A102221 Column 0 of triangular matrix A102220, which equals [2*I - A008459]^(-1).

1, 1, 5, 55, 1077, 32951, 1451723, 87054773, 6818444405, 675900963271, 82717196780955, 12248810651651333, 2158585005685222491, 446445657799551807541, 107087164031952038620481, 29487141797206760561836055, 9238158011747884080353808245

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

Cf. A000275, A008459, A102220, A102222, A192721.

Row sums of A192722.

Column k=2 of A326322.

Sequence in context: A158690 A280573 A192723 * A056600 A177819 A126456

Adjacent sequences: A102218 A102219 A102220 * A102222 A102223 A102224

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