A106268 Number triangle T(n,k) = (-1)^(n-k)*binomial(k-n, n-k) = (0^(n-k) + binomial(2*(n-k), n-k))/2 if k <= n, 0 otherwise; Riordan array (1/(2-C(x)), x) where C(x) is g.f. for Catalan numbers A000108.

1, 1, 1, 3, 1, 1, 10, 3, 1, 1, 35, 10, 3, 1, 1, 126, 35, 10, 3, 1, 1, 462, 126, 35, 10, 3, 1, 1, 1716, 462, 126, 35, 10, 3, 1, 1, 6435, 1716, 462, 126, 35, 10, 3, 1, 1, 24310, 6435, 1716, 462, 126, 35, 10, 3, 1, 1, 92378, 24310, 6435, 1716, 462, 126, 35, 10, 3, 1, 1

Offset: 0

Author

Paul Barry, Apr 28 2005

Keywords

• easy
• nonn
• tabl

Comments

Triangle includes A088218.

Inverse is A106270.

Examples

			Triangle (with rows n >= 0 and columns k >= 0) begins as follows:
    1;
    1,  1;
    3,  1,  1;
   10,  3,  1, 1;
   35, 10,  3, 1, 1;
  126, 35, 10, 3, 1, 1;
  ...
Production matrix begins:
    1, 1;
    2, 0, 1;
    5, 0, 0, 1;
   14, 0, 0, 0, 1;
   42, 0, 0, 0, 0, 1;
  132, 0, 0, 0, 0, 0, 1;
  429, 0, 0, 0, 0, 0, 0, 1;
  ... - _Philippe Deléham_, Oct 02 2014
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A000108, A024718 (row sums), A088218, A106269 (diagonal sums), A106270.

Programs

Magma

A106268:= func< n,k | k eq n select 1 else (n-k+1)*Catalan(n-k)/2 >;
[A106268(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 10 2023

Mathematica

T[n_, k_]:= (-1)^(n-k)*Binomial[k-n, n-k];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 10 2023 *)

PARI

trg(nn) = {for (n=1, nn, for (k=1, n, print1(binomial(k-n,n-k)*(-1)^(n-k), ", ");); print(););} \\ Michel Marcus, Oct 03 2014

SageMath

def A106268(n,k): return (1/2)*(0^(n-k) + (n-k+1)*catalan_number(n-k))
flatten([[A106268(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jan 10 2023

Formula

T(n, k) = (-1)^(n-k)*binomial(k-n, n-k).

T(n, k) = (1/2)*(0^(n-k) + binomial(2*(n-k), n-k)).

Sum_{k=0..n} T(n, k) = A024718(n) (row sums).

Sum_{k=0..floor(n/2)} T(n-k, k) = A106269(n) (diagonal sums).

Bivariate g.f.: Sum_{n, k >= 0} T(n,k)*x^n*y^k = (1/2) * (1/(1 - x*y)) * (1 + 1/sqrt(1 - 4*x)). - Petros Hadjicostas, Jul 15 2019