A102221 - OEIS

%I #36 May 01 2023 10:03:41

%S 1,1,5,55,1077,32951,1451723,87054773,6818444405,675900963271,

%T 82717196780955,12248810651651333,2158585005685222491,

%U 446445657799551807541,107087164031952038620481,29487141797206760561836055,9238158011747884080353808245

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

%C 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

%H Alois P. Heinz, <a href="/A102221/b102221.txt">Table of n, a(n) for n = 0..248</a>

%F a(n) = Sum_{k=0..n-1} C(n, k)^2*a(k) for n>0, with a(0)=1.

%F a(n) = A102220(n+k, k)/C(n+k, k)^2 for k>=0.

%F Sum_{n>=0} a(n)*x^n/n!^2 = 1/(2-BesselI(0,2*sqrt(x))). - _Vladeta Jovovic_, Jul 17 2006

%F 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

%F From _Geoffrey Critzer_, Apr 29 2023: (Start)

%F Sum_{n>=0} a(n)*z^n/(n!)^2 = 1/(2-E(z)) where E(z) = Sum_{n>=0} z^n/(n!)^2.

%F a(n) = Sum_{k=0..n-1} A192721(n,k)*2^k. (End)

%p b:= proc(n) option remember; `if`(n=0, 1,

%p       add(b(n-i)*binomial(n, i)/i!, i=1..n))

%p     end:

%p a:= n-> b(n)*n!:

%p seq(a(n), n=0..20);  # _Alois P. Heinz_, May 11 2016

%t Rest[CoefficientList[Series[1/(2-BesselJ[0, 2*I*Sqrt[x]]), {x, 0, 20}], x] * Range[0, 20]!^2] (* _Vaclav Kotesovec_, Mar 02 2014 *)

%t 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_ *)

%t b[n_] := b[n] = If[n==0, 1, Sum[b[n-i] Binomial[n, i]/i!, {i, 1, n}]];

%t a[n_] := b[n] n!;

%t a /@ Range[0, 20] (* _Jean-François Alcover_, Dec 03 2020, after _Alois P. Heinz_ *)

%o (PARI) a(n)=if(n==0,1,sum(k=0,n-1,binomial(n,k)^2*a(k)))

%o (Sage)

%o L = taylor(1/(1-x*hypergeometric((1,),(2,2),x)),x,0,14).list()

%o [factorial(i)^2*c for (i,c) in enumerate(L)] # _Peter Luschny_, Jul 28 2015

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

%Y Row sums of A192722.

%Y Column k=2 of A326322.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 31 2004

%E Content moved from A192723 to this sequence by _Alois P. Heinz_, Sep 11 2019