A152918 Triangle read by rows based on the Stirling numbers S1: t(n,m)=Sum[(-1)^(n + 1)* StirlingS1[n, j]*(k + 1 - j)^(n - 1), {j, 0, k + 1}].

1, 2, 5, 6, 37, 80, 24, 334, 1179, 2644, 120, 3566, 20617, 63413, 146394, 720, 44316, 413608, 1766365, 5161687, 12157088, 5040, 632052, 9362908, 55669771, 207499100, 590541383, 1411732608, 40320, 10212336, 236604140, 1953603356, 9326112285

Offset

2,2

Comments

Row sums are: {1, 7, 123, 4181, 234110, 19543784, 2275442862, 352293774104, 69988577590464,...}.

The sum algorithm is based on the Eulerian number sum with Stirling first kind substituted for the binomial.

Links

Table of n, a(n) for n=2..34.

Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Formula

t(n,m)=Sum[(-1)^(n + 1)* StirlingS1[n, j]*(k + 1 - j)^(n - 1), {j, 0, k + 1}].

Example

{1},
{2, 5},
{6, 37, 80},
{24, 334, 1179, 2644},
{120, 3566, 20617, 63413, 146394},
{720, 44316, 413608, 1766365, 5161687, 12157088},
{5040, 632052, 9362908, 55669771, 207499100, 590541383, 1411732608},
{40320, 10212336, 236604140, 1953603356, 9326112285, 32221533668, 90256527071, 218289140928},
{362880, 184767984, 6618132828, 75520418032, 462351260321, 1945272980967, 6403986114493, 17752922644079, 43341720908880}

Mathematica

Clear[t, n, k]; t[n_, k_] = Sum[(-1)^(n + 1)* StirlingS1[n, j]*(k + 1 - j)^(n - 1), {j, 0, k + 1}];
Table[Table[t[n, k], {k, 1, n - 1}], {n, 2, 10}];
Flatten[%]

Crossrefs

Sequence in context: A164805 A275285 A360085 * A276845 A365503 A055877

Adjacent sequences: A152915 A152916 A152917 * A152919 A152920 A152921

Keyword

nonn,tabl

Author

Roger L. Bagula, Dec 15 2008

Status

approved