OFFSET
0,4
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
KEYWORD
AUTHOR
Paul Barry, Apr 28 2005
STATUS
approved