A337977 Triangle T(n,m) = C(n-1,n-m)*Sum_{k=1..n} C(2*k-2,k-1)*C(n-m,m-k)/m, m>0, n>0, n>=m.

1, 1, 1, 1, 3, 2, 1, 6, 8, 5, 1, 10, 22, 26, 14, 1, 15, 50, 85, 90, 42, 1, 21, 100, 225, 348, 322, 132, 1, 28, 182, 525, 1050, 1442, 1176, 429, 1, 36, 308, 1120, 2730, 4928, 5992, 4356, 1430, 1, 45, 492, 2226, 6426, 14238, 22920, 24894, 16302, 4862

Offset: 1

Author

Vladimir Kruchinin, Oct 05 2020

Keywords

• nonn
• tabl

Examples

			1,
1, 1,
1, 3,  2,
1, 6,  8,  5,
1,10, 22, 26, 14,
1,15, 50, 85, 90, 42,
1,21,100,225,348,322,132
		

Crossrefs

T(2*n,n) is A069720.

2nd column: A000217, 3rd column: 2*A006522 or 2*(A027927-1).

Programs

Mathematica

Table[Binomial[n - 1, n - m] Sum[Binomial[2 k - 2, k - 1] Binomial[n - m, m - k]/m, {k, n}], {n, 10}, {m, n}] // Flatten (* Michael De Vlieger, Oct 05 2020 *)

Maxima

T(n,m):=(binomial(n-1,n-m)*sum(binomial(2*k-2,k-1)*binomial(n-m,m-k),k,1,n))/m;

Formula

G.f.: A(x,y) = -(sqrt((2*sqrt(-4*x^2*y+x^2-2*x+1)+3*x-2)/(4*x))-1/2).