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A076025
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G.f.: (1-3*x*C)/(1-4*x*C) where C = (1/2-1/2*(1-4*x)^(1/2))/x = g.f. for Catalan numbers A000108.
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12
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1, 1, 5, 26, 137, 726, 3858, 20532, 109361, 582782, 3106550, 16562668, 88314634, 470942044, 2511443268, 13393472616, 71428622337, 380940866574, 2031641406798, 10835261623356, 57787472903502, 308197667445204, 1643712737618748, 8766437439778776, 46754218658948922
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Contribution from Paul Barry (pbarry(AT)wit.ie), Sep 23 2009: (Start)
The Hankel transform of this sequence is 3n+1 or 1,4,7,10,... (A016777).
The Hankel transform of the aeration of this sequence is A016777 doubled, that is, 1,1,4,4,7,7,...
In general, the Hankel transform of [x^n](1-r*xc(x))/(1-(r+1)*xc(x)) is rn+1, and that of the
corresponding aerated sequence is the doubled sequence of rn+1. (End)
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REFERENCES
| L. W. Shapiro and C. J. Wang, Generating identities via 2 X 2 matrices, Congressus Numerantium, 205 (2010), 33-46.
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FORMULA
| a(n+1)=sum{k=0..n, 3^k*binomial(2n+1, n-k)*2*(k+1)/(n+k+2)} - Paul Barry (pbarry(AT)wit.ie), Jun 22 2004
a(n+1)=Sum_{k, 0<=k<=n}A039598(n,k)*3^k . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 21 2007
a(n) = Sum_{k, 0<=k<=n}A039599(n,k)*A015518(k), for n>=1 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 22 2007
Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n>=1, a(n+1)=(-1)^n*charpoly(A,-4). [From Milan R. Janjic (agnus(AT)blic.net), Jul 08 2010]
From Gary W. Adamson, Jul 25 2011: (start) a(n) = upper left term in M^(n-1), M = an infinite square production matrix as follows:
5, 1, 0, 0, 0,...
1, 1, 1, 0, 0,...
1, 1, 1, 1, 0,...
1, 1, 1, 1, 1,...
... (end)
Conjecture: 3*n*a(n) +2*(9-14*n)*a(n-1) +32*(2*n-3)*a(n-2)=0. - R. J. Mathar, Nov 14 2011
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CROSSREFS
| Cf. A000108, A001700, A049027, A076026.
Sequence in context: A052918 A018903 A083331 * A161731 A049607 A035029
Adjacent sequences: A076022 A076023 A076024 * A076026 A076027 A076028
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Oct 29 2002
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