A188144 Binomial transform A140456(n+1) (indecomposable involutions).

1, 2, 6, 20, 74, 292, 1218, 5308, 24034, 112484, 542346, 2686268, 13639226, 70863652, 376208706, 2038335580, 11259474754, 63353211332, 362819139978, 2113410084188, 12513610048154, 75274067489284, 459782361574146, 2850369932150908, 17926893505949986, 114337654086861092

Offset: 0

Author

Groux Roland, Mar 22 2011

Keywords

• nonn

Comments

a(n) is also the INVERTi transform of A005425(n+1) (self-inverse partial permutations) starting at n=2.

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A005425, A140456.

Programs

Maple

b:= proc(n) b(n):= `if`(n<2, n+1,  2*b(n-1) + (n-1)*b(n-2)) end:
g:= proc(n) g(n):= `if`(n<1, -1, -add(g(n-i)*b(i), i=1..n)) end:
a:= n-> g(n+2):
seq(a(n), n=0..28);  # Alois P. Heinz, Mar 19 2020

Mathematica

nmax = 18; A140456 = CoefficientList[ Series[1 - 1/Total[ CoefficientList[ Series[Exp[x^2/2 + x], {x, 0, nmax + 2}], x]*Range[0, nmax + 2]!* x^Range[0, nmax + 2]], {x, 0, nmax + 2}], x]; a[n_] := Sum[ Binomial[n, k]*A140456[[k + 3]], {k, 0, n}]; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Jul 03 2013 *)

PARI

seq(n)={Vec(1 - 2*x - 1/serlaplace(exp( 2*x + x^2/2 + O(x^3*x^n) )))} \\ Andrew Howroyd, Jan 06 2020

Formula

a(n) is the moment of order n for the probability density function: sqrt(2/Pi^3)*exp((x-2)^2/2)/(1+(erfi((x-2)/sqrt(2)))^2) over the interval -infinity..infinity, with erfi the imaginary error function.

G.f.: A(x) = (1 - 2*x - G(0))/x^2; G(k) = 1 - 2*x - x^2*(k+1)/G(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Jan 26 2012

Extensions

Terms a(19) and beyond from Andrew Howroyd, Jan 06 2020