A369381 Triangle read by rows: T(n,k) = binomial(n+1,k+1)*Stirling2(n+k,k).

1, 0, 1, 0, 3, 7, 0, 6, 60, 90, 0, 10, 310, 1505, 1701, 0, 15, 1260, 14490, 46620, 42525, 0, 21, 4445, 105875, 716205, 1727110, 1323652, 0, 28, 14280, 653100, 8162000, 38623200, 74570496, 49329280, 0, 36, 42924, 3591126, 77049126, 630714084, 2283709428, 3678671997, 2141764053

Offset

0,5

Comments

The triangle T(n,k) is a functional dual of the triangle A269939 in identity: B(n) = Sum_{k=0..n}(-1)^(k)*A269939(n,k)/Binomial(n+k,k) = Sum_{k=0..n}(-1)^(k)*T(n,k)/Binomial(n+k,k). Where B(n) are the Bernoulli numbers.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

I. V. Statsenko, Functional twin of triangle A269939, Innovation science No 1-2, State Ufa, Aeterna Publishing House, 2024, pp. 15-19. In Russian.

Formula

T(n,k) = binomial(n+1,k+1)*Stirling2(n+k,k).

Example

n\k  0      1       2        3        4       5
0:    1
1:    0      1
2:    0      3       7
3:    0      6      60       90
4:    0     10     310     1505     1701
5:    0     15    1260    14490    46620    42525

Maple

T:=(n, k)->((n+1)!/((k+1)!*(n-k)!))*Stirling2(n+k, k):seq(seq(T(n, k), k=0..n), n=0..10);

Prog

(PARI) T(n, k)=binomial(n+1, k+1)*stirling(n+k, k, 2) \\ Andrew Howroyd, Oct 31 2025

Crossrefs

Cf. A007820 (right diagonal), A269939.

Sequence in context: A197005 A396735 A199778 * A086729 A332527 A175576

Adjacent sequences: A369378 A369379 A369380 * A369382 A369383 A369384

Keyword

nonn,tabl

Author

Igor Victorovich Statsenko, Jan 22 2024

Extensions

More terms from Andrew Howroyd, Oct 31 2025

Status

approved