A225960 Number of permutations of [n] having exactly one strong fixed block.

0, 1, 1, 3, 9, 38, 198, 1229, 8819, 71825, 654985, 6615932, 73357572, 886078937, 11583028581, 162939646239, 2454350815033, 39415438078466, 672282146765650, 12137067564016917, 231223273420524311, 4635720862911035149, 97565878042828417209, 2150797149322137710488

Offset: 0

Author

Alois P. Heinz, May 22 2013

Keywords

• nonn

Comments

See A186373 for the definition of strong fixed blocks.

Links

• Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..450 (first 201 and last 2 terms from Alois P. Heinz)

Crossrefs

Column k=1 of A186373.

Cf. A052186, A260957.

Programs

Maple

b:= proc(n) b(n):= -`if`(n<0, 1, add(b(n-i-1)*i!, i=0..n)) end:
a:= n-> add(b(i)*add(b(j), j=0..n-i-1), i=0..n-1):
seq(a(n), n=0..25);

Mathematica

nmax = 25; A052186zero = Rest[CoefficientList[Assuming[Element[x, Reals], Series[-1/(ExpIntegralEi[1/x]/E^(1/x) + 1), {x, 0, nmax+1}]], x]]; suma = ConstantArray[0, nmax+1]; s = 0; Do[s = s + A052186zero[[j+1]]; suma[[j+1]] = s, {j, 0, nmax}]; Flatten[{0, Table[Sum[A052186zero[[i+1]]*suma[[n-i]], {i, 0, n-1}], {n, 1, nmax}]}] (* Vaclav Kotesovec, Aug 05 2015, more efficient program for big nmax *)

Formula

a(n) = Sum_{1<=i<=j<=n} A052186(i-1) * A052186(n-j).

a(n) = Sum_{i=0..n-1} A052186(i) * Sum_{j=0..n-i-1} A052186(j).

a(n) ~ 2 * (n-1)! * (1 - 1/n + 2/n^3 + 11/n^4 + 97/n^5 + 1105/n^6 + 13905/n^7 + 189633/n^8 + 2803873/n^9 + 44875599/n^10), for coefficients see A260957. - Vaclav Kotesovec, Aug 29 2014, extended Aug 05 2015