A349727 Triangle read by rows, T(n, k) = [x^(n - k)] hypergeom([-n, -1 + n], [-n], x).

1, 0, 1, 1, 1, 1, 4, 3, 2, 1, 15, 10, 6, 3, 1, 56, 35, 20, 10, 4, 1, 210, 126, 70, 35, 15, 5, 1, 792, 462, 252, 126, 56, 21, 6, 1, 3003, 1716, 924, 462, 210, 84, 28, 7, 1, 11440, 6435, 3432, 1716, 792, 330, 120, 36, 8, 1, 43758, 24310, 12870, 6435, 3003, 1287, 495, 165, 45, 9, 1

Offset

0,7

Links

Table of n, a(n) for n=0..65.

Example

Triangle starts:
[0] 1;
[1] 0,     1;
[2] 1,     1,    1;
[3] 4,     3,    2,    1;
[4] 15,    10,   6,    3,    1;
[5] 56,    35,   20,   10,   4,   1;
[6] 210,   126,  70,   35,   15,  5,   1;
[7] 792,   462,  252,  126,  56,  21,  6,   1;
[8] 3003,  1716, 924,  462,  210, 84,  28,  7,  1;
[9] 11440, 6435, 3432, 1716, 792, 330, 120, 36, 8, 1;

Maple

p := n -> hypergeom([-n, -1 + n], [-n], x):
seq(seq(coeff(simplify(p(n)), x, n - k), k = 0..n), n = 0..10);

Mathematica

(* rows[0..k], k[0..oo] *)
r={}; k=11; For[n=0, n<k+1, n++, AppendTo[r, Binomial[2*k-2-n, k-2]]]; r
(* columns [0..n], n[0..oo] *)
c={}; n=0; For[k=n, k<n+13, k++, AppendTo[c, Binomial[2*k-2-n, k-2]]]; c
(* sequence *)
s={}; For[k=0, k<13, k++, For[n=0, n<k+1, n++, AppendTo[s, Binomial[2*k-2-n, k-2]]]]; s (* Detlef Meya, Jun 26 2023 *)

Crossrefs

Row sums: A088218, alternating row sums: A091526.

Central coefficients: binomial(3*n-2, n) (cf. A117671).

T(n, 0) = binomial(2*(n-1), n) (cf. A001791).

Cf. A257635.

Sequence in context: A295673 A155172 A253902 * A129154 A055115 A294280

Adjacent sequences: A349724 A349725 A349726 * A349728 A349729 A349730

Keyword

nonn,tabl

Author

Peter Luschny, Nov 27 2021

Status

approved