OFFSET
0,4
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
EXAMPLE
Triangle begins as:
1;
1, 1;
2, 5, 1;
5, 22, 9, 1;
14, 93, 58, 13, 1;
42, 386, 325, 110, 17, 1;
132, 1586, 1686, 765, 178, 21, 1;
429, 6476, 8330, 4746, 1477, 262, 25, 1;
1430, 26333, 39796, 27314, 10654, 2525, 362, 29, 1;
4862, 106762, 185517, 149052, 69930, 20754, 3973, 478, 33, 1;
MATHEMATICA
T[n_, k_]:= If[k==0, CatalanNumber[n], (1/2)*Binomial[n, k-1]*(4^(n-k+ 1) -Binomial[2*n, n]/Binomial[2*(k-1), k-1])];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jul 28 2024 *)
PROG
(Magma)
A046527:= func< n, k | k eq 0 select Catalan(n) else (1/2)*Binomial(n, k-1)*(4^(n-k+1) - Binomial(2*n, n)/(k*Catalan(k-1))) >;
[A046527(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 28 2024
(SageMath)
def A046527(n, k):
if k==0: return catalan_number(n)
else: return (1/2)*binomial(n, k-1)*(4^(n-k+1) - binomial(2*n, n)/binomial(2*(k-1), k-1))
flatten([[A046527(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 28 2024
CROSSREFS
KEYWORD
AUTHOR
STATUS
approved