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

1, 5, 40, 524, 10776, 327732, 13920096, 788050944, 57348311040, 5215111879680, 579420628853760, 77220215372770560, 12157472554474222080, 2232192933566250681600, 472721150641130889523200, 114371049117960857921126400, 31350167446592485414541721600, 9664519305841281076219121664000, 3328880054333616589332111409152000, 1273663039174670323519439513960448000

Offset: 1

Author

Paul D. Hanna, Oct 16 2016

Keywords

• nonn

Comments

A277406(n) = (n!)^2 + a(n).

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.

Examples

			Illustration of initial terms.
a(1) = 1, the constant term of (1+x);
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.
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.
etc.
Alternatively,
a(2) = 5 = Sum_{i=1..2} (1+i),
a(3) = 40 = Sum_{i=1..3, j=1..3, j<>i} (1 + i*(1+j)),
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.
		

Crossrefs

Cf. A277406, A277407.

Programs

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)))}
for(n=1, 20, print1(a(n), ", "))

Formula

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).