A247493 Triangle read by rows: T(n, k) = C(n, k)*C(2*k, k)/(k+1) - sum(j = 0..k, (-1)^j*(1-j)^n*C(k, j)/k!), 0<=k<=n.

0, 0, 0, 0, 1, 1, 0, 2, 6, 4, 0, 3, 11, 22, 13, 0, 4, 20, 45, 75, 41, 0, 5, 29, 110, 190, 261, 131, 0, 6, 42, 154, 560, 826, 938, 428, 0, 7, 55, 322, 749, 2646, 3570, 3452, 1429, 0, 8, 72, 335, 2499, 3885, 12012, 15198, 12897, 4861, 0, 9, 89, 770, 650, 16947, 21693, 53880, 63915, 48655, 16795, 0, 10, 110, 484, 11660, -8338, 97482

Offset: 0

Author

Peter Luschny, Oct 02 2014

Keywords

• sign
• tabl

Comments

First negative value appears at T(11,5). - Indranil Ghosh, Mar 04 2017

Examples

			0;
0, 0;
0, 1, 1;
0, 2, 6, 4;
0, 3, 11, 22, 13;
0, 4, 20, 45, 75, 41;
0, 5, 29, 110, 190, 261, 131;
0, 6, 42, 154, 560, 826, 938, 428;
		

Links

• Indranil Ghosh, Rows 0..100, flattened

Crossrefs

Cf. A001453, A098474, A105794, A247491.

Programs

Maple

T := proc(n, k) binomial(n,k)*binomial(2*k,k)/(k+1) - add((-1)^j*(1-j)^n /(j!*(k-j)!), j = 0..k) end:
for n from 0 to 12 do seq(T(n,k), k=0..n) od;

Mathematica

Flatten[Table[(Binomial[n,k] * Binomial[2k,k] / (k+1)) - Sum[(-1)^j*(1-j)^n*Binomial[k,j]/k!,{j,0,k}],{n,0,10},{k,0,n}]] (* Indranil Ghosh, Mar 04 2017 *)

PARI

tabl(nn) = {for (n=0, nn, for (k=0, n, print1((binomial(n,k)*binomial(2*k,k)/(k+1))-sum(j=0, k, (-1)^j*(1-j)^n*binomial(k,j)/k!),", ",);); print(););};
tabl(10); \\ Indranil Ghosh, Mar 04 2017

Formula

A105794(n, k) = (-1)^(n-k)*(C(n, k)*Catalan(k) - T(n, k)).

A247491(n) = Sum(k=0..n, (-1)^(n-k+1)*T(n, k)).

A001453(n) = T(n, n).

T(n,k) = A098474 (n,k) - A105794 (n,k). - Michel Marcus, Mar 04 2017