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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). 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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

LINKS

Table of n, a(n) for n=0..48.

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)!).

EXAMPLE

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;

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

CROSSREFS

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

Sequence in context: A202183 A227612 A221273 * A202185 A304334 A303535

Adjacent sequences:  A352604 A352605 A352606 * A352608 A352609 A352610

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny and Mélika Tebni, Mar 23 2022

STATUS

approved

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Last modified October 4 21:22 EDT 2022. Contains 357240 sequences. (Running on oeis4.)