A155796 Triangle read by rows: first define the Narayana numbers: Y(n,m)=Binomial[n, m]*Binomial[n + 1, m + 1]/(n - m + 1); then t(n,m)=Sum[(-1)^j *Y(n + 1, j)*(k + 1 - j)^n, {j, 0, k + 1}]

1, 1, -2, 1, -2, -33, 1, 1, -109, -209, 1, 11, -324, -894, 1641, 1, 36, -867, -4262, 12951, 85926, 1, 92, -2085, -20516, 74369, 625164, 1435939, 1, 211, -4419, -93989, 344617, 4306671, 9337441, -7280909, 1, 457, -7652, -402971, 1253140, 28570687

Offset

0,3

Comments

Row sums are: {0, 1, -1, -34, -316, 435, 93785, 2112964, 6609624, -1422008070, -59772756330,...}.

Links

Table of n, a(n) for n=0..41.

Formula

Narayana numbers:

Y(n,m)=Binomial[n, m]*Binomial[n + 1, m + 1]/(n - m + 1);

Eulerian sum form:

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

Example

{1},
{1, -2},
{1, -2, -33},
{1, 1, -109, -209},
{1, 11, -324, -894, 1641},
{1, 36, -867, -4262, 12951, 85926},
{ 1, 92, -2085, -20516, 74369, 625164, 1435939},
{1, 211, -4419, -93989, 344617, 4306671, 9337441, -7280909},
{1, 457, -7652, -402971, 1253140, 28570687, 62681044, -180621929, -1333480847},
{1, 958, -7325, -1618693, 2748769, 182235307, 459030881, -2310249875, -15651462716, -42453433637}

Mathematica

A[n_, m_] = Binomial[n, m]*Binomial[n + 1, m + 1]/(n - m + 1);
t[n_, k_] = Sum[(-1)^j *A[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}];
Table[Table[t[n, k], {k, 0, n - 1}], {n, 1, 10}];
Flatten[%]

Crossrefs

Sequence in context: A281129 A136156 A191657 * A341530 A141238 A094690

Adjacent sequences: A155793 A155794 A155795 * A155797 A155798 A155799

Keyword

sign,tabl

Author

Roger L. Bagula, Jan 27 2009

Extensions

Edited by N. J. A. Sloane, Jan 31 2009

Status

approved