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A110180 Triangle of generalized central trinomial coefficients. 4
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 (list; table; graph; refs; listen; history; text; internal format)
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 A005765 A263159

Adjacent sequences:  A110177 A110178 A110179 * A110181 A110182 A110183

KEYWORD

easy,nonn,tabl

AUTHOR

Paul Barry, Jul 14 2005

STATUS

approved

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Last modified May 18 01:37 EDT 2021. Contains 343992 sequences. (Running on oeis4.)