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

Author

Wolfdieter Lang

Keywords

• easy
• nonn
• tabl

Examples

			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;
		

Links

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

Crossrefs

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

Row sums: A046885.

Cf. A000302.

Programs

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

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 *)

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

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).