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A025567
a(n) = T(n,n+1), where T is the array defined in A025564.
2
1, 4, 13, 40, 120, 356, 1050, 3088, 9069, 26620, 78133, 229384, 673699, 1979628, 5820195, 17121312, 50394579, 148413996, 437324919, 1289330520, 3803175474, 11223840012, 33139076292, 97889042384, 289276841475, 855205791076, 2529279459099
OFFSET
1,2
LINKS
Jean-Luc Baril, Richard Genestier, Sergey Kirgizov, Pattern distributions in Dyck paths with a first return decomposition constrained by height, arXiv:1911.03119 [math.CO], 2019.
Luca Ferrari and Emanuele Munarini, Enumeration of edges in some lattices of paths, arXiv preprint arXiv:1203.6792 [math.CO], 2012 and J. Int. Seq. 17 (2014) #14.1.5
FORMULA
G.f.: (x^2-1-sqrt(1+x)*(x^2+2*x-1)/sqrt(1-3*x))/(2*x^3). - Mark van Hoeij, May 01 2013
Conjecture: (n+3)*a(n) +4*(-n-2)*a(n-1) +2*a(n-2) +8*(n-1)*a(n-3) +3*(n-3)*a(n-4)=0. - R. J. Mathar, Apr 03 2015
Conjecture: (n-1)*(n-2)*(n+3)*a(n) -2*n*(n-2)*(n+2)*a(n-1) -3*n*(n-1)^2*a(n-2)=0. - R. J. Mathar, Apr 03 2015
a(n) ~ 2 * 3^(n + 1/2) / sqrt(Pi*n). - Vaclav Kotesovec, May 02 2024
MATHEMATICA
T[_, 0] = 1; T[1, 1] = 2; T[n_, k_] /; 0 <= k <= 2n := T[n, k] = T[n-1, k-2] + T[n-1, k-1] + T[n-1, k]; T[_, _] = 0;
a[n_] := T[n+1, n+3];
Array[a, 27] (* Jean-François Alcover, Oct 30 2018 *)
PROG
(PARI) x='x+O('x^66); Vec((x^2-1-sqrt(1+x)*(x^2+2*x-1)/sqrt(1-3*x))/(2*x^3)) \\ Joerg Arndt, May 01 2013
CROSSREFS
Pairwise sums of A014531.
Sequence in context: A027130 A027121 A238846 * A003462 A076040 A261547
KEYWORD
nonn
STATUS
approved