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A277405 a(n) equals the constant term in the sum of all permutations of compositions of functions (1 + k*x) for k=1..n.

%I A277405 #26 Mar 08 2025 01:35:46
%S A277405 1,5,40,524,10776,327732,13920096,788050944,57348311040,5215111879680,
%T A277405 579420628853760,77220215372770560,12157472554474222080,
%U A277405 2232192933566250681600,472721150641130889523200,114371049117960857921126400,31350167446592485414541721600,9664519305841281076219121664000,3328880054333616589332111409152000,1273663039174670323519439513960448000
%N A277405 a(n) equals the constant term in the sum of all permutations of compositions of functions (1 + k*x) for k=1..n.
%C A277405 A277406(n) = (n!)^2 + a(n).
%C A277405 The sum of all permutations of the compositions of functions (1+k*x), for k=1..n, equals: (n!)^2*x + a(n); this sequence gives the constant term.
%F A277405 a(n) = Sum_{k=0..n-1} k!*(n-k)! * Sum_{i=0..n-k+1} (-1)^(n-i+1) * Stirling2(i,n-k+1) * Stirling1(n+1,i).
%e A277405 Illustration of initial terms.
%e A277405 a(1) = 1, the constant term of (1+x);
%e A277405 a(2) = 5, the constant term of the sum of permutations of compositions of functions (1+x) and (1+2*x): (1+x)o(1+2*x) + (1+2*x)o(1+x) = (2*x + 2) + (2*x + 3) = 4*x + 5.
%e A277405 a(3) = 40, the constant term of the sum of permutations of compositions of functions (1+x), (1+2*x), and (1+3*x): (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 A277405 etc.
%e A277405 Alternatively,
%e A277405 a(2) = 5 = Sum_{i=1..2} (1+i),
%e A277405 a(3) = 40 = Sum_{i=1..3, j=1..3, j<>i} (1 + i*(1+j)),
%e A277405 a(4) = 524 = Sum_{i=1..4, j=1..4, k=1..4, k<>j<>i, k<>i} (1 + i*(1 + j*(1+k))), etc.
%o A277405 (PARI) {a(n) = sum(k=0, n-1, 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 A277405 for(n=1, 20, print1(a(n), ", "))
%Y A277405 Cf. A277406, A277407.
%K A277405 nonn
%O A277405 1,2
%A A277405 _Paul D. Hanna_, Oct 16 2016