A307804 Triangle T(n,k) read by rows: number of labeled 2-regular digraphs (multiple arcs and loops allowed) on n nodes with k components.

1, 2, 1, 14, 6, 1, 201, 68, 12, 1, 4704, 1285, 200, 20, 1, 160890, 36214, 4815, 460, 30, 1, 7538040, 1422288, 160594, 13755, 910, 42, 1, 462869190, 74416131, 7151984, 535864, 33110, 1624, 56, 1, 36055948320, 5016901734, 413347787, 26821368, 1490664, 70686, 2688, 72, 1, 3474195588360

Offset

1,2

Links

Table of n, a(n) for n=1..46.

E. N. Gilbert, Enumeration of labelled graphs, Can. J. Math. 8 (1956) 405-411.

Richard J. Mathar, 2-regular Digraphs of the Lovelock Lagrangian, arXiv:1903.12477 [math.GM], 2019.

Formula

T(n,1) = A123543(n).

T(n,k) = Sum_{Compositions n=n_1+n_2+...n_k, n_i>=1} multinomial(n; n_1,n_2,..,n_k) * T(n_1,1) * T(n_2,1)*... *T(n_k,1)/ k!.

E.g.f.: sum_{n,k>=0} T(n,k)*x^n*t^k /n!= exp(t*E123543(x)) where E123543(x) = sum_{n>=1} A123543(n)*x^n/t^n. [Gilbert]. - R. J. Mathar, May 08 2019

Example

The triangle starts:
1;
2,1;
14,6,1;
201,68,12,1;
4704,1285,200,20,1;
160890,36214,4815,460,30,1;
7538040,1422288,160594,13755,910,42,1;

Crossrefs

Cf. A123543 (column k=1), A000681 (row sums).

Sequence in context: A288298 A288762 A187920 * A338207 A063613 A245733

Adjacent sequences: A307801 A307802 A307803 * A307805 A307806 A307807

Keyword

nonn,tabl,easy

Author

R. J. Mathar, Apr 29 2019

Status

approved