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%I #62 Mar 03 2024 14:30:51
%S 1,6,30,144,685,3258,15533,74280,356283,1713690,8263596,39938616,
%T 193419915,938430990,4560542550,22195961280,108171753355,527816696850,
%U 2578310320610,12607504827600,61706212037295,302275142049870,1481908332595625,7270432009471224
%N a(n) is the number of integer strings s(0),...,s(n) counted by array T in A026374 that have s(n)=2; also a(n) = T(2n,n-1).
%C Number of Schroeder paths (i.e., consisting of steps U=(1,1), D=(1,-1) and H=(2,0) and never going below the x-axis) from (0,0) to (2n+2,0), with exactly one peak at an even level. E.g., a(2)=6 because we have UUDDH, HUUDD, UDUUDD, UUDDUD, UUDHD and UHUDD. - _Emeric Deutsch_, Dec 28 2003
%C Number of left steps in all skew Dyck paths of semilength n+1. A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps. Example: a(2)=6 because in the 10 (=A002212(3)) skew Dyck paths of semilength 3 ( namely UDUUDL, UUUDLD, UUDUDL, UUUDDL, UUUDLL and five Dyck paths that have no left steps) we have altogether 6 left steps. - _Emeric Deutsch_, Aug 05 2007
%C From _Gary W. Adamson_, May 17 2009: (Start)
%C Equals A026378 (1, 4, 17, 75, ...) convolved with A007317 (1, 2, 5, 15, 51, ...).
%C Equals A081671 (1, 3, 11, 45, ...) convolved with A002212 (1, 3, 10, 36, 137, ...).
%C (End)
%H Vincenzo Librandi, <a href="/A026376/b026376.txt">Table of n, a(n) for n = 1..200</a>
%H Emeric Deutsch, Emanuele Munarini, and Simone Rinaldi, <a href="http://dx.doi.org/10.1016/j.jspi.2010.01.015">Skew Dyck paths</a>, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.
%H Ricardo Gómez Aíza, <a href="https://arxiv.org/abs/2402.16111">Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis</a>, arXiv:2402.16111 [math.CO], 2024. See p. 18.
%H Toufik Mansour and José Luis Ramírez, <a href="https://doi.org/10.33039/ami.2022.01.001">Enumeration of Fuss-skew paths</a>, Ann. Math. Inform. (2022) Vol. 55, 125-136. See p. 129.
%F E.g.f.: exp(3x)*I_1(2x), where I_1 is Bessel function. - _Michael Somos_, Sep 09 2002
%F G.f.: (1 - 3*z - t)/(2*z*t) where t = sqrt(1-6*z+5*z^2). - _Emeric Deutsch_, May 25 2003
%F a(n) = [t^(n+1)](1+3t+t^2)^n. a := n -> Sum_{j=ceiling((n+1)/2)..n} 3^(2j-n-1)*binomial(n, j)*binomial(j, n+1-j). - _Emeric Deutsch_, Jan 30 2004
%F a(n) = Sum_{k=0..n} binomial(n, k)*binomial(2k, k+1). - _Paul Barry_, Sep 20 2004
%F a(n) = n*A002212(n). - _Emeric Deutsch_, Aug 05 2007
%F D-finite with recurrence (n+1)*a(n) - 9*n*a(n-1) + (23*n-27)*a(n-2) + 15*(-n+2)*a(n-3) = 0. - _R. J. Mathar_, Dec 02 2012
%F a(n) ~ 5^(n+1/2) / (2*sqrt(Pi*n)). - _Vaclav Kotesovec_, Feb 13 2014
%F a(n) = n*hypergeometric([1, 3/2, 1-n],[1, 3],-4). - _Peter Luschny_, Sep 16 2014
%F a(n) = GegenbauerC(n-1, -n, -3/2). - _Peter Luschny_, May 09 2016
%p a := n -> simplify(GegenbauerC(n-1, -n, -3/2)):
%p seq(a(n), n=1..24); # _Peter Luschny_, May 09 2016
%t Rest[CoefficientList[Series[(1-3*x-Sqrt[1-6*x+5*x^2])/(2*x*Sqrt[1-6*x+5*x^2]), {x, 0, 20}], x]] (* _Vaclav Kotesovec_, Feb 13 2014 *)
%o (PARI) a(n)=if(n<0,0,polcoeff((1+3*x+x^2)^n,n-1))
%o (Sage)
%o A026376 = lambda n : n*hypergeometric([1, 3/2, 1-n], [1, 3], -4)
%o [round(A026376(n).n(100)) for n in (1..24)] # _Peter Luschny_, Sep 16 2014
%o (Sage) # Recurrence:
%o def A026376():
%o x, y, n = 1, 1, 1
%o while True:
%o x, y = y, ((6*n + 3)*y - (5*n - 5)*x) / (n + 2)
%o yield n*x
%o n += 1
%o a = A026376()
%o [next(a) for i in (1..24)] # _Peter Luschny_, Sep 16 2014
%Y Cf. A006318, A002212, A081671, A007317, A026378.
%K nonn
%O 1,2
%A _Clark Kimberling_
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