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A084603
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Coefficients of 1/(1-2x-11x^2)^(1/2); also, a(n) is the central coefficient of (1+x+3x^2)^n.
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6
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1, 1, 7, 19, 91, 331, 1441, 5797, 24739, 103411, 441397, 1876777, 8047909, 34533253, 148803487, 642228139, 2778852979, 12043194163, 52286516821, 227323871929, 989675651041, 4313712072241, 18822940658947, 82215245701519
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 5th binomial transform of 2^n*LegendreP(n,-2) (signed version of A069835). - Paul Barry (pbarry(AT)wit.ie), Sep 03 2004
Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the U (or D) steps come in three colors. - Nour-Eddine Fahssi (fahssin(AT)yahoo.fr), Feb 05 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, k)binomial(n, k)3^k}. - Paul Barry (pbarry(AT)wit.ie), Aug 26 2004
Binomial transform is A084609. Hankel transform is 6^n*3^C(n,2). - Paul Barry (pbarry(AT)wit.ie), Sep 16 2006
a(n)=(1/pi)*Int(x^n/sqrt(-x^2+2x+11),x,1-2sqrt(3),1+2sqrt(3)); - Paul Barry (pbarry(AT)wit.ie), Sep 16 2006
a(n)=sum{k=0..floor(n/2), C(n,2k)*C(2k,k)*3^k}; a(n)=sum{k=0..floor(n/2), C(n,k)*C(n-k,k)*3^k}. - Paul Barry (pbarry(AT)wit.ie), Sep 16 2006
a(n) is also the central coefficient of (3+x+x^2)^n; a(n)=sum_{k=0..n} 2^(n-k) C(n,k) T(k,n), where T(k,n) is the triangle of trinomial coefficients = Coefficient of x^n of (1+x+x^2)^k : A027907 - Nour-Eddine Fahssi (fahssin(AT)yahoo.fr), Feb 05 2008
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PROG
| (PARI) for(n=0, 30, t=polcoeff((1+x+3*x^2)^n, n, x); print1(t", "))
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CROSSREFS
| Cf. A002426, A084600-A084602, A084604-A084615.
Sequence in context: A088988 A109879 A109880 * A088883 A026574 A091149
Adjacent sequences: A084600 A084601 A084602 * A084604 A084605 A084606
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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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