A089900 Square array, read by antidiagonals, where the n-th row is the n-th binomial transform of the factorials, starting with row 0: {1!,2!,3!,...}.

1, 2, 1, 6, 3, 1, 24, 11, 4, 1, 120, 49, 18, 5, 1, 720, 261, 92, 27, 6, 1, 5040, 1631, 536, 159, 38, 7, 1, 40320, 11743, 3552, 1029, 256, 51, 8, 1, 362880, 95901, 26608, 7353, 1848, 389, 66, 9, 1, 3628800, 876809, 223456, 58095, 14384, 3125, 564, 83, 10, 1

Offset: 0

Author

Paul D. Hanna, Nov 14 2003

Keywords

• nonn
• tabl

Comments

Row 1 is A001339, antidiagonal sums form A089902 and the main diagonal is A089901; the next lower diagonal forms {1,4,27,256,..,n^n,..}, which is the hyperbinomial transform (cf. A088956) of the main diagonal.

Examples

			Note secondary diagonal: {(n+1)^(n+1)}; rows begin:
1, 2,. 6,. 24,. 120,.. 720,.. 5040,..
1, 3, 11,. 49,. 261,. 1631,. 11743,..
1,_4, 18,. 92,. 536,. 3552,. 26608,..
1, 5,_27, 159, 1029,. 7353,. 58095,..
1, 6, 38,_256, 1848, 14384, 121264,..
1, 7, 51, 389,_3125, 26595, 241015,..
1, 8, 66, 564, 5016,_46656, 456048,..
1, 9, 83, 787, 7701, 78077,_823543,..
		

Crossrefs

Cf. A001339, A088956, A089901, A089902.

Programs

Mathematica

t[n_, k_] := (n^(k+2) - Exp[n]*(n-k-1)*Gamma[k+2, n])/(k+1) // Round; Table[t[n-k, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jun 24 2013 *)

PARI

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

Formula

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

E.g.f.: 1/((1-y*exp(x))*(1-x)^2). E.g.f. (n-th row): exp(n*x)/(1-x)^2.