OFFSET
1,5
COMMENTS
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
LINKS
Paul D. Hanna, Rows n = 0..45, flattened.
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 (2017) Table 80.
FORMULA
T(n,k) = binomial(k^2+n-k-1, n-k).
Row sums form A178325.
Central terms form A214400.
T(n,n-2) = A037270(n-2). - R. J. Mathar, Aug 03 2017
T(n,n-3) = (n^2-6*n+11)*(n^2-6*n+10)*(n-3)^2 /6. - R. J. Mathar, Aug 03 2017
EXAMPLE
Triangle begins:
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, 36, 1;
1, 84, 1287, 3876, 2925, 666, 49, 1;
1, 120, 3003, 15504, 20475, 8436, 1225, 64, 1;
1, 165, 6435, 54264, 118755, 82251, 20825, 2080, 81, 1;
1, 220, 12870, 170544, 593775, 658008, 270725, 45760, 3321, 100, 1; ...
MAPLE
A214398 := proc(n, k)
binomial(k^2+n-k-1, n-k) ;
end proc:
seq(seq(A214398(n, k), k=1..n), n=1..10) ; # R. J. Mathar, Aug 03 2017
MATHEMATICA
nmax = 11;
T[n_, k_] := SeriesCoefficient[1/(1-x)^(k^2), {x, 0, n-k}];
Table[T[n, k], {n, 1, nmax}, {k, 1, n}] // Flatten
PROG
(PARI) T(n, k)=binomial(k^2+n-k-1, n-k)
for(n=1, 11, for(k=1, n, print1(T(n, k), ", ")); print(""))
CROSSREFS
KEYWORD
AUTHOR
Paul D. Hanna, Jul 15 2012
STATUS
approved