OFFSET
0,3
COMMENTS
a(n) is the number of ways to form an ordered pair of n-permutations and then choose a subset of its common descent set. Cf. A192721. - Geoffrey Critzer, Apr 29 2023
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..248
FORMULA
a(n) = Sum_{k=0..n-1} C(n, k)^2*a(k) for n>0, with a(0)=1.
a(n) = A102220(n+k, k)/C(n+k, k)^2 for k>=0.
Sum_{n>=0} a(n)*x^n/n!^2 = 1/(2-BesselI(0,2*sqrt(x))). - Vladeta Jovovic, Jul 17 2006
a(n) ~ c * (n!)^2 / r^n, where r = 0.81712266563155429332453954757369795... is the root of the equation BesselJ(0, 2*I*sqrt(x))=2, and c = 0.833570458821600548332410448635741072476086046022299770387... = 1/(sqrt(r) * BesselI(1, 2*sqrt(r))). - Vaclav Kotesovec, Mar 02 2014, updated Apr 01 2018
From Geoffrey Critzer, Apr 29 2023: (Start)
Sum_{n>=0} a(n)*z^n/(n!)^2 = 1/(2-E(z)) where E(z) = Sum_{n>=0} z^n/(n!)^2.
a(n) = Sum_{k=0..n-1} A192721(n,k)*2^k. (End)
MAPLE
b:= proc(n) option remember; `if`(n=0, 1,
add(b(n-i)*binomial(n, i)/i!, i=1..n))
end:
a:= n-> b(n)*n!:
seq(a(n), n=0..20); # Alois P. Heinz, May 11 2016
MATHEMATICA
Rest[CoefficientList[Series[1/(2-BesselJ[0, 2*I*Sqrt[x]]), {x, 0, 20}], x] * Range[0, 20]!^2] (* Vaclav Kotesovec, Mar 02 2014 *)
m = 20; CoefficientList[1/(2 - BesselI[0, 2 Sqrt[x]]) + O[x]^m, x] Range[0, m - 1]!^2 (* Jean-François Alcover, Jun 11 2019, after Vladeta Jovovic *)
b[n_] := b[n] = If[n==0, 1, Sum[b[n-i] Binomial[n, i]/i!, {i, 1, n}]];
a[n_] := b[n] n!;
a /@ Range[0, 20] (* Jean-François Alcover, Dec 03 2020, after Alois P. Heinz *)
PROG
(PARI) a(n)=if(n==0, 1, sum(k=0, n-1, binomial(n, k)^2*a(k)))
(Sage)
L = taylor(1/(1-x*hypergeometric((1, ), (2, 2), x)), x, 0, 14).list()
[factorial(i)^2*c for (i, c) in enumerate(L)] # Peter Luschny, Jul 28 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 31 2004
EXTENSIONS
Content moved from A192723 to this sequence by Alois P. Heinz, Sep 11 2019
STATUS
approved