cp's OEIS Frontend

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

%I A188144 #28 Feb 27 2022 15:41:54
%S A188144 1,2,6,20,74,292,1218,5308,24034,112484,542346,2686268,13639226,
%T A188144 70863652,376208706,2038335580,11259474754,63353211332,362819139978,
%U A188144 2113410084188,12513610048154,75274067489284,459782361574146,2850369932150908,17926893505949986,114337654086861092
%N A188144 Binomial transform A140456(n+1) (indecomposable involutions).
%C A188144 a(n) is also the INVERTi transform of A005425(n+1) (self-inverse partial permutations) starting at n=2.
%H A188144 Andrew Howroyd, <a href="/A188144/b188144.txt">Table of n, a(n) for n = 0..200</a>
%F A188144 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.
%F A188144 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
%p A188144 b:= proc(n) b(n):= `if`(n<2, n+1,  2*b(n-1) + (n-1)*b(n-2)) end:
%p A188144 g:= proc(n) g(n):= `if`(n<1, -1, -add(g(n-i)*b(i), i=1..n)) end:
%p A188144 a:= n-> g(n+2):
%p A188144 seq(a(n), n=0..28);  # _Alois P. Heinz_, Mar 19 2020
%t A188144 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 *)
%o A188144 (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
%Y A188144 Cf. A005425, A140456.
%K A188144 nonn
%O A188144 0,2
%A A188144 _Groux Roland_, Mar 22 2011
%E A188144 Terms a(19) and beyond from _Andrew Howroyd_, Jan 06 2020