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A084609
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Coefficients of 1/(1-4x-8x^2)^(1/2); also, a(n) is the central coefficient of (1+2x+3x^2)^n.
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7
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1, 2, 10, 44, 214, 1052, 5284, 26840, 137638, 710828, 3692140, 19266920, 100932220, 530479640, 2795917960, 14771797424, 78210099718, 414862155980, 2204273582236, 11729283976136, 62496686731924, 333400654676168
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| 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 3 colors and H can have 2 colors. - Nour-Eddine Fahssi (fahssin(AT)yahoo.fr), Mar 30 2008
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REFERENCES
| 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
| a(n)=sum{k=0..floor(n/2), Binomial(n, k)Binomial(2(n-k), n)2^k} - Paul Barry (pbarry(AT)wit.ie), Sep 08 2004
a(n)=sum{k=0..floor(n/2), C(n,2k)*C(2k,k)*3^k*2^(n-2k)}; a(n)=sum{k=0..floor(n/2), C(n,k)*C(n-k,k)*3^k*2^(n-2k)}. - Paul Barry (pbarry(AT)wit.ie), Sep 19 2006
E.g.f. : exp(2x) Bessel_I(0,2sqrt(3)x)
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MATHEMATICA
| f[n_] := Sum[Binomial[n, k] Binomial[2 (n - k), n] 2^k, {k, 0, n/2}] (* Or *)
f[n_] := CoefficientList[ Expand[(1 + 2 x + 3 x^2)^n], x][[n + 1]]; Array[f, 22, 0] (* Or *)
CoefficientList[ Series[ 1/Sqrt[1 - 4 x - 8 x^2], {x, 0, 21}], x] (* Or *)
Range[0, 21]! CoefficientList[ Series[ Exp[ 2x] BesselI[0, Sqrt[12] x], {x, 0, 21}], x] (* RGWv *)
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PROG
| (PARI) for(n=0, 30, t=polcoeff((1+2*x+3*x^2)^n, n, x); print1(t", "))
(Maxima) a(n):=coeff(expand((1+2*x+3*x^2)^n), x, n);
makelist(a(n), n, 0, 12);
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CROSSREFS
| Cf. A002426, A084600-A084608, A084610-A084615.
Sequence in context: A099919 A100397 A084059 * A105485 A151313 A144896
Adjacent sequences: A084606 A084607 A084608 * A084610 A084611 A084612
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KEYWORD
| nonn
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Jun 01 2003
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