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
KEYWORD
nonn,tabl
AUTHOR
Henry Bottomley, Oct 19 2002
STATUS
approved