A076571 Binomial triangle based on factorials.

1, 1, 2, 2, 3, 5, 6, 8, 11, 16, 24, 30, 38, 49, 65, 120, 144, 174, 212, 261, 326, 720, 840, 984, 1158, 1370, 1631, 1957, 5040, 5760, 6600, 7584, 8742, 10112, 11743, 13700, 40320, 45360, 51120, 57720, 65304, 74046, 84158, 95901, 109601

Offset: 0

Author

Henry Bottomley, Oct 19 2002

Keywords

• nonn
• tabl

Examples

			Rows start:
    1;
    1,   2;
    2,   3,   5;
    6,   8,  11,  16;
   24,  30,  38,  49,  65;
  120, 144, 174, 212, 261, 326;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened
• E. Biondi, L. Divieti, and G. Guardabassi, Counting paths, circuits, chains and cycles in graphs: A unified approach, Canad. J. Math. 22 1970 22-35. See Table I.
• D. Dumont, Matrices d'Euler-Seidel, Sem. Loth. Comb. B05c (1981) 59-78.

Crossrefs

Columns include A000142, A001048, A001344, A001345, A001346, A001347.

Right hand columns include A000522, A001339, A001340, A001341, A001342.

Cf. A002627 (row sums), A099022.

Programs

Magma

A076571:= func< n,k| (&+[Binomial(k,j)*Factorial(n-j): j in [0..k]]) >;
[A076571(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 05 2023

Mathematica

A076571[n_, k_]:= n!*Hypergeometric1F1[-k,-n,1];
Table[A076571[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 05 2023 *)

SageMath

def A076571(n,k): return sum(binomial(k,j)*factorial(n-j) for j in range(k+1))
flatten([[A076571(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 05 2023

Formula

T(n, k) = Sum_{j=0..k} binomial(k, j)*(n-j)!.

T(n, k) = T(n, k-1) + T(n-1, k-1) with T(n, 0) = n!.

T(n, n) = A000522(n).

Sum_{k=0..n} T(n, k) = A002627(n+1).

From G. C. Greubel, Oct 05 2023: (Start)

T(n, k) = n! * Hypergeometric1F1([-k], [-n], 1).

T(2*n, n) = A099022(n). (End)