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Search: id:A084770
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| A084770 |
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Coefficients of 1/(1-4x-16x^2)^(1/2); also, a(n) is the central coefficient of (1+2x+5x^2)^n. |
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+0
7
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| 1, 2, 14, 68, 406, 2332, 13964, 83848, 509926, 3118892, 19194724, 118654648, 736365436, 4584612632, 28623792344, 179142212368, 1123532958086, 7059622447052, 44431918660724, 280059644507608, 1767597777222676
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), U can have 5 colors and H can have 2 colors. - Nour-Eddine Fahssi (fahssin(AT)yahoo.fr), Mar 30 2008
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REFERENCES
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Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
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.
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FORMULA
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E.g.f.: exp(2*x)*BesselI(0, 2*sqrt(5)*x). More generally, e.g.f.: exp(b*x)*BesselI(0, 2*sqrt(c)*x) yields central coefficients of (1+b*x+c*x^2)^n. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Mar 21 2004
a(n)=sum{k=0..floor(n/2), binomial(n, k)binomial(2(n-k), n)4^k} - Paul Barry (pbarry(AT)wit.ie), Sep 08 2004
Define Q(n, x)=sum{k=0..floor(n/2), binomial(n, k)binomial(2(n-k), n)x^(n-2k)}. A084770(n) is 2^n*Q(n, 1/2). - Paul Barry (pbarry(AT)wit.ie), Sep 08 2004
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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) for(n=0, 30, t=polcoeff((1+2*x+5*x^2)^n, n, x); print1(t", "))
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CROSSREFS
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Sequence in context: A095376 A109869 A084132 this_sequence A086243 A072888 A094583
Adjacent sequences: A084767 A084768 A084769 this_sequence A084771 A084772 A084773
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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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