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

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

Offset: 1

Author

Paul D. Hanna, Jul 15 2012

Keywords

• nonn
• tabl
• easy

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

Examples

			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; ...
		

Links

• Paul D. Hanna, Rows n = 0..45, flattened.
• R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 (2017) Table 80.

Crossrefs

Cf. A214400 (central terms), A178325 (row sums), A054688, A000290 (1st subdiagonal), A037270 (2nd subdiagonal).

Cf. A230049.

Programs

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

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(""))

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