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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). 4

%I

%S 1,1,12,148,2523,48996,1127354,29348080,849632392,27096593838,

%T 943340417806,35501579861404,1434531966551084,61939404662074706,

%U 2844544965703554566,138338597978951126666,7098617731036257970895

%N 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 CONJECTURE: For all integer 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 a(n) = (1/n!)*Sum_{k=0..n} Stirling1(n,k)*A167137(k)^2. [From _Vladeta Jovovic_, Nov 08 2009]

%e G.f.: A(x) = 1 + x + 12*x^2 + 148*x^3 + 2523*x^4 +...

%e Illustrate A(x) = Sum_{n>=0} A167137(n)^2*log(1+x)^n/n!:

%e 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 where P(x), the partition function of A000041, is generated by:

%e 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 (PARI) {Stirling2(n, k)=if(k<0|k>n,0, sum(i=0,k,(-1)^i*binomial(k, i)/k!*(k-i)^n))}

%o {A167137(n)=sum(k=0,n,numbpart(k)*Stirling2(n, k)*k!)}

%o {a(n)=polcoeff(sum(m=0,n,A167137(m)^2*log(1+x+x*O(x^n))^m/m!),n)}

%Y Cf. A167137, A000041.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 03 2009

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Last modified January 26 23:09 EST 2022. Contains 350601 sequences. (Running on oeis4.)