A348539 Triangle T(n, m) = binomial(n+2, m)*binomial(n+2, n-m), read by rows.

1, 3, 3, 6, 16, 6, 10, 50, 50, 10, 15, 120, 225, 120, 15, 21, 245, 735, 735, 245, 21, 28, 448, 1960, 3136, 1960, 448, 28, 36, 756, 4536, 10584, 10584, 4536, 756, 36, 45, 1200, 9450, 30240, 44100, 30240, 9450, 1200, 45, 55, 1815, 18150, 76230, 152460, 152460, 76230, 18150, 1815, 55

Offset: 0

Author

Vladimir Kruchinin, Oct 21 2021

Keywords

• nonn
• tabl

Examples

			Triangle starts:
  [0]  1;
  [1]  3,   3;
  [2]  6,  16,    6;
  [3] 10,  50,   50,   10;
  [4] 15, 120,  225,  120,   15;
  [5] 21, 245,  735,  735,  245,  21;
  [6] 28, 448, 1960, 3136, 1960, 448, 28.
  ...
Taylor series: 1 + 3*x*(y + 1) + 2*x^2*(3*y^2 + 8*y + 3) + 10*x^3*(y^3 + 5*y^2 + 5*y + 1) + 15*x^4 (y^4 + 8*y^3 + 15*y^2 + 8*y + 1) + O(x^5)
		

Crossrefs

Cf. A001263, A000217, A002694 (with offset 0 are row sums).

Programs

Maple

T := (n, k) -> binomial(n+2, k) * binomial(n+2, n-k):
for n from 0 to 9 do seq(T(n, k), k = 0..n) od; # Peter Luschny, Oct 22 2021

Mathematica

T[n_, m_] := Binomial[n + 2, m] * Binomial[n + 2, n - m]; Table[T[n, m], {n, 0, 9}, {m, 0, n}] // Flatten (* Amiram Eldar, Oct 22 2021 *)

Maxima

T(n,m):=binomial(n+2,m)*binomial(n+2,n-m);

Formula

G.f.: (x^2*y^2 - 2*x*y + x^2 - 2*x + 1)/(2*x^4*y^2*sqrt(x^2*y^2 + (-2*x^2-2*x)*y + x^2 - 2*x + 1)) + (x*y + x - 1)/(2*x^4*y^2).

G.f.: diff(N(x,y),x)*N(x,y)/(x*y^2), where N(x,y) is the g.f. of the Narayana numbers A001263.