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Search: id:A084771
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| A084771 |
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Coefficients of 1/sqrt(1-10*x+9*x^2); also, a(n) is the central coefficient of (1+5x+4x^2)^n. |
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+0
6
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| 1, 5, 33, 245, 1921, 15525, 127905, 1067925, 9004545, 76499525, 653808673, 5614995765, 48416454529, 418895174885, 3634723102113, 31616937184725, 275621102802945, 2407331941640325, 21061836725455905, 184550106298084725
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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The Hankel transform of this sequence gives A103488 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 02 2007
Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the U steps come in four colors and the H steps come in five colors. - Nour-Eddine Fahssi (fahssin(AT)yahoo.fr), Mar 30 2008
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REFERENCES
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Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
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FORMULA
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Binomial transform of A059304. G.f.: Sum_{k>=0} binomial(2*k, k)*(2*x)^k/(1-x)^(k+1). E.g.f.: exp(5*x)*BesselI(0, 4*x). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 20 2003
G.f.: 1/sqrt(1-10x+9x^2).
a(n)=sum{k=0..n, sum{j=0..n-k, C(n,j)C(n-j,k)C(2n-2j,n-j)}}; - Paul Barry (pbarry(AT)wit.ie), May 19 2006
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EXAMPLE
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G.f.: 1/sqrt(1-2*b*x+(b^2-4*c)*x^2) yields central coefficients of (1+b*x+c*x^2)^n.
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PROGRAM
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(PARI) a(n)=sum(k=0, n, binomial(n, k)^2*4^k)
(PARI) a(n)=if(n<0, 0, polcoeff((1+5*x+4*x^2)^n, n))
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CROSSREFS
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Adjacent sequences: A084768 A084769 A084770 this_sequence A084772 A084773 A084774
Sequence in context: A093427 A142989 A084131 this_sequence A034015 A056159 A061253
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Jun 10 2003
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