A089902 Antidiagonal sums of array A089900.

1, 3, 10, 40, 193, 1107, 7412, 56960, 495055, 4805327, 51540462, 605360184, 7726837413, 106484488843, 1575591323104, 24910186990320, 419042540060243, 7472730215908551, 140804433625595626, 2795108750920323336

Offset: 0

Author

Paul D. Hanna, Nov 14 2003

Keywords

• nonn

Comments

The n-th row of array A089900 is the n-th binomial transform of the factorials found in row 0: {1!,2!,3!,..,(n+1)!,..}. The hyperbinomial transform of the main diagonal gives: {1,4,27,..,(n+1)^(n+1),..}, which is the next lower diagonal in array A089900.

Crossrefs

Cf. A089900, A089901, A000312.

Programs

PARI

a(n)=if(n<0,0,sum(k=0,n,sum(i=0,k,(n-k)^(k-i)*binomial(k,i)*(i+1)!)))

PARI

a(n)=sum(k=0,n,sum(i=0,k,(n-k)^(k-i)*binomial(k,i)*(i+1)!));

PARI

a(n)=polcoeff(sum(m=0,2*n,sum(k=1,2*n,k!*x^(m+k-1)/(1-m*x)^k),x*O(x^n)),n);

Formula

a(n) = sum_{k=0..n} sum_{i=0..k} (n-k)^(k-i)*binomial(k, i)*(i+1)!

O.g.f.: Sum_{m>=0, n>=1} n!*x^(m+n-1)/(1-m*x)^n - Vladeta Jovovic, Nov 18 2003