Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #11 Dec 28 2023 09:23:14
%S 1,1,0,1,2,0,1,4,3,0,1,6,11,4,0,1,8,24,26,5,0,1,10,42,82,57,6,0,1,12,
%T 65,188,261,120,7,0,1,14,93,360,787,804,247,8,0,1,16,126,614,1870,
%U 3204,2440,502,9,0,1,18,164,966,3810,9476,12900,7356,1013,10,0
%N Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} k^(n-j) * binomial(j+k-1,j).
%F G.f. of column k: 1/((1-k*x) * (1-x)^k).
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, 1, ...
%e 0, 2, 4, 6, 8, 10, 12, ...
%e 0, 3, 11, 24, 42, 65, 93, ...
%e 0, 4, 26, 82, 188, 360, 614, ...
%e 0, 5, 57, 261, 787, 1870, 3810, ...
%e 0, 6, 120, 804, 3204, 9476, 23112, ...
%e 0, 7, 247, 2440, 12900, 47590, 139134, ...
%o (PARI) T(n, k) = sum(j=0, n, k^(n-j)*binomial(j+k-1, j));
%Y Columns k=0..3 give A000007, A000027(n+1), A125128(n+1), A052150.
%Y Main diagonal gives A293574.
%Y Cf. A008949, A368487.
%K nonn,tabl
%O 0,5
%A _Seiichi Manyama_, Dec 27 2023