cp's OEIS Frontend

A167139 G.f.: Sum_{n>=0} A005649(n)^2 * log(1+x)^n/n! where 1/(1-x)^2 = Sum_{n>=0} A005649(n)*log(1+x)^n/n!.

%I A167139 #13 Sep 15 2024 01:57:04
%S A167139 1,4,30,292,3497,49488,806504,14860032,305261640,6914828176,
%T A167139 171186477632,4597513706496,133116705145408,4133143450593536,
%U A167139 136981118139314688,4826352390162440704,180139085757269111824
%N A167139 G.f.: Sum_{n>=0} A005649(n)^2 * log(1+x)^n/n! where 1/(1-x)^2 = Sum_{n>=0} A005649(n)*log(1+x)^n/n!.
%C A167139 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 A167139 a(n) = (1/n!)*Sum_{k=0..n} Stirling1(n,k)*A005649(k)^2, cf. A101370. - _Vladeta Jovovic_, Nov 09 2009
%e A167139 G.f.: A(x) = 1 + 4*x + 30*x^2 + 292*x^3 + 3497*x^4 + 49488*x^5 + ...
%e A167139 Illustrate A(x) = Sum_{n>=0} A005649(n)^2 * log(1+x)^n/n!:
%e A167139 A(x) = 1 + 2^2*log(1+x) + 8^2*log(1+x)^2/2! + 44^2*log(1+x)^3/3! + 308^2*log(1+x)^4/4! + 2612^2*log(1+x)^5/5! + ... + A005649(n)^2*log(1+x)^n/n! + ...
%e A167139 where the g.f. of A005649 is 1/(2 - exp(x))^2:
%e A167139 1/(1-x)^2 = 1 + 2*log(1+x) + 8*log(1+x)^2/2! + 44*log(1+x)^3/3! + 308*log(1+x)^4/4! + 2612*log(1+x)^5/5! + ... + A005649(n)*log(1+x)^n/n! + ...
%o A167139 (PARI) {A005649(n)=sum(k=0,n,(k+1)*stirling(n, k, 2)*k!)}
%o A167139 {a(n)=polcoef(sum(m=0,n,A005649(m)^2*log(1+x+x*O(x^n))^m/m!),n)}
%Y A167139 Cf. A167138, A005649.
%K A167139 nonn
%O A167139 0,2
%A A167139 _Paul D. Hanna_, Nov 03 2009