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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+5*x+4*x^2)^n.
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5
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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; 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), 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
Number of lattice paths from (0,0) to (n,n) using steps (1,0), (0,1), and three kinds of steps (1,1). [Joerg Arndt, Jul 01 2011]
Sums of squares of coefficients of (1+2*x)^n. [Joerg Arndt, Jul 06 2011]
The Hankel transform of this sequence gives A103488 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 02 2007
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REFERENCES
| 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
| G.f.: 1/sqrt(1-10*x+9*x^2).
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.rs), Aug 20 2003
a(n) = sum(k=0..n, sum(j=0..n-k, C(n,j)*C(n-j,k)*C(2*n-2*j,n-j) ) ). - Paul Barry, May 19 2006
a(n) = sum(k=0..n, 4^k*(C(n,k))^2 ) [From heruneedollar (heruneedollar(AT)gmail.com), Mar 20 2010]
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EXAMPLE
| 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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PROG
| (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))
(PARI) /* as lattice paths: same as in A092566 but use */
steps=[[1, 0], [0, 1], [1, 1], [1, 1], [1, 1]]; /* note the triple [1, 1] */
/* Joerg Arndt, Jul 01 2011 */
(PARI) a(n)={local(v=Vec((1+2*x)^n)); sum(k=1, #v, v[k]^2); } /* Joerg Arndt, Jul 06 2011 */
(PARI) a(n)={local(v=Vec((1+2*I*x)^n)); sum(k=1, #v, real(v[k])^2+imag(v[k])^2); } /* Joerg Arndt, Jul 06 2011 */
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CROSSREFS
| Sequence in context: A093427 A142989 A084131 * A153398 A034015 A056159
Adjacent sequences: A084768 A084769 A084770 * A084772 A084773 A084774
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KEYWORD
| nonn
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Jun 10 2003
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