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

%I #12 Jan 12 2023 06:16:25

%S 2,3,3,7,10,7,21,29,29,21,71,86,106,86,71,253,277,352,352,277,253,925,

%T 960,1149,1324,1149,960,925,3433,3481,3873,4657,4657,3873,3481,3433,

%U 12871,12934,13654,16006,17770,16006,13654,12934,12871,48621,48701,49916,55676,64496,64496,55676,49916,48701,48621

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

%H G. C. Greubel, <a href="/A157531/b157531.txt">Rows n = 0..50 of the triangle, flattened</a>

%F 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 From _G. C. Greubel_, Dec 09 2021: (Start)

%F Sum_{k=0..n} T(n, k) = (n+2)*binomial(2*n, n).

%F T(n, k) = T(n, n-k).

%F T(n, 0) = 1 + binomial(2*n, n) = A323230(n+1).

%F T(2*n, n) = 2*A036910(n). (End)

%e Triangle begins as:

%e 2;

%e 3, 3;

%e 7, 10, 7;

%e 21, 29, 29, 21;

%e 71, 86, 106, 86, 71;

%e 253, 277, 352, 352, 277, 253;

%e 925, 960, 1149, 1324, 1149, 960, 925;

%e 3433, 3481, 3873, 4657, 4657, 3873, 3481, 3433;

%e 12871, 12934, 13654, 16006, 17770, 16006, 13654, 12934, 12871;

%e 48621, 48701, 49916, 55676, 64496, 64496, 55676, 49916, 48701, 48621;

%p A157531 := proc(n,k)

%p binomial(2*n,n)+binomial(n,k)^2 ;

%p end proc:

%p seq(seq(A157531(n,k),k=0..n),n=0..12) ; # _R. J. Mathar_, Jan 12 2023

%t T[n_, k_]:= T[n,k]= Sum[Binomial[n, j]^2, {j,0,k}] + Sum[Binomial[n, j]^2, {j, 0, n-k}];

%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten

%o (Magma) [Binomial(2*n, n) + Binomial(n, k)^2: k in [0..n], n in [0..12]]; // _G. C. Greubel_, Dec 09 2021

%o (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 Cf. A000984, A036910, A323230.

%K nonn,tabl

%O 0,1

%A _Roger L. Bagula_, Mar 02 2009

%E Edited by _G. C. Greubel_, Dec 09 2021