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A341470 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} binomial(k*n,n-j) * binomial(k*n+j,j).

%I A341470 #22 Feb 16 2025 08:34:01
%S A341470 1,1,1,1,3,1,1,5,13,1,1,7,41,63,1,1,9,85,377,321,1,1,11,145,1159,3649,
%T A341470 1683,1,1,13,221,2625,16641,36365,8989,1,1,15,313,4991,50049,246047,
%U A341470 369305,48639,1,1,17,421,8473,118721,982729,3707509,3800305,265729,1
%N A341470 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} binomial(k*n,n-j) * binomial(k*n+j,j).
%H A341470 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DelannoyNumber.html">Delannoy Number</a>.
%F A341470 T(n,k) = A008288(n,k*n).
%F A341470 T(n,k) = Sum_{j=0..n} 2^j * binomial(n,j) * binomial(k*n,j).
%e A341470 Square array begins:
%e A341470   1,    1,     1,      1,      1,       1, ...
%e A341470   1,    3,     5,      7,      9,      11, ...
%e A341470   1,   13,    41,     85,    145,     221, ...
%e A341470   1,   63,   377,   1159,   2625,    4991, ...
%e A341470   1,  321,  3649,  16641,  50049,  118721, ...
%e A341470   1, 1683, 36365, 246047, 982729, 2908411, ...
%o A341470 (PARI) T(n, k) = sum(j=0, n, binomial(k*n, n-j)*binomial(k*n+j, j));
%o A341470 (PARI) T(n, k) = sum(j=0, n, 2^j*binomial(n, j)*binomial(k*n, j));
%Y A341470 Columns k=0..5 give A000012, A001850, A026000, A026001, A331329, A341491.
%Y A341470 Rows n=0..2 give A000012, A005408, A102083.
%Y A341470 Main diagonal gives A181675(n+1).
%Y A341470 Cf. A008288.
%K A341470 nonn,tabl
%O A341470 0,5
%A A341470 _Seiichi Manyama_, Feb 13 2021