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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.

%I A307804 #42 Apr 25 2025 04:29:44
%S A307804 1,0,1,0,2,1,0,14,6,1,0,201,68,12,1,0,4704,1285,200,20,1,0,160890,
%T A307804 36214,4815,460,30,1,0,7538040,1422288,160594,13755,910,42,1,0,
%U A307804 462869190,74416131,7151984,535864,33110,1624,56,1,0,36055948320,5016901734,413347787,26821368,1490664,70686,2688,72,1
%N 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.
%H A307804 Alois P. Heinz, <a href="/A307804/b307804.txt">Rows n = 0..140, flattened</a>
%H A307804 Tom Copeland, <a href="https://mathoverflow.net/a/490571/231922">Production matrix for certain family of integer coefficients</a>, answer to question on MathOverflow (2025).
%H A307804 E. N. Gilbert, <a href="https://doi.org/10.4153/CJM-1956-046-2">Enumeration of labelled graphs</a>, Can. J. Math. 8 (1956) 405-411.
%H A307804 Richard J. Mathar, <a href="https://arxiv.org/abs/1903.12477">2-regular Digraphs of the Lovelock Lagrangian</a>, arXiv:1903.12477 [math.GM], 2019.
%F A307804 T(n,1) = A123543(n).
%F A307804 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!.
%F A307804 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
%F A307804 Conjectures from _Mikhail Kurkov_, Mar 22 2025: (Start)
%F A307804 Recursion for the k-th column (independently of other columns): T(n,k) = (1/(n-k))*Sum_{j=2..n-k+1} c(j-1)*binomial(n,j)*T(n-j+1,k) for 1 <= k < n with T(n,n) = 1 where b(n) = A123543(n), c(n) = n*b(n+1) - Sum_{j=1..n-1} binomial(n+1,j+1)*b(n-j+1)*c(j) for n > 0.
%F A307804 Production matrix is binomial(n,k)*d(n-k) (starting from the first row) for 0 <= k <= n, 0 otherwise where d(n) = E_n^{(-1)} from A356145 with a_k = b(k+1) for k > 0 (see _Tom Copeland_ link).
%F A307804 The same things seems to work for any b(n) with b(1) = 1 (I mean that it works for e.g.f. exp(t*F(x)) where F(x) = Sum_{n>=1} b(n)*x^n/n!). (End)
%e A307804 Triangle T(n,k) starts:
%e A307804   1;
%e A307804   0,       1;
%e A307804   0,       2,       1;
%e A307804   0,      14,       6,      1;
%e A307804   0,     201,      68,     12,     1;
%e A307804   0,    4704,    1285,    200,    20,   1;
%e A307804   0,  160890,   36214,   4815,   460,  30,  1;
%e A307804   0, 7538040, 1422288, 160594, 13755, 910, 42, 1;
%e A307804   ...
%p A307804 b:= proc(n) option remember; `if`(n<2, 1,
%p A307804       n^2*b(n-1)-n*(n-1)^2*b(n-2)/2)
%p A307804     end:
%p A307804 a:= proc(n) option remember; `if`(n=0, 0, b(n)-
%p A307804       add(j*binomial(n, j)*b(n-j)*a(j), j=1..n-1)/n)
%p A307804     end:
%p A307804 g:= proc(n, k) option remember; `if`(n=0, x^k/k!,
%p A307804       add(g(n-j, k+1)*a(j)*binomial(n,j), j=1..n))
%p A307804     end:
%p A307804 T:= (n,k)-> coeff(g(n, 0), x, k):
%p A307804 seq(seq(T(n, k), k=0..n), n=0..10);  # _Alois P. Heinz_, Mar 22 2025
%t A307804 b[n_] := b[n] = If[n < 2, 1, n^2*b[n - 1] - n*(n - 1)^2*b[n - 2]/2];
%t A307804 a[n_] := a[n] = If[n == 0, 0, b[n] - Sum[j*Binomial[n, j]*b[n - j]*a[j], {j, 1, n - 1}]/n];
%t A307804 g[n_, k_] := g[n, k] = If[n == 0, x^k/k!, Sum[g[n - j, k + 1]*a[j]* Binomial[n, j], {j, 1, n}]];
%t A307804 T[n_, k_] := Coefficient[g[n, 0], x, k];
%t A307804 Table[Table[T[n, k], { k, 0, n}], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Apr 16 2025, after _Alois P. Heinz_ *)
%Y A307804 Cf. A123543 (column k=1), A000681 (row sums).
%K A307804 nonn,tabl,easy
%O A307804 0,5
%A A307804 _R. J. Mathar_, Apr 29 2019