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%I
%S 1,4,17,77,371,1890,10095,56040,320795,1881524,11250827,68330773,
%T 420314629,2612922694,16389162537,103587298965,659071002195,
%U 4217699773140,27129590096595,175303621195647,1137400502295081,7406899253418414,48396105031873197,317180187174490902,2084542632685363221
%N Expansion of exp(5x)*(BesselI(0,2x)-BesselI(1,2x)).
%C Third binomial transform of A000108. In general, the k-th binomial transform of A000108 will have g.f. (1-sqrt((1-(k+4)x)/(1-kx)))/(2x), e.g.f. exp((k+2)x)(BesselI(0,2x)-BesselI(1,2x)) and a(n)=sum{i=0..n, C(n,i) C(i) k^(n-i)}.
%C Hankel transform of this sequence gives A000012 = [1,1,1,1,1,1,1,...] . - _Philippe DELEHAM_, Oct 24 2007
%C In general, the k-th binomial transform of A000108 can be generated from M^n, M = the production matrix of the form shown in the formula section, with a diagonal (k+1, k+1, k+1,...). - _Gary W. Adamson_, Jul 21 2011.
%C a(n) is the number of Schroeder paths of semilength n in which the H=(2,0) steps come in 3 colors and having no (2,0)-steps at levels 1,3,5,... [_José Luis Ramírez Ramírez_, Mar 30 2013]
%H Vincenzo Librandi, <a href="/A104455/b104455.txt">Table of n, a(n) for n = 0..200</a>
%F G.f.: (1-sqrt((1-7x)/(1-3x)))/(2x).
%F a(n) = sum{k=0..n, C(n, k)*C(k)*3^(n-k)}.
%F a(n) = 3^n+Sum_{k=0..n-1} a(k)*a(n-1-k), a(0)=1. [_Philippe DELEHAM_, Dec 12 2009]
%F From _Gary W. Adamson_, Jul 21 2011: (Start)
%F a(n) = upper left term of M^n, M = an infinite square production matrix as follows:
%F 4, 1, 0, 0,...
%F 1, 4, 1, 0,...
%F 1, 1, 4, 1,...
%F 1, 1, 1, 4,...
%F ... (end)
%F Recurrence: (n+1)*a(n) = 2*(5*n-1)*a(n-1) - 21*(n-1)*a(n-2). - _Vaclav Kotesovec_, Oct 17 2012
%F a(n) ~ 7^(n+3/2)/(8*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 17 2012
%t CoefficientList[Series[(1-Sqrt[(1-7*x)/(1-3*x)])/(2*x), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 17 2012 *)
%o (PARI) x='x+O('x^66); Vec((1-sqrt((1-7*x)/(1-3*x)))/(2*x)) \\ _Joerg Arndt_, Mar 31 2013
%Y Cf. A007317, A064613.
%K easy,nonn,changed
%O 0,2
%A _Paul Barry_, Mar 08 2005
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