A352607 Triangle read by rows. T(n, k) = Bell(k)*Sum_{j=0..k}(-1)^(k+j)*binomial(n, n-k+j)*Stirling2(n-k+j, j) for n >= 0 and 0 <= k <= floor(n/2).

1, 0, 0, 1, 0, 1, 0, 1, 6, 0, 1, 20, 0, 1, 50, 75, 0, 1, 112, 525, 0, 1, 238, 2450, 1575, 0, 1, 492, 9590, 18900, 0, 1, 1002, 34125, 141750, 49140, 0, 1, 2024, 114675, 854700, 900900, 0, 1, 4070, 371580, 4544925, 9909900, 2110185

Offset: 0

Author

Peter Luschny and Mélika Tebni, Mar 23 2022

Keywords

• nonn
• tabl

Examples

			Triangle starts:
[0] 1;
[1] 0;
[2] 0, 1;
[3] 0, 1;
[4] 0, 1,   6;
[5] 0, 1,  20;
[6] 0, 1,  50,   75;
[7] 0, 1, 112,  525;
[8] 0, 1, 238, 2450,  1575;
[9] 0, 1, 492, 9590, 18900;
		

Crossrefs

Cf. A028248 (row sums), A052515 (column 2), A081066, A008299, A000110, A137375.

Programs

Maple

A352607 := (n, k) -> combinat:-bell(k)*add((-1)^(k+j)*binomial(n, n-k+j)* Stirling2(n-k+j, j), j = 0..k):
seq(seq(A352607(n, k), k = 0..n/2), n = 0..12);
# Second program:
egf := k -> combinat[bell](k)*(exp(x) - 1 - x)^k/k!:
A352607 := (n, k) -> n! * coeff(series(egf(k), x, n+1), x, n):
seq(print(seq(A352607(n, k), k = 0..n/2)), n=0..12);
# Recurrence:
A352607 := proc(n, k) option remember;
if k > n/2 then 0 elif k = 0 then k^n else k*A352607(n-1, k) +
combinat[bell](k)/combinat[bell](k-1)*(n-1)*A352607(n-2, k-1) fi end:
seq(print(seq(A352607(n, k), k=0..n/2)), n=0..12); # Mélika Tebni, Mar 24 2022

Mathematica

T[n_, k_] := BellB[k]*Sum[(-1)^(k+j)*Binomial[n, n-k+j]*StirlingS2[n-k+j, j], {j, 0, k}]; Table[T[n, k], {n, 0, 12}, {k, 0, Floor[n/2]}] // Flatten (* Jean-François Alcover, Oct 21 2023 *)

Formula

T(n, k) = (-1)^k*A000110(k)*A137375(n, k) = A000110(k)*A008299(n, k).

T(2*n, n) = A081066(n).

E.g.f. column k: Bell(k)*(exp(x) - 1 - x)^k / k!, k >= 0.

T(n, k) = Bell(k)*Sum_{j=0..k} Sum_{i=0..j} ((-1)^j*(k-j)^(n-i)*binomial(n, i)) / ((k - j)!*(j - i)!).