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A006442
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Expansion of 1/sqrt(1-10*x+x^2).
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1
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1, 5, 37, 305, 2641, 23525, 213445, 1961825, 18205345, 170195525, 1600472677, 15122515985, 143457011569, 1365435096485, 13033485491077, 124715953657025, 1195966908404545, 11490534389896325
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Number of Delannoy paths from (0,0) to (n,n) with steps U(0,1), H(1,0) and D(1,1) where H can choose from two colors. - Paul Barry (pbarry(AT)wit.ie), May 25 2005
Number of lattice paths from (0,0) to (n,n) using steps (0,1), (1,1), and two kinds of steps (1,0). [Joerg Arndt, Jul 01 2011]
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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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LINKS
| T. D. Noe, Table of n, a(n) for n = 0..200
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FORMULA
| Legendre polynomial evaluated at 5. - Michael Somos, Dec 04, 2001
G.f.: 1/sqrt(1-10*x+x^2)
a(n) = central coefficient of (1+5*x+6*x^2)^n. - Paul D. Hanna, Jun 03 2003
a(n) equals the (n+1)-th term of the binomial transform of 1/(1-2x)^(n+1). - Paul D. Hanna, Sep 29 2003
a(n)=sum(k=0, n, 2^k*binomial(n, k)*binomial(n+k, k)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 13 2004
a(n) = Sum_{k=0..n} binomial(n,k)^2*2^k*3^(n-k). [From Paul D. Hanna, Feb 04 2012]
E.g.f.: exp(5*x)*Bessel_I(0, 2*sqrt(6)*x); - Paul Barry (pbarry(AT)wit.ie), May 25 2005
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PROG
| (PARI) a(n)=subst(pollegendre(n), x, 5)
(PARI) /* as lattice paths: same as in A092566 but use */
steps=[[1, 0], [1, 0], [0, 1], [1, 1]]; /* note the double [1, 0] */
/* Joerg Arndt, Jul 01 2011 */
(PARI) {a(n)=sum(k=0, n, binomial(n, k)^2*2^k*3^(n-k))} /* Paul D. Hanna */
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CROSSREFS
| Sequence in context: A091126 A066381 A078253 * A084212 A176818 A199562
Adjacent sequences: A006439 A006440 A006441 * A006443 A006444 A006445
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KEYWORD
| nonn,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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