A174119 - OEIS

%I #12 Feb 12 2021 07:52:26

%S 1,1,1,1,1,1,1,5,5,1,1,14,70,14,1,1,30,420,420,30,1,1,55,1650,4620,

%T 1650,55,1,1,91,5005,30030,30030,5005,91,1,1,140,12740,140140,300300,

%U 140140,12740,140,1,1,204,28560,519792,2042040,2042040,519792,28560,204,1,1,285,58140,1627920,10581480,19399380,10581480,1627920,58140,285,1

%N Triangle T(n, k) = ((n-k)/6)*binomial(n-1, k-1)*binomial(2*n, 2*k) with T(n, 0) = T(n, n) = 1, read by rows.

%H G. C. Greubel, <a href="/A174119/b174119.txt">Rows n = 0..100 of the triangle, flattened</a>

%F T(n, k) = c(n)/(c(k)*c(n-k)) where c(n) = Product_{j=2..n} j*(j-1)*(2*j-1)/6 for n > 2 otherwise 1.

%F From _G. C. Greubel_, Feb 11 2021: (Start)

%F T(n, k) = ((n-k)/6)*binomial(n-1, k-1)*binomial(2*n, 2*k) with T(n, 0) = T(n, n) =1.

%F Sum_{k=0..n} T(n, k) = (n*(n-1)*(2*n-1)/6)*HypergeometricPFQ[{1-n, 3/2-n, 2-n}, {3/2, 2}, -1] + 2 - [n=0] (n*(n-1)*(2*n-1)/6)*A196148[n-2] + 2 - [n=0]. (End)

%e Triangle begins as:

%e   1;

%e   1,   1;

%e   1,   1,     1;

%e   1,   5,     5,       1;

%e   1,  14,    70,      14,        1;

%e   1,  30,   420,     420,       30,        1;

%e   1,  55,  1650,    4620,     1650,       55,        1;

%e   1,  91,  5005,   30030,    30030,     5005,       91,       1;

%e   1, 140, 12740,  140140,   300300,   140140,    12740,     140,     1;

%e   1, 204, 28560,  519792,  2042040,  2042040,   519792,   28560,   204,   1;

%e   1, 285, 58140, 1627920, 10581480, 19399380, 10581480, 1627920, 58140, 285, 1;

%t (* First program *)

%t c[n_]:= If[n<2, 1, Product[j*(j-1)*(2*j-1)/6, {j, 2, n}]];

%t T[n_, k_]:= c[n]/(c[k]*c[n-k]);

%t Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten

%t (* Second program *)

%t T[n_, k_]:= If[k==0 || k==n, 1, ((n-k)/6)*Binomial[n-1, k-1]*Binomial[2*n, 2*k]];

%t Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Feb 11 2021 *)

%o (Sage)

%o def T(n,k): return 1 if (k==0 or k==n) else ((n-k)/6)*binomial(n-1, k-1)*binomial(2*n, 2*k)

%o flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 11 2021

%o (Magma)

%o T:= func< n,k | k eq 0 or k eq n select 1 else ((n-k)/6)*Binomial(n-1, k-1)*Binomial(2*n, 2*k) >;

%o [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 11 2021

%Y Cf. A174116, A174117, A174124, A174125.

%Y Cf. A111910, A196148.

%K nonn,tabl,easy

%O 0,8

%A _Roger L. Bagula_, Mar 08 2010

%E Edited by _G. C. Greubel_, Feb 11 2021