A156588 A triangle of q factorial type based on Stirling first polynomials: t(n,k)=If[m == 0, n!, Product[Sum[(-1)^(i + k)*StirlingS1[k - 1, i]*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]].

1, 1, 1, 1, -1, 2, 1, -1, 2, 6, 1, -1, 3, -12, 24, 1, -1, 4, -36, 288, 120, 1, -1, 5, -80, 2160, -34560, 720, 1, -1, 6, -150, 9600, -777600, 24883200, 5040, 1, -1, 7, -252, 31500, -8064000, 1959552000, -125411328000, 40320, 1, -1, 8, -392, 84672, -52920000

Offset

0,6

Comments

Row sums are:

{1, 2, 2, 8, 15, 376, -31755, 24120096, -123459768425, 5017134314247168,

-1827769039991244222327,...}.

Links

Table of n, a(n) for n=0..50.

Formula

t(n,k)=If[m == 0, n!, Product[Sum[(-1)^(i + k)*StirlingS1[k - 1, i]*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]];

out_(n,k)=Antidiagonal(t(n,k)).

Example

{1},
{1, 1},
{1, -1, 2},
{1, -1, 2, 6},
{1, -1, 3, -12, 24},
{1, -1, 4, -36, 288, 120},
{1, -1, 5, -80, 2160, -34560, 720},
{1, -1, 6, -150, 9600, -777600, 24883200, 5040},
{1, -1, 7, -252, 31500, -8064000, 1959552000, -125411328000, 40320},
{1, -1, 8, -392, 84672, -52920000, 54190080000, -39504568320000, 5056584744960000, 362880},
{1, -1, 9, -576, 197568, -256048128, 800150400000, -3277416038400000, 7167708875980800000, -1834933472251084800000, 3628800}

Mathematica

Clear[t, n, m, i, k, a, b];
t[n_, m_] = If[m == 0, n!, Product[Sum[(-1)^(i + k)*StirlingS1[k - 1, i]*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]];
a = Table[Table[t[n, m], {n, 0, 10}], {m, 0, 10}];
b = Table[Table[a[[m, n - m + 1]], {m, n, 1, -1}], {n, 1, Length[a]}];
Flatten[%]

Crossrefs

A009963

Sequence in context: A346422 A245405 A233543 * A278543 A113186 A206497

Adjacent sequences: A156585 A156586 A156587 * A156589 A156590 A156591

Keyword

sign,tabl,uned

Author

Roger L. Bagula, Feb 10 2009

Status

approved