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

%I A102221 #36 May 01 2023 10:03:41
%S A102221 1,1,5,55,1077,32951,1451723,87054773,6818444405,675900963271,
%T A102221 82717196780955,12248810651651333,2158585005685222491,
%U A102221 446445657799551807541,107087164031952038620481,29487141797206760561836055,9238158011747884080353808245
%N A102221 Column 0 of triangular matrix A102220, which equals [2*I - A008459]^(-1).
%C A102221 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 A102221 Alois P. Heinz, <a href="/A102221/b102221.txt">Table of n, a(n) for n = 0..248</a>
%F A102221 a(n) = Sum_{k=0..n-1} C(n, k)^2*a(k) for n>0, with a(0)=1.
%F A102221 a(n) = A102220(n+k, k)/C(n+k, k)^2 for k>=0.
%F A102221 Sum_{n>=0} a(n)*x^n/n!^2 = 1/(2-BesselI(0,2*sqrt(x))). - _Vladeta Jovovic_, Jul 17 2006
%F A102221 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 A102221 From _Geoffrey Critzer_, Apr 29 2023: (Start)
%F A102221 Sum_{n>=0} a(n)*z^n/(n!)^2 = 1/(2-E(z)) where E(z) = Sum_{n>=0} z^n/(n!)^2.
%F A102221 a(n) = Sum_{k=0..n-1} A192721(n,k)*2^k. (End)
%p A102221 b:= proc(n) option remember; `if`(n=0, 1,
%p A102221       add(b(n-i)*binomial(n, i)/i!, i=1..n))
%p A102221     end:
%p A102221 a:= n-> b(n)*n!:
%p A102221 seq(a(n), n=0..20);  # _Alois P. Heinz_, May 11 2016
%t A102221 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 A102221 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 A102221 b[n_] := b[n] = If[n==0, 1, Sum[b[n-i] Binomial[n, i]/i!, {i, 1, n}]];
%t A102221 a[n_] := b[n] n!;
%t A102221 a /@ Range[0, 20] (* _Jean-François Alcover_, Dec 03 2020, after _Alois P. Heinz_ *)
%o A102221 (PARI) a(n)=if(n==0,1,sum(k=0,n-1,binomial(n,k)^2*a(k)))
%o A102221 (Sage)
%o A102221 L = taylor(1/(1-x*hypergeometric((1,),(2,2),x)),x,0,14).list()
%o A102221 [factorial(i)^2*c for (i,c) in enumerate(L)] # _Peter Luschny_, Jul 28 2015
%Y A102221 Cf. A000275, A008459, A102220, A102222, A192721.
%Y A102221 Row sums of A192722.
%Y A102221 Column k=2 of A326322.
%K A102221 nonn
%O A102221 0,3
%A A102221 _Paul D. Hanna_, Dec 31 2004
%E A102221 Content moved from A192723 to this sequence by _Alois P. Heinz_, Sep 11 2019