A110180 Triangle of generalized central trinomial coefficients.

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 7, 5, 1, 1, 1, 19, 13, 7, 1, 1, 1, 51, 49, 19, 9, 1, 1, 1, 141, 161, 91, 25, 11, 1, 1, 1, 393, 581, 331, 145, 31, 13, 1, 1, 1, 1107, 2045, 1441, 561, 211, 37, 15, 1, 1, 1, 3139, 7393, 5797, 2841, 851, 289, 43, 17, 1, 1

Offset

0,8

Comments

Rows sums are A110181. Diagonal sums are A110182. Columns include central trinomial coefficients A002426, A084601, A084603, A084605, A098264. T(n,k) = central coefficient (1 + x + kx^2)^n.

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Formula

Number triangle T(n, k) = Sum_{j=0..floor((n-k)/2)} C(n-k, j)*C(n-k-j, j)*k^j.

Example

Rows begin
  1;
  1,  1;
  1,  1,  1;
  1,  3,  1,  1;
  1,  7,  5,  1,  1;
  1, 19, 13,  7,  1,  1;

Mathematica

T[n_, 0] := 1; T[n_, k_] := Sum[Binomial[n - k, j]*Binomial[n - k - j, j]*k^j, {j, 0, Floor[(n - k)/2]}]; Table[T[n, k], {n, 0, 49}, {k, 0, n}] // Flatten (* G. C. Greubel, Mar 05 2017 *)

Crossrefs

Sequence in context: A325969 A325826 A081297 * A328718 A362897 A005765

Adjacent sequences: A110177 A110178 A110179 * A110181 A110182 A110183

Keyword

easy,nonn,tabl

Author

Paul Barry, Jul 14 2005

Status

approved