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

0,2

Links

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

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.

Example

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)

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 *)

Prog

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

Crossrefs

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

Sequence in context: A006807 A298180 A119460 * A095356 A123104 A038076

Adjacent sequences: A348536 A348537 A348538 * A348540 A348541 A348542

Keyword

nonn,tabl

Author

Vladimir Kruchinin, Oct 21 2021

Status

approved