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A214398 Triangle where the g.f. of column k is 1/(1-x)^(k^2) for k>=1, as read by rows n>=1.

%I A214398 #25 Dec 14 2017 07:38:31
%S A214398 1,1,1,1,4,1,1,10,9,1,1,20,45,16,1,1,35,165,136,25,1,1,56,495,816,325,
%T A214398 36,1,1,84,1287,3876,2925,666,49,1,1,120,3003,15504,20475,8436,1225,
%U A214398 64,1,1,165,6435,54264,118755,82251,20825,2080,81,1,1,220,12870,170544
%N A214398 Triangle where the g.f. of column k is 1/(1-x)^(k^2) for k>=1, as read by rows n>=1.
%C A214398 This is also the array A(n,k) read upwards antidiagonals, where the entry in row n and column k counts the vertex-labeled digraphs with n arcs and k vertices, allowing multi-edges and multi-loops (labeled analog to A138107). The binomial formula counts the weak compositions of distributing n arcs over the k^2 positions in the adjacency matrix. - _R. J. Mathar_, Aug 03 2017
%H A214398 Paul D. Hanna, <a href="/A214398/b214398.txt">Rows n = 0..45, flattened.</a>
%H A214398 R. J. Mathar, <a href="http://arxiv.org/abs/1709.09000">Statistics on Small Graphs</a>, arXiv:1709.09000 (2017) Table 80.
%F A214398 T(n,k) = binomial(k^2+n-k-1, n-k).
%F A214398 Row sums form A178325.
%F A214398 Central terms form A214400.
%F A214398 T(n,n-2) = A037270(n-2). - _R. J. Mathar_, Aug 03 2017
%F A214398 T(n,n-3) = (n^2-6*n+11)*(n^2-6*n+10)*(n-3)^2 /6. - _R. J. Mathar_, Aug 03 2017
%e A214398 Triangle begins:
%e A214398 1;
%e A214398 1, 1;
%e A214398 1, 4, 1;
%e A214398 1, 10, 9, 1;
%e A214398 1, 20, 45, 16, 1;
%e A214398 1, 35, 165, 136, 25, 1;
%e A214398 1, 56, 495, 816, 325, 36, 1;
%e A214398 1, 84, 1287, 3876, 2925, 666, 49, 1;
%e A214398 1, 120, 3003, 15504, 20475, 8436, 1225, 64, 1;
%e A214398 1, 165, 6435, 54264, 118755, 82251, 20825, 2080, 81, 1;
%e A214398 1, 220, 12870, 170544, 593775, 658008, 270725, 45760, 3321, 100, 1; ...
%p A214398 A214398 := proc(n,k)
%p A214398     binomial(k^2+n-k-1,n-k) ;
%p A214398 end proc:
%p A214398 seq(seq(A214398(n,k),k=1..n),n=1..10) ; # _R. J. Mathar_, Aug 03 2017
%t A214398 nmax = 11;
%t A214398 T[n_, k_] := SeriesCoefficient[1/(1-x)^(k^2), {x, 0, n-k}];
%t A214398 Table[T[n, k], {n, 1, nmax}, {k, 1, n}] // Flatten
%o A214398 (PARI) T(n,k)=binomial(k^2+n-k-1,n-k)
%o A214398 for(n=1,11,for(k=1,n,print1(T(n,k),", "));print(""))
%Y A214398 Cf. A214400 (central terms), A178325 (row sums), A054688, A000290 (1st subdiagonal), A037270 (2nd subdiagonal).
%Y A214398 Cf. A230049.
%K A214398 nonn,tabl,easy
%O A214398 1,5
%A A214398 _Paul D. Hanna_, Jul 15 2012