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 A006442 Expansion of 1/sqrt(1-10*x+x^2). 7
 1, 5, 37, 305, 2641, 23525, 213445, 1961825, 18205345, 170195525, 1600472677, 15122515985, 143457011569, 1365435096485, 13033485491077, 124715953657025, 1195966908404545, 11490534389896325, 110584004488276645, 1065853221648055025 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 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, 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 LINKS T. D. Noe, Table of n, a(n) for n = 0..200 Hacène Belbachir, Abdelghani Mehdaoui, László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5. Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7. T. R. S. Walsh, Number of sensed planar maps with n edges and m vertices 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, Apr 13 2004 a(n) = Sum_{k=0..n} binomial(n,k)^2 * 2^k * 3^(n-k). - Paul D. Hanna, Feb 04 2012 E.g.f.: exp(5*x)*Bessel_I(0, 2*sqrt(6)*x); - Paul Barry, May 25 2005 D-finite with recurrence: n*a(n) -5*(2n-1)*a(n-1)+(n-1)*a(n-2)=0 [Eq (4) in the T. D. Noe article]. R. J. Mathar, Jun 26 2012 a(n) ~ (5+2*sqrt(6))^n/(2*sqrt(Pi*n)*sqrt(5*sqrt(6)-12)). - Vaclav Kotesovec, Oct 05 2012 a(n) = hypergeom([-n, n+1], [1], -2). - Peter Luschny, May 23 2014 a(n) = Sum_{k=0..n} 2^k * C(2*k, k) * C(n+k, n-k). - Paul D. Hanna, Aug 17 2014 a(n) = Sum_{k=0..n} (k+1) * 3^k * (-1)^(n-k) * binomial(n,k) * binomial(n+k+1,n) / (n+k+1). - Vladimir Kruchinin, Nov 23 2014 MAPLE seq(orthopoly[P](n, 5), n = 0 .. 20); # Robert Israel, Aug 18 2014 MATHEMATICA Table[LegendreP[n, 5], {n, 0, 19}] (* Arkadiusz Wesolowski, Aug 13 2012 *) CoefficientList[Series[1 / Sqrt[1 - 10 x + x^2], {x, 0, 20}], x] (* Vincenzo Librandi, Nov 23 2014 *) 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 */ (PARI) {a(n) = sum(k=0, n, 2^k * binomial(2*k, k) * binomial(n+k, n-k) )} for(n=0, 25, print1(a(n), ", ")) \\ Paul D. Hanna, Aug 17 2014 CROSSREFS Cf. A243943 (a(n)^2). Sequence in context: A091126 A066381 A078253 * A208675 A084212 A323219 Adjacent sequences:  A006439 A006440 A006441 * A006443 A006444 A006445 KEYWORD nonn AUTHOR STATUS approved

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Last modified May 10 05:10 EDT 2021. Contains 343748 sequences. (Running on oeis4.)