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A350202 Number T(n,k) of nodes in the k-th connected component of all endofunctions on [n] when components are ordered by increasing size; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

%I A350202 #17 Mar 18 2022 04:32:56
%S A350202 1,7,1,61,19,1,709,277,37,1,9911,4841,811,61,1,167111,91151,19706,
%T A350202 1876,91,1,3237921,1976570,486214,60229,3739,127,1,71850913,47203241,
%U A350202 13110749,1892997,152937,6721,169,1,1780353439,1257567127,380291461,62248939,5971291,340729,11197,217,1
%N A350202 Number T(n,k) of nodes in the k-th connected component of all endofunctions on [n] when components are ordered by increasing size; triangle T(n,k), n>=1, 1<=k<=n, read by rows.
%H A350202 Alois P. Heinz, <a href="/A350202/b350202.txt">Rows n = 1..141, flattened</a>
%e A350202 Triangle T(n,k) begins:
%e A350202          1;
%e A350202          7,        1;
%e A350202         61,       19,        1;
%e A350202        709,      277,       37,       1;
%e A350202       9911,     4841,      811,      61,      1;
%e A350202     167111,    91151,    19706,    1876,     91,    1;
%e A350202    3237921,  1976570,   486214,   60229,   3739,  127,   1;
%e A350202   71850913, 47203241, 13110749, 1892997, 152937, 6721, 169, 1;
%e A350202   ...
%p A350202 g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
%p A350202 b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i>n, 0,
%p A350202       add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))(g(i)^j*
%p A350202         b(n-i*j, i+1, max(0, t-j))/j!*combinat[multinomial]
%p A350202          (n, i$j, n-i*j)), j=0..n/i)))
%p A350202     end:
%p A350202 T:= (n, k)-> b(n, 1, k)[2]:
%p A350202 seq(seq(T(n, k), k=1..n), n=1..10);
%t A350202 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A350202 g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
%t A350202 b[n_, i_, t_] := b[n, i, t] = If[n == 0, {1, 0}, If[i > n, {0, 0}, Sum[ Function[p, p + If[t > 0 && t - j < 1, {0, p[[1]]*i}, {0, 0}]][g[i]^j*b[n - i*j, i + 1, Max[0, t - j]]/j!*multinomial[n, Append[Table[i, {j}], n - i*j]]], {j, 0, n/i}]]];
%t A350202 T[n_, k_] := b[n, 1, k][[2]];
%t A350202 Table[Table[T[n, k], {k, 1, n}], {n, 1, 10}] // Flatten (* _Jean-François Alcover_, Mar 18 2022, after _Alois P. Heinz_ *)
%Y A350202 Column k=1 gives A350157.
%Y A350202 Row sums give A007778.
%Y A350202 T(n+1,n) gives A003215 for n>=1.
%Y A350202 Cf. A001865, A319298, A322383.
%K A350202 nonn,tabl
%O A350202 1,2
%A A350202 _Alois P. Heinz_, Dec 19 2021