where Sum_{n>=0} A167137(n)*log(1+x)^n/n! = g.f. of the partition numbers (A000041).

%I A167138 #13 Sep 15 2024 01:57:16
%S A167138 1,1,12,148,2523,48996,1127354,29348080,849632392,27096593838,
%T A167138 943340417806,35501579861404,1434531966551084,61939404662074706,
%U A167138 2844544965703554566,138338597978951126666,7098617731036257970895
%N A167138 G.f.: Sum_{n>=0} A167137(n)^2 * log(1+x)^n/n! where Sum_{n>=0} A167137(n)*log(1+x)^n/n! = g.f. of the partition numbers (A000041).
%C A167138 Conjecture: For all integers m > 0, Sum_{n>=0} L(n)^m * log(1+x)^n/n! is an integer series whenever Sum_{n>=0} L(n)*log(1+x)^n/n! is an integer series.
%F A167138 a(n) = (1/n!)*Sum_{k=0..n} Stirling1(n,k)*A167137(k)^2. - _Vladeta Jovovic_, Nov 08 2009
%e A167138 G.f.: A(x) = 1 + x + 12*x^2 + 148*x^3 + 2523*x^4 + ...
%e A167138 Illustrate A(x) = Sum_{n>=0} A167137(n)^2*log(1+x)^n/n!:
%e A167138 A(x) = 1 + log(1+x) + 5^2*log(1+x)^2/2! + 31^2*log(1+x)^3/3! + 257^2*log(1+x)^4/4! + ...
%e A167138 where P(x), the partition function of A000041, is generated by:
%e A167138 P(x) = 1 + log(1+x) + 5*log(1+x)^2/2! + 31*log(1+x)^3/3! + 257*log(1+x)^4/4! + ...
%o A167138 (PARI) {A167137(n)=sum(k=0,n,numbpart(k)*stirling(n, k, 2)*k!)}
%o A167138 {a(n)=polcoef(sum(m=0,n,A167137(m)^2*log(1+x+x*O(x^n))^m/m!),n)}
%Y A167138 Cf. A167137, A000041.
%K A167138 nonn
%O A167138 0,3
%A A167138 _Paul D. Hanna_, Nov 03 2009