A046527 A triangle related to A000108 (Catalan) and A000302 (powers of 4).
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
Offset: 0
Examples
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;
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Crossrefs
Programs
-
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
-
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 *) -
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