A124926 Triangle read by rows: T(n,k) = binomial(n,k)*r(k), where r(k) are the Riordan numbers (r(k) = A005043(k); 0 <= k <= n).

1, 1, 0, 1, 0, 1, 1, 0, 3, 1, 1, 0, 6, 4, 3, 1, 0, 10, 10, 15, 6, 1, 0, 15, 20, 45, 36, 15, 1, 0, 21, 35, 105, 126, 105, 36, 1, 0, 28, 56, 210, 336, 420, 288, 91, 1, 0, 36, 84, 378, 756, 1260, 1296, 819, 232, 1, 0, 45, 120, 630, 1512, 3150, 4320, 4095, 2320, 603

Offset: 0

Author

Gary W. Adamson, Nov 12 2006

Keywords

• nonn
• tabl

Comments

Row sums = Catalan numbers, A000108: (1, 1, 2, 5, 14, 42...); e.g., sum of row 4 terms = A000108(4) = 14 = (1 + 0 + 6 + 4 + 3). A005043 is the inverse binomial transform of the Catalan numbers.

Examples

			First few rows of the triangle:
  1;
  1,  0;
  1,  0,  1;
  1,  0,  3,  1;
  1,  0,  6,  4,  3;
  1,  0, 10, 10, 15,  6;
  1,  0, 15, 20, 45, 36, 15;
  ...
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened
• Chao-Jen Wang, Applications of the Goulden-Jackson cluster method to counting Dyck paths by occurrences of subwords, Dissertation, Brandeis University, 2011.

Crossrefs

Cf. A000108, A005043.

Programs

GAP

B:=Binomial;;  Flat(List([0..12], n-> List([0..n], k-> B(n,k)* Sum([0..k], j-> (-1)^j*B(k+1,j)*B(2*(k-j), k-j))/(k+1) ))); # G. C. Greubel, Nov 19 2019

Magma

B:=Binomial; [B(n,k)*(&+[(-1)^j*B(k+1,j)*B(2*(k-j), k-j): j in [0..k]])/(k+1): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 19 2019

Maple

r:=n->(1/(n+1))*sum((-1)^i*binomial(n+1,i)*binomial(2*n-2*i,n-i),i=0..n): T:=(n,k)->r(k)*binomial(n,k): for n from 0 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

Mathematica

T[n_, k_]:= T[n, k]= Binomial[n, k]*Sum[(-1)^j*Binomial[k+1, j]* Binomial[2*(k-j), k-j], {j,0,k}]/(k+1); Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 19 2019 *)

PARI

T(n,k) = b=binomial; b(n,k)*sum(j=0,k, (-1)^j*b(k+1,j)*b(2*(k-j), k-j))/(k+1); \\ G. C. Greubel, Nov 19 2019

Sage

b=binomial; [[b(n,k)*sum((-1)^j*b(k+1,j)*b(2*(k-j), k-j) for j in (0..k))/(k+1) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 19 2019

Extensions

Edited by N. J. A. Sloane, Nov 29 2006