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A104455
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Expansion of exp(5x)*(BesselI(0,2x)-BesselI(1,2x)).
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8
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| 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 (kolotoko(AT)wanadoo.fr), 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
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FORMULA
| 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. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 12 2009]
Contribution 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)
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CROSSREFS
| 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
| easy,nonn
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Mar 08 2005
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