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

0,4

Comments

Triangle includes A088218.

Inverse is A106270.

Links

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

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

Example

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

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 *)

Prog

(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
(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
(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

Crossrefs

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

Sequence in context: A267392 A267553 A268115 * A267655 A263864 A060543

Adjacent sequences: A106265 A106266 A106267 * A106269 A106270 A106271

Keyword

easy,nonn,tabl

Author

Paul Barry, Apr 28 2005

Status

approved