A176190 Triangle T(n,m) read by rows: (1 + binomial(n, m)*binomial(n + 1, m)/(m + 1) )^n, 0<=m<=n.

1, 2, 2, 4, 16, 4, 8, 343, 343, 8, 16, 14641, 194481, 14641, 16, 32, 1048576, 345025251, 345025251, 1048576, 32, 64, 113379904, 1418519112256, 29721861554176, 1418519112256, 113379904, 64, 128, 17249876309, 11514990476898413

Offset

0,2

Comments

Row sums are 1, 4, 24, 702, 223795, 692147718, 32559126538624, 13782459932899570762,

101546283389149890712153383, 6529719086356437527412019094222406,

8094296318445604076821089587379253402373981,...

The entry T(n,m) is also the sum of the coefficients of the polynomial (1 + (binomiral(n, m)*binomial(n + 1, m)/(m + 1))*x)^n.

Links

Table of n, a(n) for n=0..30.

Formula

T(n,m) = T(n,n-m), symmetric.

T(n,m) = (1+A001263(n+1,m+1))^n - R. J. Mathar, May 21 2025

Example

1;
2, 2;
4, 16, 4;
8, 343, 343, 8;
16, 14641, 194481, 14641, 16;
32, 1048576, 345025251, 345025251, 1048576, 32;
64, 113379904, 1418519112256, 29721861554176, 1418519112256, 113379904, 64;
128, 17249876309, 11514990476898413, 6879714958723010531, 6879714958723010531, 11514990476898413, 17249876309, 128;
256, 3512479453921, 166356282569519253121, 3683084668584351607174081, 94179781339409023500390625, 3683084668584351607174081, 166356282569519253121, 3512479453921, 256;

Mathematica

p[x_, n_, m_] := (1 + (Binomial[n, m]*Binomial[n + 1, m]/(m + 1))*x)^n
Table[Table[Apply[Plus, CoefficientList[p[x, n, m], x]], {m, 0, n}], {n, 0, 10}];
Flatten[%]

Crossrefs

Cf. A001263.

Sequence in context: A279069 A257609 A087783 * A106241 A395972 A392124

Adjacent sequences: A176187 A176188 A176189 * A176191 A176192 A176193

Keyword

nonn,tabl

Author

Roger L. Bagula, Apr 11 2010

Status

approved