A167139 - OEIS

%I #13 Sep 15 2024 01:57:04

%S 1,4,30,292,3497,49488,806504,14860032,305261640,6914828176,

%T 171186477632,4597513706496,133116705145408,4133143450593536,

%U 136981118139314688,4826352390162440704,180139085757269111824

%N 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 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 a(n) = (1/n!)*Sum_{k=0..n} Stirling1(n,k)*A005649(k)^2, cf. A101370. - _Vladeta Jovovic_, Nov 09 2009

%e G.f.: A(x) = 1 + 4*x + 30*x^2 + 292*x^3 + 3497*x^4 + 49488*x^5 + ...

%e Illustrate A(x) = Sum_{n>=0} A005649(n)^2 * log(1+x)^n/n!:

%e 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 where the g.f. of A005649 is 1/(2 - exp(x))^2:

%e 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 (PARI) {A005649(n)=sum(k=0,n,(k+1)*stirling(n, k, 2)*k!)}

%o {a(n)=polcoef(sum(m=0,n,A005649(m)^2*log(1+x+x*O(x^n))^m/m!),n)}

%Y Cf. A167138, A005649.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Nov 03 2009