A338523 Triangle T(n,m) = (2*m*n+2*n-2*m^2+1)*C(2*n+2,2*m+1)/(4*n+2).

1, 2, 2, 3, 14, 3, 4, 44, 44, 4, 5, 100, 238, 100, 5, 6, 190, 828, 828, 190, 6, 7, 322, 2233, 4092, 2233, 322, 7, 8, 504, 5096, 14872, 14872, 5096, 504, 8, 9, 744, 10332, 43992, 70070, 43992, 10332, 744, 9, 10, 1050, 19176, 112200, 260780, 260780, 112200, 19176, 1050, 10

Offset: 0

Author

Vladimir Kruchinin, Nov 01 2020

Keywords

• nonn
• tabl

Examples

			1,
2, 2,
3, 14, 3,
4, 44, 44, 4,
5, 100, 238, 100, 5,
6, 190, 828, 828, 190, 6,
7, 322, 2233, 4092, 2233, 322, 7
		

Crossrefs

Cf. A045623, A086645.

2nd column=2*A002412.

Programs

Mathematica

Table[Sum[Binomial[n + 1, 2 k + 1] Binomial[n - 2 k, m - k] (k + 1)*4^k, {k, 0, n} ], {n, 0, 9}, {m, 0, n}] // Flatten (* Michael De Vlieger, Nov 04 2020 *)

Maxima

T(n,m):=((2*m*n+2*n-2*m^2+1)*binomial(2*n+2,2*m+1))/(4*n+2);

Formula

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

T(n,m) = Sum_{k=0..n} C(n+1,2*k+1)*C(n-2*k,m-k)*(k+1)*4^k.

A045563(n) = (Sum_{m=0..n} T(n,m))/2^n.