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%I #54 Jun 05 2021 04:38:31
%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 e.g.f. exp(5*x)*(BesselI(0,2*x) - BesselI(1,2*x)).
%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-k*x)))/(2*x), e.g.f. exp((k+2)*x)*(BesselI(0,2*x) - BesselI(1,2*x)) 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 Deléham_, 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
%C From _Tom Copeland_, Nov 08 2014: (Start)
%C This array is one of a family of Catalan arrays related by compositions of the special fractional linear (Möbius) transformations P(x,t) = x/(1-t*x); its inverse Pinv(x,t) = P(x,-t); and an o.g.f. of the Catalan numbers A000108, C(x) = [1-sqrt(1-4*x)]/2; and its inverse Cinv(x) = x*(1-x). (Cf A091867.)
%C O.g.f.: G(x) = C[P[P[P(x,-1),-1]]-1] = C[P(x,-3)] = [1-sqrt(1-4*x/(1-3*x)]/2 = x*A104455(x).
%C Ginv(x) = Pinv[Cinv(x),-3]= P[Cinv(x),3] = x*(1-x)/[1+3*x*(1-x)] = (x-x^2)/[1+3(x-x^2)] = x*A125145(-x). (Cf. A030528.) (End)
%H Vincenzo Librandi, <a href="/A104455/b104455.txt">Table of n, a(n) for n = 0..200</a>
%F G.f.: (1-sqrt((1-7*x)/(1-3*x)))/(2*x).
%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 Deléham_, 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 D-finite with 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
%F G.f. A(x) satisfies: A(x) = 1/(1 - 3*x) + x * A(x)^2. - _Ilya Gutkovskiy_, Jun 30 2020
%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.
%Y Cf. A000108, A091867, A125145, A030528.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Mar 08 2005
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