A136596 Column 2 of triangle A136595.

1, -3, 31, -375, 5911, -113463, 2571031, -67170855, 1987919671, -65731585623, 2401646633431, -96089053104135, 4178215255335031, -196193483904124983, 9894077286353278231, -533334378459657706215, 30602112192036616407991

Offset: 2

Author

Paul D. Hanna, Jan 10 2008

Keywords

• sign

Crossrefs

Cf. A136595; A048287, A136597.

Programs

PARI

{a(n)=n!* sum(i=0,n-1,(-1)^i*polcoeff(((exp(x+x*O(x^n))-1)^(2+i)),n)*binomial(2*i+2,i)/(2*i+2))}
for(n=2,20,print1(a(n),", "))

PARI

/* Define Stirling2: */
{Stirling2(n,k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!,n)}
/* Define Catalan(m,n) = [x^n] C(x)^m: */
{Catalan(m,n)=binomial(2*n+m,n)*m/(2*n+m)}
/* Define this sequence: */
{a(n)=sum(i=0,n-1,(-1)^i*(2+i)!*Stirling2(n,2+i)*Catalan(2,i)/2!)}
for(n=2,20,print1(a(n),", "))

Formula

a(n) = Sum_{i=0..n-1} (-1)^i*(2+i)!*Stirling2(n,2+i)*Catalan(2,i)/2!, where Stirling2(n,k) = A008277(n,k); Catalan(k,i) = binomial(2*i+k,i)*k/(2*i+k) = coefficient of x^i in C(x)^k with C(x) = (1-sqrt(1-4x))/(2x).

a(n) = (1+(-1)^n*A048287(n))/2. - Vladeta Jovovic, Jan 27 2008