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A104455
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Expansion of exp(5x)*(BesselI(0,2x)-BesselI(1,2x)).
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9
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1, 4, 17, 77, 371, 1890, 10095, 56040, 320795, 1881524, 11250827, 68330773, 420314629, 2612922694, 16389162537, 103587298965, 659071002195, 4217699773140, 27129590096595, 175303621195647, 1137400502295081, 7406899253418414, 48396105031873197, 317180187174490902, 2084542632685363221
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OFFSET
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0,2
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COMMENTS
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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)}.
Hankel transform of this sequence gives A000012 = [1,1,1,1,1,1,1,...] . - Philippe DELEHAM, Oct 24 2007
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.
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]
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..200
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FORMULA
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G.f.: (1-sqrt((1-7x)/(1-3x)))/(2x).
a(n) = sum{k=0..n, C(n, k)*C(k)*3^(n-k)}.
a(n) = 3^n+Sum_{k=0..n-1} a(k)*a(n-1-k), a(0)=1. [Philippe DELEHAM, Dec 12 2009]
From Gary W. Adamson, Jul 21 2011: (Start)
a(n) = upper left term of M^n, M = an infinite square production matrix as follows:
4, 1, 0, 0,...
1, 4, 1, 0,...
1, 1, 4, 1,...
1, 1, 1, 4,...
... (end)
Recurrence: (n+1)*a(n) = 2*(5*n-1)*a(n-1) - 21*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 17 2012
a(n) ~ 7^(n+3/2)/(8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012
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MATHEMATICA
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CoefficientList[Series[(1-Sqrt[(1-7*x)/(1-3*x)])/(2*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 17 2012 *)
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PROG
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(PARI) x='x+O('x^66); Vec((1-sqrt((1-7*x)/(1-3*x)))/(2*x)) \\ Joerg Arndt, Mar 31 2013
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CROSSREFS
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Cf. A007317, A064613.
Sequence in context: A081922 A124325 A151248 * A123952 A005494 A193782
Adjacent sequences: A104452 A104453 A104454 * A104456 A104457 A104458
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KEYWORD
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easy,nonn,changed
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AUTHOR
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Paul Barry, Mar 08 2005
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STATUS
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approved
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