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

0,2

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

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

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 *)

Prog

(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

Crossrefs

Cf. A005425, A140456.

Sequence in context: A150157 A371712 A145867 * A245734 A150158 A034010

Adjacent sequences: A188141 A188142 A188143 * A188145 A188146 A188147

Keyword

nonn

Author

Groux Roland, Mar 22 2011

Extensions

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

Status

approved