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

0,4

Links

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

Formula

T(n, k) = binomial(n, k-1)*( 4^(n-k+1) - binomial(2*n, n)/binomial(2*(k-1), k-1) )/2, for n >= k >= 0, with T(n, 0) = A000108(n).

G.f. for column k: c(x)*(x/(1-4*x))^m, where c(x) = g.f. for Catalan numbers (A000108).

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

Column sequences are: A000108 (k=0), A000346 (k=1), A018218 (k=2), A042941 (k=3), A042985 (k=4), A045505 (k=5), A045622 (k=6).

Row sums: A046814.

Sequence in context: A021803 A356273 A077382 * A378764 A008343 A257264

Adjacent sequences: A046524 A046525 A046526 * A046528 A046529 A046530

Keyword

easy,nonn,tabl

Author

Wolfdieter Lang

Status

approved