A367023 Triangle read by rows, T(n, k) = [x^k] p(n), where p(n) = hypergeom([1/2, -n - 1, -n], [2, 2], 4*x).

1, 1, 1, 1, 3, 2, 1, 6, 12, 5, 1, 10, 40, 50, 14, 1, 15, 100, 250, 210, 42, 1, 21, 210, 875, 1470, 882, 132, 1, 28, 392, 2450, 6860, 8232, 3696, 429, 1, 36, 672, 5880, 24696, 49392, 44352, 15444, 1430, 1, 45, 1080, 12600, 74088, 222264, 332640, 231660, 64350, 4862

Offset

0,5

Links

Table of n, a(n) for n=0..54.

Formula

T(2*n, n) = Sum_{k=0..n} CatalanNumber(n)^2 * binomial(n + k, k).

From Detlef Meya, Nov 22 2023: (Start)

T(n, k) = binomial(n, k)*binomial(n+1, k)*binomial(2*k, k)/(k+1)^2.

T(n, k) = A001263(n+1, k+1)*binomial(2*k, k)/(k+1). (End)

Example

Triangle T(n, k) starts:
  [0] 1;
  [1] 1,  1;
  [2] 1,  3,    2;
  [3] 1,  6,   12,     5;
  [4] 1, 10,   40,    50,    14;
  [5] 1, 15,  100,   250,   210,     42;
  [6] 1, 21,  210,   875,  1470,    882,    132;
  [7] 1, 28,  392,  2450,  6860,   8232,   3696,    429;
  [8] 1, 36,  672,  5880, 24696,  49392,  44352,  15444,  1430;
  [9] 1, 45, 1080, 12600, 74088, 222264, 332640, 231660, 64350, 4862;

Maple

p := n -> hypergeom([1/2, -n - 1, -n], [2, 2], 4*x):
T := (n, k) -> coeff(simplify(p(n)), x, k):
seq(seq(T(n, k), k = 0..n), n = 0..9);

Mathematica

T[n_, k_]:=Binomial[n, k]*Binomial[n+1, k]*Binomial[2*k, k]/(k+1)^2; Flatten[Table[T[n, k], {n, 0, 9}, {k, 0, n}]]
(* Detlef Meya, Nov 22 2023 *)

Crossrefs

Cf. A128088 (row sums), A358368 (central terms), A367022.

Sequence in context: A094638 A196844 A196843 * A143778 A200536 A164645

Adjacent sequences: A367020 A367021 A367022 * A367024 A367025 A367026

Keyword

nonn,tabl

Author

Peter Luschny, Nov 06 2023

Status

approved