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A330798 Triangle read by rows, interpolating between the central binomial coefficients and the central coefficients of the Catalan triangle. T(n, k) for 0 <= k <= n.

%I A330798 #11 May 24 2023 03:57:01
%S A330798 1,2,2,6,15,9,20,84,112,48,70,420,900,825,275,252,1980,5940,8580,6006,
%T A330798 1638,924,9009,35035,70070,76440,43316,9996,3432,40040,192192,495040,
%U A330798 742560,651168,310080,62016,12870,175032,1002456,3174444,6104700,7325640,5372136,2206413,389367
%N A330798 Triangle read by rows, interpolating between the central binomial coefficients and the central coefficients of the Catalan triangle. T(n, k) for 0 <= k <= n.
%H A330798 G. C. Greubel, <a href="/A330798/b330798.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A330798 T(n, k) := ((n+1)/(2*n+1))*binomial(2*n+1, n+k+1)*binomial(2*n+k, k).
%F A330798 T(n, 0) = A000984(n).
%F A330798 T(n, n) = A174687(n).
%F A330798 Sum_{k=0..n} T(n, k) = A330801(n).
%F A330798 Sum_{k=0..n} (-1)^k*T(n, k) = 0^n. - _G. C. Greubel_, May 23 2023
%e A330798 Triangle starts:
%e A330798   n\k  [0]    [1]     [2]     [3]     [4]     [5]     [6]   [7]
%e A330798   [0]    1
%e A330798   [1]    2,     2
%e A330798   [2]    6,    15,      9
%e A330798   [3]   20,    84,    112,     48
%e A330798   [4]   70,   420,    900,    825,    275
%e A330798   [5]  252,  1980,   5940,   8580,   6006,   1638
%e A330798   [6]  924,  9009,  35035,  70070,  76440,  43316,   9996
%e A330798   [7] 3432, 40040, 192192, 495040, 742560, 651168, 310080, 6201
%p A330798 alias(C=binomial): T := (n, k) -> ((n+1)/(2*n+1))*C(2*n+1, n+k+1)*C(2*n+k, k):
%p A330798 seq(seq(T(n,k), k=0..n), n=0..8);
%t A330798 T[n_, k_]:= ((n+1)/(n+k+1))*Binomial[n,k]*Binomial[2*n+k,n];
%t A330798 Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, May 23 2023 *)
%o A330798 (Magma)
%o A330798 A330798:= func< n,k | ((n+1)/(n+k+1))*Binomial(n,k)*Binomial(2*n+k,n) >;
%o A330798 [A330798(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, May 23 2023
%o A330798 (SageMath)
%o A330798 def A330798(n,k): return ((n+1)/(n+k+1))*binomial(n, k)*binomial(2*n+k, n)
%o A330798 flatten([[A330798(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, May 23 2023
%Y A330798 Cf. A000984, A033184, A174687, A330801.
%K A330798 nonn,tabl
%O A330798 0,2
%A A330798 _Peter Luschny_, Jan 02 2020