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

0,3

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.

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)

Example

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

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 *)

Prog

(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
(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

Crossrefs

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

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

Cf. A002627 (row sums), A099022.

Sequence in context: A118399 A278298 A178927 * A084783 A265853 A129838

Adjacent sequences: A076568 A076569 A076570 * A076572 A076573 A076574

Keyword

nonn,tabl

Author

Henry Bottomley, Oct 19 2002

Status

approved