A046658 Triangle related to A001700 and A000302 (powers of 4).

1, 3, 1, 10, 7, 1, 35, 38, 11, 1, 126, 187, 82, 15, 1, 462, 874, 515, 142, 19, 1, 1716, 3958, 2934, 1083, 218, 23, 1, 6435, 17548, 15694, 7266, 1955, 310, 27, 1, 24310, 76627, 80324, 44758, 15086, 3195, 418, 31, 1, 92378, 330818, 397923, 259356, 105102, 27866, 4867, 542, 35, 1

Offset

1,2

Links

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Formula

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

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

Example

Triangle begins as:
      1;
      3,     1;
     10,     7,     1;
     35,    38,    11,     1;
    126,   187,    82,    15,     1;
    462,   874,   515,   142,    19,    1;
   1716,  3958,  2934,  1083,   218,   23,   1;
   6435, 17548, 15694,  7266,  1955,  310,  27,  1;
  24310, 76627, 80324, 44758, 15086, 3195, 418, 31, 1;

Mathematica

T[n_, k_]:= (1/2)*Binomial[n, k-1]*(Binomial[2*n, n]/Binomial[2*(k-1), k -1] - 4^(n-k+1)*(k-1)/n);
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jul 28 2024 *)

Prog

(Magma)
A046658:= func< n, k | Binomial(n, k)*(Binomial(n+1, 2)*Catalan(n )/Catalan(k-1) -4^(n-k+1)*Binomial(k, 2))/(n*(n-k+1)) >;
[A046658(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 28 2024
(SageMath)
def A046658(n, k): return (1/2)*binomial(n, k-1)*(binomial(2*n, n)/binomial(2*(k-1), k-1) - 4^(n-k+1)*(k-1)/n)
flatten([[A046658(n, k) for k in range(1, n+1)] for n in range(1, 13)]) # G. C. Greubel, Jul 28 2024

Crossrefs

Column sequences for m=1..6: A001700, A000531, A029887, A045724, A045492, A045530.

Row sums: A046885.

Cf. A000302.

Sequence in context: A337273 A116384 A117207 * A124574 A322383 A295856

Adjacent sequences: A046655 A046656 A046657 * A046659 A046660 A046661

Keyword

easy,nonn,tabl

Author

Wolfdieter Lang

Status

approved