OFFSET
0,1
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
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).
From G. C. Greubel, Dec 09 2021: (Start)
Sum_{k=0..n} T(n, k) = (n+2)*binomial(2*n, n).
T(n, k) = T(n, n-k).
T(n, 0) = 1 + binomial(2*n, n) = A323230(n+1).
T(2*n, n) = 2*A036910(n). (End)
EXAMPLE
Triangle begins as:
2;
3, 3;
7, 10, 7;
21, 29, 29, 21;
71, 86, 106, 86, 71;
253, 277, 352, 352, 277, 253;
925, 960, 1149, 1324, 1149, 960, 925;
3433, 3481, 3873, 4657, 4657, 3873, 3481, 3433;
12871, 12934, 13654, 16006, 17770, 16006, 13654, 12934, 12871;
48621, 48701, 49916, 55676, 64496, 64496, 55676, 49916, 48701, 48621;
MAPLE
A157531 := proc(n, k)
binomial(2*n, n)+binomial(n, k)^2 ;
end proc:
seq(seq(A157531(n, k), k=0..n), n=0..12) ; # R. J. Mathar, Jan 12 2023
MATHEMATICA
T[n_, k_]:= T[n, k]= Sum[Binomial[n, j]^2, {j, 0, k}] + Sum[Binomial[n, j]^2, {j, 0, n-k}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten
PROG
(Magma) [Binomial(2*n, n) + Binomial(n, k)^2: k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 09 2021
(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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Mar 02 2009
EXTENSIONS
Edited by G. C. Greubel, Dec 09 2021
STATUS
approved