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A156052 Triangle read by rows: T(n, k) = binomial(n, k)/Beta(n+1, n-k+1) + binomial(n, n-k)/Beta(n+1, k+1).

%I A156052 #7 Sep 08 2022 08:45:41
%S A156052 2,8,8,33,48,33,144,240,240,144,635,1240,1260,1240,635,2778,6510,6720,
%T A156052 6720,6510,2778,12019,33600,38430,33600,38430,33600,12019,51488,
%U A156052 168672,223776,184800,184800,223776,168672,51488,218799,824400,1275120,1119888,900900,1119888,1275120,824400,218799
%N A156052 Triangle read by rows: T(n, k) = binomial(n, k)/Beta(n+1, n-k+1) + binomial(n, n-k)/Beta(n+1, k+1).
%C A156052 Row sums are 2*A108666(n+1): {2, 16, 114, 768, 5010, 32016, 201698, 1257472, 7777314, 47800080, 292292946, ...}.
%H A156052 G. C. Greubel, <a href="/A156052/b156052.txt">Rows n = 0..100 of triangle, flattened</a>
%F A156052 T(n, k) = binomial(n, k)/Beta(n+1, n-k+1) + binomial(n, n-k)/Beta(n+1, k+1).
%F A156052 T(n, k) = (k+1)*binomial(n+1, k+1)*( binomial(2*n-k+1, n+1) + binomial(n+k+1, n+1) ). - _G. C. Greubel_, Dec 01 2019
%e A156052 Triangle begins as:
%e A156052       2;
%e A156052       8,      8;
%e A156052      33,     48,     33;
%e A156052     144,    240,    240,    144;
%e A156052     635,   1240,   1260,   1240,    635;
%e A156052    2778,   6510,   6720,   6720,   6510,   2778;
%e A156052   12019,  33600,  38430,  33600,  38430,  33600,  12019;
%e A156052   51488, 168672, 223776, 184800, 184800, 223776, 168672, 51488;
%p A156052 b:=binomial; seq(seq( (k+1)*b(n+1, k+1)*( b(2*n-k+1, n+1) + b(n+k+1, n+1) ), k=0..n), n=0..10); # _G. C. Greubel_, Dec 01 2019
%t A156052 Table[Binomial[n, k]/Beta[n+1, n-k+1] + Binomial[n, n-k]/Beta[n+1, k+1], {n, 0, 10}, {k, 0, n}]//FlattenTable[(k+1)*Binomial[n+1, k+1]*(Binomial[n+k+1, n+1] + Binomial[2*n-k+1, n+1]), {n, 0, 10}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Dec 01 2019 *)
%o A156052 (PARI) T(n, k) = my(b=binomial); (k+1)*b(n+1, k+1)*( b(2*n-k+1, n+1) + b(n+k+1, n+1) ); \\ _G. C. Greubel_, Dec 01 2019
%o A156052 (Magma) B:=Binomial; [(k+1)*B(n+1, k+1)*( B(2*n-k+1, n+1) + B(n+k+1, n+1) ): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Dec 01 2019
%o A156052 (Sage) b=binomial; [[(k+1)*b(n+1, k+1)*( b(2*n-k+1, n+1) + b(n+k+1, n+1) ) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Dec 01 2019
%o A156052 (GAP) B:=Binomial;; Flat(List([0..10], n-> List([0..n], k-> (k+1)*B(n+1, k+1)*( B(2*n-k+1, n+1) + B(n+k+1, n+1) ) ))); # _G. C. Greubel_, Dec 01 2019
%K A156052 nonn,tabl
%O A156052 0,1
%A A156052 _Roger L. Bagula_, Feb 02 2009