A257488 Triangle, read by rows, T(n,k) = k*Sum_{i=0..n-k} C(2*i+2*k,i)*C(n-i-1,k-1)/(i+k) for 1 <= k <= n.

1, 3, 1, 8, 6, 1, 22, 25, 9, 1, 64, 92, 51, 12, 1, 196, 324, 237, 86, 15, 1, 625, 1128, 996, 484, 130, 18, 1, 2055, 3934, 3966, 2377, 860, 183, 21, 1, 6917, 13812, 15335, 10744, 4845, 1392, 245, 24, 1, 23713, 48884, 58359, 46068, 24603, 8859, 2107, 316, 27, 1

Offset: 1

Author

Vladimir Kruchinin, Apr 26 2015

Keywords

• nonn
• tabl
• easy

Examples

			Triangle starts:
1;
3,   1;
8,   6,  1;
22, 25,  9,  1;
64, 92, 51, 12, 1;
		

Crossrefs

Cf. A014138.

Programs

Mathematica

Flatten@ Table[k Sum[Binomial[2 i + 2 k, i] Binomial[n - i - 1, k - 1]/(i + k), {i, 0, n - k}], {n, 10}, {k, n}] (* Michael De Vlieger, Apr 27 2015 *)

Maxima

T(n,k):=k*sum((binomial(2*i+2*k,i)*binomial(n-i-1,k-1))/(i+k),i,0,n-k);

PARI

T(n,k)=k*sum(i=0,n-k,(binomial(2*i+2*k,i)*binomial(n-i-1,k-1))/(i+k))
for(n=1,10,for(k=1,n,print1(T(n,k),", "))) \\ Derek Orr, Apr 27 2015

Formula

G.f.: 1/(1-(C(x)-1)/(1-x)*y)-1, where C(x) is g.f. of Catalan numbers (A000108).

T(n,n-1) = 3*(n-1) for n > 1. - Derek Orr, Apr 27 2015

T(n,n-2) = A062728(n-2) for n > 2. - Derek Orr, Apr 27 2015

T(n,1) = A014138(n). - Derek Orr, Apr 27 2015