A167141 - OEIS

%I #8 Sep 15 2024 01:56:47

%S 1,4,48,864,20880,632448,23018688,978179328,47529084096,2598928566336,

%T 157937795847936,10559489876375040,770269715428025088,

%U 60876094422772800000,5181654464327251948032,472584847824904789910016

%N G.f.: Sum_{n>=0} A004123(n)^2*log(1+x)^n/n! where 1/(1-2x) = Sum_{n>=0} A004123(n)*log(1+x)^n/n!.

%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.

%C In this case, m=2 and L(n) = A004123(n), which is the number of generalized weak orders on n points.

%e G.f.: A(x) = 1 + 4*x + 48*x^2 + 864*x^3 + 20880*x^4 + 632448*x^5 +...

%e Illustrate A(x) = Sum_{n>=0} A004123(n)^2 * log(1+x)^n/n!:

%e A(x) = 1 + 2^2*log(1+x) + 10^2*log(1+x)^2/2! + 74^2*log(1+x)^3/3! + 730^2*log(1+x)^4/4! + 9002^2*log(1+x)^5/5! +...+ A004123(n)^2*log(1+x)^n/n! +...

%e where the e.g.f. of A004123 is 1/(3 - 2*exp(x)) and thus:

%e 1/(1-2x) = 1 + 2*log(1+x) + 10*log(1+x)^2/2! + 74*log(1+x)^3/3! + 730*log(1+x)^4/4! + 9002*log(1+x)^5/5! +...+ A004123(n)*log(1+x)^n/n! +...

%o (PARI) {A004123(n)=sum(k=0,n,2^k*stirling(n, k, 2)*k!)}

%o {a(n)=polcoeff(sum(m=0,n,A004123(m)^2*log(1+x+x*O(x^n))^m/m!),n)}

%Y Cf. A004123, variants: A167139, A167138, A101370.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Nov 04 2009