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A156763 Triangle T(n, k) = binomial(2*k, k)*binomial(n+k, n-k) + binomial(2*n-k, k)*binomial(2*(n-k), n-k), read by rows.

%I A156763 #11 Sep 08 2022 08:45:41
%S A156763 2,3,3,7,12,7,21,42,42,21,71,160,180,160,71,253,660,770,770,660,253,
%T A156763 925,2814,3570,3360,3570,2814,925,3433,12068,17388,15750,15750,17388,
%U A156763 12068,3433,12871,51552,85344,81312,69300,81312,85344,51552,12871
%N A156763 Triangle T(n, k) = binomial(2*k, k)*binomial(n+k, n-k) + binomial(2*n-k, k)*binomial(2*(n-k), n-k), read by rows.
%D A156763 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 66.
%H A156763 G. C. Greubel, <a href="/A156763/b156763.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A156763 T(n, k) = binomial(2*k, k)*binomial(n+k, n-k) + binomial(2*n-k, k)*binomial(2*(n-k), n-k).
%F A156763 T(n, k) = A063007(n, k) + A063007(n, n-k).
%F A156763 Sum_{k=0..n} T(n, k) = 2*A001850(n). - _G. C. Greubel_, Jun 15 2021
%e A156763 Triangle begins as:
%e A156763       2;
%e A156763       3,      3;
%e A156763       7,     12,      7;
%e A156763      21,     42,     42,     21;
%e A156763      71,    160,    180,    160,     71;
%e A156763     253,    660,    770,    770,    660,    253;
%e A156763     925,   2814,   3570,   3360,   3570,   2814,    925;
%e A156763    3433,  12068,  17388,  15750,  15750,  17388,  12068,   3433;
%e A156763   12871,  51552,  85344,  81312,  69300,  81312,  85344,  51552,  12871;
%e A156763   48621, 218880, 413820, 438900, 342342, 342342, 438900, 413820, 218880, 48621;
%t A156763 T[n_, k_]:= Binomial[n+k, n-k]*Binomial[2*k, k] + Binomial[2*(n-k), n-k]*Binomial[ 2*n-k, k];
%t A156763 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jun 15 2021 *)
%o A156763 (Magma)
%o A156763 A063007:= func< n,k | Binomial(n, k)*Binomial(n+k, k) >;
%o A156763 A156763:= func< n,k | A063007(n,k) + A063007(n,n-k) >;
%o A156763 [A156763(n,k): k in [0..n]. n in [0..12]]; // _G. C. Greubel_, Jun 15 2021
%o A156763 (Sage)
%o A156763 def A063007(n, k): return binomial(n+k, n-k)*binomial(2*k, k)
%o A156763 def A156763(n, k): return A063007(n,k) + A063007(n,n-k)
%o A156763 flatten([[A156763(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 15 2021
%Y A156763 Cf. A001850, A063007.
%K A156763 nonn,tabl
%O A156763 0,1
%A A156763 _Roger L. Bagula_, Feb 15 2009
%E A156763 Edited by _G. C. Greubel_, Jun 15 2021