A057552 a(n) = Sum_{k=0..n} C(2k+2,k).
1, 5, 20, 76, 286, 1078, 4081, 15521, 59279, 227239, 873885, 3370029, 13027729, 50469889, 195892564, 761615284, 2965576714, 11563073314, 45141073924, 176423482324, 690215089744, 2702831489824, 10593202603774, 41550902139550, 163099562175850, 640650742051802
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Jean-Luc Baril, Pamela E. Harris, Kimberly J. Harry, Matt McClinton, and José L. Ramírez, Enumerating runs, valleys, and peaks in Catalan words, arXiv:2404.05672 [math.CO], 2024. See p. 21.
- Jean-Luc Baril, Sergey Kirgizov, and Mehdi Naima, A lattice on Dyck paths close to the Tamari lattice, arXiv:2309.00426 [math.CO], 2023.
- A. V. Kitaev and A. Vartanian, Algebroid Solutions of the Degenerate Third Painlevé Equation for Vanishing Formal Monodromy Parameter, arXiv:2304.05671 [math.CA], 2023. See p. 59.
Programs
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Maple
a:= n->add(binomial(2*j+2, j), j=0..n): seq(a(n), n=0..24); # Zerinvary Lajos, Oct 25 2006
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Mathematica
Table[Sum[Binomial[2k+2,k],{k,0,n}],{n,0,20}] (* or *) Table[SeriesCoefficient[1/2*(2*x+(1-4*x)^(1/2)-1)/(1-4*x)^(1/2)/x^2/(-1+x),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 11 2012 *) Table[(CatalanNumber[n + 1] (4 n + 6 - (n + 2) Hypergeometric2F1[1, -n-1, -n-1/2, 1/4]) - 1)/2, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 03 2016 *) -
PARI
a(n) = sum(k=0, n, binomial(2*k+2, k)); \\ Michel Marcus, Oct 04 2016
Formula
G.f.: 1/2*(2*x+(1-4*x)^(1/2)-1)/(1-4*x)^(1/2)/x^2/(-1+x). - Vladeta Jovovic, Sep 10 2003
D-finite with recurrence: n*(n+2)*a(n) = (5*n^2+8*n+2)*a(n-1) - 2*(n+1)*(2*n+1)*a(n-2). - Vaclav Kotesovec, Oct 11 2012
a(n) ~ 2^(2*n+4)/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 11 2012
a(n) = Sum_{k=1..n+1} k*A000108(k) = Sum_{k=1..n+1} A001791(k) = (A000108(n+1) * (4*n + 6 - (n+2)*hypergeom([1,-n-1], [-n-1/2], 1/4)) - 1)/2.
a(n) = Sum_{k=1..n+1} Sum_{i=1..k} C(i+k-1,k). - Wesley Ivan Hurt, Sep 19 2017
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(2*n+3-k, n-2*k). - Michael Weselcouch, Jun 17 2025
a(n) = binomial(3+2*n, n)*hypergeom([1, (1-n)/2, -n/2], [-3-2*n, 4+n], 4). - Stefano Spezia, Jun 18 2025