A046658 - OEIS

%I #16 Aug 09 2024 05:18:17

%S 1,3,1,10,7,1,35,38,11,1,126,187,82,15,1,462,874,515,142,19,1,1716,

%T 3958,2934,1083,218,23,1,6435,17548,15694,7266,1955,310,27,1,24310,

%U 76627,80324,44758,15086,3195,418,31,1,92378,330818,397923,259356,105102,27866,4867,542,35,1

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

%H G. C. Greubel, <a href="/A046658/b046658.txt">Rows n = 1..50 of the triangle, flattened</a>

%F 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.

%F 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).

%e Triangle begins as:

%e       1;

%e       3,     1;

%e      10,     7,     1;

%e      35,    38,    11,     1;

%e     126,   187,    82,    15,     1;

%e     462,   874,   515,   142,    19,    1;

%e    1716,  3958,  2934,  1083,   218,   23,   1;

%e    6435, 17548, 15694,  7266,  1955,  310,  27,  1;

%e   24310, 76627, 80324, 44758, 15086, 3195, 418, 31, 1;

%t 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);

%t Table[T[n, k], {n,12}, {k,n}]//Flatten (* _G. C. Greubel_, Jul 28 2024 *)

%o (Magma)

%o 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)) >;

%o [A046658(n,k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Jul 28 2024

%o (SageMath)

%o 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)

%o flatten([[A046658(n,k) for k in range(1,n+1)] for n in range(1,13)]) # _G. C. Greubel_, Jul 28 2024

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

%Y Row sums: A046885.

%Y Cf. A000302.

%K easy,nonn,tabl

%O 1,2

%A _Wolfdieter Lang_