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 A084771 Coefficients of expansion of 1/sqrt(1-10*x+9*x^2); also, a(n) is the central coefficient of (1+5*x+4*x^2)^n. 11
 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; text; internal format)
 OFFSET 0,2 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. - N-E. Fahssi, 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 Deléham, Dec 02 2007 Partial sums of A085363. - J. M. Bergot, Jun 12 2013 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 T. Amdeberhan, In search of multiple expressions for a sequence Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8. Curtis Greene, Posets of shuffles, Journal of Combinatorial Theory, Series A 47.2 (1988): 191-206. See Eq. (30). Christopher Huffaker, Nathan McCue, Cameron N. Miller, Kayla S. Miller, The M&M Game: From Morsels to Modern Mathematics, arXiv:1508.06542 [math.HO], (24-August-2015) Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7. 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. 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, 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] Asymptotic: a(n) ~ 3^(2*n+1)/(2*sqrt(2*Pi*n)). [Vaclav Kotesovec, Sep 11 2012] Conjecture: n*a(n) +5*(-2*n+1)*a(n-1) +9*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 26 2012 a(n) = hypergeom([-n,1/2], [1], -8). - Peter Luschny, Apr 26 2016 From Michael Somos, Jun 01 2017: (Start) a(n) = -3 * 9^n * a(-1-n) for all n in Z. 0 = a(n)*(+81*a(n+1) -135*a(n+2) +18*a(n+3)) +a(n+1)*(-45*a(n+1) +100*a(n+2) -15*a(n+3)) +a(n+2)*(-5*a(n+2) +a(n+3)) for all n in Z. (End) 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. MAPLE seq(simplify(hypergeom([-n, 1/2], [1], -8)), n=0..19); # Peter Luschny, Apr 26 2016 MATHEMATICA Table[n! SeriesCoefficient[E^(5 x) BesselI[0, 4 x], {x, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, May 10 2013 *) Table[Hypergeometric2F1[-n, -n, 1, 4], {n, 0, 19}] (* Vladimir Reshetnikov, Nov 29 2013 *) CoefficientList[Series[1/Sqrt[1-10x+9x^2], {x, 0, 30}], x] (* Harvey P. Dale, Mar 08 2016 *) PROG (PARI) {a(n) = if( n<0, -3 * 9^n * a(-1-n), sum(k=0, n, binomial(n, k)^2 * 4^k))}; /* _Michael Somos, Oct 08 2003 */ (PARI) {a(n) = if( n<0, -3 * 9^n * a(-1-n), polcoeff((1 + 5*x + 4*x^2)^n, n))}; /* _Michael Somos, Oct 08 2003 */ (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 */ CROSSREFS Cf. A246923 (a(n)^2). Sequence in context: A093427 A142989 A084131 * A153398 A242522 A034015 Adjacent sequences:  A084768 A084769 A084770 * A084772 A084773 A084774 KEYWORD nonn AUTHOR Paul D. Hanna, Jun 10 2003 STATUS approved

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