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

1, 2, 9, 76, 1100, 25176, 846132, 39321696, 2413753344, 189030205440, 18383301319680, 2172771551093760, 306662748175330560, 50933260598106862080, 9832247390118248121600, 2182733403365330313523200, 552134185815355910465126400, 157863713952139655599757721600, 50654908373638564216229105664000

Offset

0,2

Links

Table of n, a(n) for n=0..18.

Formula

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)).

a(n) = (n!)^2 + A277405(n).

a(n) = (n+1) * A277407(n).

a(n) = Sum_{k=0..n} A277408(n,k).

Example

Illustration of initial terms.
a(0) = 1, by convention;
a(1) = 2, the function (1+x) evaluated at x=1;
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.
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.
etc.
Alternatively,
a(1) = 2 = Sum_{i=1..1} (1+i),
a(2) = 9 = Sum_{i=1..2, j=1..2, j<>i} (1 + i*(1+j)),
a(3) = 76 = Sum_{i=1..3, j=1..3, k=1..3, i,j,k distinct} (1 + i*(1 + j*(1+k))),
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.

Mathematica

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 *)

Prog

(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)))}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A277405, A277407, A277408.

Sequence in context: A108992 A058164 A006059 * A182947 A006041 A192551

Adjacent sequences: A277403 A277404 A277405 * A277407 A277408 A277409

Keyword

nonn

Author

Paul D. Hanna, Oct 16 2016

Status

approved