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A157531 Triangle T(n, k) = binomial(2*n, n) + binomial(n, k)^2, read by rows.

%I A157531 #12 Jan 12 2023 06:16:25
%S A157531 2,3,3,7,10,7,21,29,29,21,71,86,106,86,71,253,277,352,352,277,253,925,
%T A157531 960,1149,1324,1149,960,925,3433,3481,3873,4657,4657,3873,3481,3433,
%U A157531 12871,12934,13654,16006,17770,16006,13654,12934,12871,48621,48701,49916,55676,64496,64496,55676,49916,48701,48621
%N A157531 Triangle T(n, k) = binomial(2*n, n) + binomial(n, k)^2, read by rows.
%H A157531 G. C. Greubel, <a href="/A157531/b157531.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A157531 T(n, k) = Sum_{j=0..k} binomial(n,j)*binomial(n,n-j) + Sum_{j=0..n-k} binomial(n,j)*binomial(n,n-j).
%F A157531 From _G. C. Greubel_, Dec 09 2021: (Start)
%F A157531 Sum_{k=0..n} T(n, k) = (n+2)*binomial(2*n, n).
%F A157531 T(n, k) = T(n, n-k).
%F A157531 T(n, 0) = 1 + binomial(2*n, n) = A323230(n+1).
%F A157531 T(2*n, n) = 2*A036910(n). (End)
%e A157531 Triangle begins as:
%e A157531       2;
%e A157531       3,     3;
%e A157531       7,    10,     7;
%e A157531      21,    29,    29,    21;
%e A157531      71,    86,   106,    86,    71;
%e A157531     253,   277,   352,   352,   277,   253;
%e A157531     925,   960,  1149,  1324,  1149,   960,   925;
%e A157531    3433,  3481,  3873,  4657,  4657,  3873,  3481,  3433;
%e A157531   12871, 12934, 13654, 16006, 17770, 16006, 13654, 12934, 12871;
%e A157531   48621, 48701, 49916, 55676, 64496, 64496, 55676, 49916, 48701, 48621;
%p A157531 A157531 := proc(n,k)
%p A157531     binomial(2*n,n)+binomial(n,k)^2 ;
%p A157531 end proc:
%p A157531 seq(seq(A157531(n,k),k=0..n),n=0..12) ; # _R. J. Mathar_, Jan 12 2023
%t A157531 T[n_, k_]:= T[n,k]= Sum[Binomial[n, j]^2, {j,0,k}] + Sum[Binomial[n, j]^2, {j, 0, n-k}];
%t A157531 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten
%o A157531 (Magma) [Binomial(2*n, n) + Binomial(n, k)^2: k in [0..n], n in [0..12]]; // _G. C. Greubel_, Dec 09 2021
%o A157531 (Sage) flatten([[binomial(2*n, n) + binomial(n, k)^2 for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Dec 09 2021
%Y A157531 Cf. A000984, A036910, A323230.
%K A157531 nonn,tabl
%O A157531 0,1
%A A157531 _Roger L. Bagula_, Mar 02 2009
%E A157531 Edited by _G. C. Greubel_, Dec 09 2021