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 A085362 a(0)=1; for n>0, a(n) = 2*5^(n-1) - (1/2)*Sum_{i=1..n-1} a(i)*a(n-i). 23
 1, 2, 8, 34, 150, 678, 3116, 14494, 68032, 321590, 1528776, 7301142, 35003238, 168359754, 812041860, 3926147730, 19022666310, 92338836390, 448968093320, 2186194166950, 10659569748370, 52037098259090, 254308709196660 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Number of bilateral Schroeder paths (i.e. lattice paths consisting of steps U=(1,1), D=(1,-1) and H=(2,0)) from (0,0) to (2n,0) and with no H-steps at even (zero, positive or negative) levels. Example: a(2)=8 because we have UDUD, UUDD, UHD, UDDU and their reflections in the x-axis. First differences of A026375. - Emeric Deutsch, Jan 28 2004 From G. C. Greubel, May 22 2020: (Start) This sequence is part of a class of sequences, for m >= 0, with the properties: a(n) = 2*m*(4*m+1)^(n-1) - (1/2)*Sum_{k=1..n-1} a(k)*a(n-k). a(n) = Sum_{k=0..n} m^k * binomial(n-1, n-k) * binomial(2*k, k). a(n) = (2*m) * Hypergeometric2F1(-n+1, 3/2; 2; -4*m), for n > 0. n*a(n) = 2*((2*m+1)*n - (m+1))*a(n-1) - (4*m+1)*(n-2)*a(n-2). (4*m + 1)^n = Sum_{k=0..n} Sum_{j=0..k} a(j)*a(k-j). G.f.: sqrt( (1 - t)/(1 - (4*m+1)*t) ). This sequence is the case of m=1. (End) LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 László Németh, Tetrahedron trinomial coefficient transform, arXiv:1905.13475 [math.CO], 2019. FORMULA G.f.: sqrt((1-x)/(1-5*x)). Sum_{i=0..n} (Sum_{j=0..i} a(j)*a(i-j)) = 5^n. D-finite with recurrence: a(n) = (2*(3*n-2)*a(n-1)-5*(n-2)*a(n-2)])n; a(0)=1, a(1)=2. - Emeric Deutsch, Jan 28 2004 a(n) ~ 2*5^(n-1/2)/sqrt(Pi*n). - Vaclav Kotesovec, Oct 14 2012 G.f.: G(0), where G(k)= 1 + 4*x*(4*k+1)/( (4*k+2)*(1-x) - 2*x*(1-x)* (2*k+1)*(4*k+3)/(x*(4*k+3) + (1-x)*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 22 2013 a(n) = Sum_{k=0..n} binomial(2*k,k)*binomial(n-1,n-k). - Vladimir Kruchinin, May 30 2016 a(n) = 2*hypergeom([3/2, 1-n], [2], -4) for n>0. - Peter Luschny, Jan 30 2017 a(0) = 1; a(n) = (2/n) * Sum_{k=0..n-1} (n+k) * a(k). - Seiichi Manyama, Mar 28 2023 MAPLE a := n -> `if`(n=0, 1, 2*hypergeom([3/2, 1-n], [2], -4)): seq(simplify(a(n)), n=0..22); # Peter Luschny, Jan 30 2017 MATHEMATICA CoefficientList[Series[Sqrt[(1-x)/(1-5x)], {x, 0, 25}], x] PROG (PARI) my(x='x+O('x^66)); Vec(sqrt((1-x)/(1-5*x))) \\ Joerg Arndt, May 10 2013 (Magma) R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( Sqrt((1-x)/(1-5*x)) )); // G. C. Greubel, May 23 2020 (Sage) def A085362_list(prec): P. = PowerSeriesRing(ZZ, prec) return P( sqrt((1-x)/(1-5*x)) ).list() A085362_list(30) # G. C. Greubel, May 23 2020 CROSSREFS Bisection of A026392. Cf. A000351 (5^n), A026375, A026387, A085363, A085364. Cf. A110170, A162478, A359758, A360132. Sequence in context: A245090 A151829 A026387 * A150889 A150890 A150891 Adjacent sequences: A085359 A085360 A085361 * A085363 A085364 A085365 KEYWORD nonn,easy AUTHOR Mario Catalani (mario.catalani(AT)unito.it), Jun 25 2003 STATUS approved

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Last modified November 28 19:28 EST 2023. Contains 367419 sequences. (Running on oeis4.)