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 A111160 G.f.: C - Z; where C is the g.f. for the Catalan numbers (A000108) and Z is the g.f. for A055113 with offset 0. 5
 0, 1, 1, 4, 9, 31, 91, 309, 1009, 3481, 11956, 42065, 148655, 532039, 1915369, 6950452, 25357233, 93034813, 342888250, 1269246437, 4715945712, 17583623988, 65766726906, 246694006971, 927801717255, 3497918129001, 13217196871126, 50046561077947 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Expressible in terms of ballot numbers. Number of positive walks with n steps {-2,-1,1,2} starting at the origin, ending at altitude 2, and staying strictly above the x-axis. - David Nguyen, Dec 16 2016 LINKS T. D. Noe, Table of n, a(n) for n=0..200 C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, Explicit formulas for enumeration of lattice paths: basketball and the kernel method, arXiv preprint arXiv:1609.06473 [math.CO], 2016. Jean-Luc Baril and José L. Ramírez, Knight's paths towards Catalan numbers, Univ. Bourgogne Franche-Comté (2022). D. G. Rogers, Comments on A111160, A055113 and A006013 FORMULA Let C := (1 - sqrt(1 - 4*x)) / (2*x), Z := (- 1/4 - (1/4)*(1 - 4*x)^(1/2) + (1/4)*(2 + 2*(1 - 4*x)^(1/2) + 12*x)^(1/2))/x; g.f. is W := C - Z. G.f.: -((-3 + sqrt(1 - 4x) + sqrt(2)*sqrt(1 + sqrt(1 - 4x) + 6x))/(4x)). a(n) = sum(j=0..n+1, binomial(n+2*j-1,j)*(-1)^(n+j+1)*binomial(2*n+1,j+n))/(2*n+1). [Vladimir Kruchinin, Feb 15 2013] a(n) ~ (1+1/sqrt(5))*2^(2*n-1)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 13 2013 Recurrence: 2*n*(n+1)*(2*n+1)*(5*n-8)*a(n) = n*(115*n^3 - 344*n^2 + 299*n - 82)*a(n-1) + 4*(2*n-3)*(5*n^3 + 27*n^2 - 74*n + 30)*a(n-2) - 36*(n-2)*(2*n-5)*(2*n-3)*(5*n-3)*a(n-3). - Vaclav Kotesovec, Aug 13 2013 a(n) = Sum_{j=0..(n+1)/2}(binomial(n-j,n-2*j+1)*binomial(2*n+1,j))/(2*n+1). - Vladimir Kruchinin, Oct 05 2015 a(n) = (-1)^(n+1)*C(2*n+1,n)*hypergeom([-n-1,n/2+1/2,n/2],[n,n+1],4)/(2*n+1) for n>0. - Peter Luschny, Oct 06 2015 MAPLE a := n -> (-1)^(n+1)*binomial(2*n+1, n)*hypergeom([-n-1, n/2+1/2, n/2], [n, n+1], 4)/ (2*n+1); [0, op([seq(round(evalf(a(n), 32)), n=1..27)])]; # Peter Luschny, Oct 06 2015 MATHEMATICA CoefficientList[ Series[ -((-3 + Sqrt[1 - 4*x] + Sqrt[2]*Sqrt[1 + Sqrt[1 - 4x] + 6x])/(4x)), {x, 0, 10}], x] (* Robert G. Wilson v *) PROG (PARI) a(n) = if(n==0, 0, sum(k=0, (n+1)/2, binomial(n-k, n-2*k+1)*binomial(2*n+1, k))/(2*n+1)); \\ Altug Alkan, Oct 05 2015 (Magma) I:=[1, 1, 4]; [0] cat [n le 3 select I[n] else (n*(115*n^3 - 344*n^2 + 299*n - 82)*Self(n-1) + 4*(2*n-3)*(5*n^3 + 27*n^2 - 74*n + 30)*Self(n-2) - 36*(n-2)*(2*n-5)*(2*n-3)*(5*n-3)*Self(n-3))/(2*n*(n+1)*(2*n+1)*(5*n-8)): n in [1..30]]; // Vincenzo Librandi, Oct 06 2015 CROSSREFS Cf. A000108, A055113, A006013, A187430, A276901, A306668. Sequence in context: A206960 A145543 A141043 * A192876 A201689 A356650 Adjacent sequences: A111157 A111158 A111159 * A111161 A111162 A111163 KEYWORD nonn AUTHOR N. J. A. Sloane, Oct 22 2005 STATUS approved

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Last modified May 29 22:39 EDT 2023. Contains 363044 sequences. (Running on oeis4.)