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A155497 Triangle T(n, k) = binomial(n, k)*binomial(n+1, k+1)*binomial(2*n, 2*k)/(n-k+1), read by rows.

%I A155497 #13 Feb 03 2025 02:23:19
%S A155497 1,1,1,1,18,1,1,90,90,1,1,280,1400,280,1,1,675,10500,10500,675,1,1,
%T A155497 1386,51975,161700,51975,1386,1,1,2548,196196,1471470,1471470,196196,
%U A155497 2548,1,1,4320,611520,9417408,22702680,9417408,611520,4320,1,1,6885,1652400,46781280,231567336,231567336,46781280,1652400,6885,1
%N A155497 Triangle T(n, k) = binomial(n, k)*binomial(n+1, k+1)*binomial(2*n, 2*k)/(n-k+1), read by rows.
%H A155497 G. C. Greubel, <a href="/A155497/b155497.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A155497 T(n, k) = binomial(n, k)*binomial(2*n, 2*k)*f(n)/(f(k)*f(n-k)), where f(n) = (n+1)!.
%F A155497 T(n, k) = binomial(n, k)*binomial(n+1, k+1)*binomial(2*n, 2*k)/(n-k+1).
%F A155497 From _G. C. Greubel_, May 29 2021: (Start)
%F A155497 Sum_{k=0..n} T(n, k) = Hypergeometric4F3([-n, -n, -n-1, -n-1/2], [1/2, 1, 2], 1).
%F A155497 T(n, k) = binomial(n, k)*A155495(n, k)/(n-k+1).
%F A155497 T(n, k) = binomial(2*n, 2*k)*A103371(n, k)/(n-k+1). (End)
%e A155497 Triangle begins as:
%e A155497   1;
%e A155497   1,    1;
%e A155497   1,   18,       1;
%e A155497   1,   90,      90,        1;
%e A155497   1,  280,    1400,      280,         1;
%e A155497   1,  675,   10500,    10500,       675,         1;
%e A155497   1, 1386,   51975,   161700,     51975,      1386,        1;
%e A155497   1, 2548,  196196,  1471470,   1471470,    196196,     2548,       1;
%e A155497   1, 4320,  611520,  9417408,  22702680,   9417408,   611520,    4320,    1;
%e A155497   1, 6885, 1652400, 46781280, 231567336, 231567336, 46781280, 1652400, 6885, 1;
%t A155497 T[n_, k_]:= Binomial[n,k]*Binomial[n+1,k+1]*Binomial[2*n,2*k]/(n-k+1);
%t A155497 Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, May 29 2021 *)
%o A155497 (Magma) [Binomial(n, k)*Binomial(n+1, k+1)*Binomial(2*n, 2*k)/(n-k+1): k in [0..n], n in [0..12]]; // _G. C. Greubel_, May 29 2021
%o A155497 (Sage) flatten([[binomial(n, k)*binomial(n+1, k+1)*binomial(2*n, 2*k)/(n-k+1) for k in (0..12)] for n in (0..12)]) # _G. C. Greubel_, May 29 2021
%Y A155497 Cf. A103371, A155495.
%K A155497 nonn,tabl,easy
%O A155497 0,5
%A A155497 _Roger L. Bagula_, Jan 23 2009
%E A155497 Edited by _G. C. Greubel_, May 29 2021