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A277406 a(n) equals the sum of all permutations of compositions of functions (1 + k*x) for k=1..n, evaluated at x=1.

%I A277406 #16 Mar 08 2025 01:35:50
%S A277406 1,2,9,76,1100,25176,846132,39321696,2413753344,189030205440,
%T A277406 18383301319680,2172771551093760,306662748175330560,
%U A277406 50933260598106862080,9832247390118248121600,2182733403365330313523200,552134185815355910465126400,157863713952139655599757721600,50654908373638564216229105664000
%N A277406 a(n) equals the sum of all permutations of compositions of functions (1 + k*x) for k=1..n, evaluated at x=1.
%F A277406 a(n) = Sum_{k=0..n} k!*(n-k)! * Sum_{i=0..n-k+1} (-1)^(n-i+1) * Stirling2(i,n-k+1) * Stirling1(n+1,i).
%F A277406 a(n) = (n!)^2 + A277405(n).
%F A277406 a(n) = (n+1) * A277407(n).
%F A277406 a(n) = Sum_{k=0..n} A277408(n,k).
%e A277406 Illustration of initial terms.
%e A277406 a(0) = 1, by convention;
%e A277406 a(1) = 2, the function (1+x) evaluated at x=1;
%e A277406 a(2) = 9, the sum of permutations of compositions of functions (1+x) and (1+2*x), evaluated at x=1: (1+x)o(1+2*x) + (1+2*x)o(1+x) = (2*x + 2) + (2*x + 3) = 4*x + 5.
%e A277406 a(3) = 76, the sum of permutations of compositions of functions (1+x), (1+2*x), and (1+3*x), evaluated at x=1: (1+x)o(1+2*x)o(1+3*x) + (1+x)o(1+3*x)o(1+2*x) + (1+2*x)o(1+1*x)o(1+3*x) + (1+2*x)o(1+3*x)o(1+1*x) + (1+3*x)o(1+1*x)o(1+2*x) + (1+3*x)o(1+2*x)o(1+1*x) = (6*x + 4) + (6*x + 5) + (6*x + 5) + (6*x + 9) + (6*x + 7) + (6*x + 10) = 36*x + 40.
%e A277406 etc.
%e A277406 Alternatively,
%e A277406 a(1) = 2 = Sum_{i=1..1} (1+i),
%e A277406 a(2) = 9 = Sum_{i=1..2, j=1..2, j<>i} (1 + i*(1+j)),
%e A277406 a(3) = 76 = Sum_{i=1..3, j=1..3, k=1..3, i,j,k distinct} (1 + i*(1 + j*(1+k))),
%e A277406 a(4) = 1100 = Sum_{i=1..4, j=1..4, k=1..4, m=1..4, i,j,k,m distinct} (1 + i*(1 + j*(1 + k*(1+m)))), etc.
%t A277406 Table[Sum[k!*(n-k)! * Sum[(-1)^(n-i+1) * StirlingS2[i, n-k+1] * StirlingS1[n+1, i], {i, 0, n-k+1}], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Oct 27 2016 *)
%o A277406 (PARI) {a(n) = sum(k=0,n, k!*(n-k)! * sum(i=0, n-k+1, (-1)^(n-i+1) * stirling(i, n-k+1, 2) * stirling(n+1, i, 1)))}
%o A277406 for(n=0,20,print1(a(n),", "))
%Y A277406 Cf. A277405, A277407, A277408.
%K A277406 nonn
%O A277406 0,2
%A A277406 _Paul D. Hanna_, Oct 16 2016