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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).

%I A085362 #66 Aug 27 2025 10:58:23
%S A085362 1,2,8,34,150,678,3116,14494,68032,321590,1528776,7301142,35003238,
%T A085362 168359754,812041860,3926147730,19022666310,92338836390,448968093320,
%U A085362 2186194166950,10659569748370,52037098259090,254308709196660
%N 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).
%C A085362 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
%C A085362 From _G. C. Greubel_, May 22 2020: (Start)
%C A085362 This sequence is part of a class of sequences, for m >= 0, with the properties:
%C A085362 a(n) = 2*m*(4*m+1)^(n-1) - (1/2)*Sum_{k=1..n-1} a(k)*a(n-k).
%C A085362 a(n) = Sum_{k=0..n} m^k * binomial(n-1, n-k) * binomial(2*k, k).
%C A085362 a(n) = (2*m) * Hypergeometric2F1(-n+1, 3/2; 2; -4*m), for n > 0.
%C A085362 n*a(n) = 2*((2*m+1)*n - (m+1))*a(n-1) - (4*m+1)*(n-2)*a(n-2).
%C A085362 (4*m + 1)^n = Sum_{k=0..n} Sum_{j=0..k} a(j)*a(k-j).
%C A085362 G.f.: sqrt( (1 - t)/(1 - (4*m+1)*t) ).
%C A085362 This sequence is the case of m=1. (End)
%H A085362 Vincenzo Librandi, <a href="/A085362/b085362.txt">Table of n, a(n) for n = 0..200</a>
%H A085362 László Németh, <a href="https://arxiv.org/abs/1905.13475">Tetrahedron trinomial coefficient transform</a>, arXiv:1905.13475 [math.CO], 2019.
%F A085362 G.f.: sqrt((1-x)/(1-5*x)).
%F A085362 Sum_{i=0..n} (Sum_{j=0..i} a(j)*a(i-j)) = 5^n.
%F A085362 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
%F A085362 a(n) ~ 2*5^(n-1/2)/sqrt(Pi*n). - _Vaclav Kotesovec_, Oct 14 2012
%F A085362 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
%F A085362 a(n) = Sum_{k=0..n} binomial(2*k,k)*binomial(n-1,n-k). - _Vladimir Kruchinin_, May 30 2016
%F A085362 a(n) = 2*hypergeom([3/2, 1-n], [2], -4) for n>0. - _Peter Luschny_, Jan 30 2017
%F A085362 a(0) = 1; a(n) = (2/n) * Sum_{k=0..n-1} (n+k) * a(k). - _Seiichi Manyama_, Mar 28 2023
%F A085362 From _Seiichi Manyama_, Aug 22 2025: (Start)
%F A085362 a(n) = (1/4)^n * Sum_{k=0..n} 5^k * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
%F A085362 a(n) = Sum_{k=0..n} (-1)^k * 5^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n-1,n-k). (End)
%p A085362 a := n -> `if`(n=0,1,2*hypergeom([3/2, 1-n], [2], -4)):
%p A085362 seq(simplify(a(n)), n=0..22); # _Peter Luschny_, Jan 30 2017
%t A085362 CoefficientList[Series[Sqrt[(1-x)/(1-5x)], {x, 0, 25}], x]
%o A085362 (PARI) my(x='x+O('x^66)); Vec(sqrt((1-x)/(1-5*x))) \\ _Joerg Arndt_, May 10 2013
%o A085362 (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( Sqrt((1-x)/(1-5*x)) )); // _G. C. Greubel_, May 23 2020
%o A085362 (Sage)
%o A085362 def A085362_list(prec):
%o A085362     P.<x> = PowerSeriesRing(ZZ, prec)
%o A085362     return P( sqrt((1-x)/(1-5*x)) ).list()
%o A085362 A085362_list(30) # _G. C. Greubel_, May 23 2020
%Y A085362 Bisection of A026392.
%Y A085362 Cf. A000351 (5^n), A026375, A085363, A085364, A085455.
%Y A085362 Cf. A110170, A162478, A359758, A360132.
%Y A085362 Cf. A063886, A360317, A360318, A360319, A360321, A360322.
%Y A085362 Essentially the same as A026387.
%K A085362 nonn,easy,changed
%O A085362 0,2
%A A085362 Mario Catalani (mario.catalani(AT)unito.it), Jun 25 2003