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A126120
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Catalan numbers (A000108) interpolated with 0's.
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58
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1, 0, 1, 0, 2, 0, 5, 0, 14, 0, 42, 0, 132, 0, 429, 0, 1430, 0, 4862, 0, 16796, 0, 58786, 0, 208012, 0, 742900, 0, 2674440, 0, 9694845, 0, 35357670, 0, 129644790, 0, 477638700, 0, 1767263190, 0, 6564120420, 0, 24466267020, 0, 91482563640, 0, 343059613650, 0
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OFFSET
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0,5
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COMMENTS
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Inverse binomial transform of A001006.
The Hankel transform of this sequence gives A000012 = [1,1,1,1,1,...].
Counts returning walks (excursions) of length n on a 1-d integer lattice with step set {+1,-1} which stay in the chamber x >= 0. - Andrew V. Sutherland, Feb 29 2008
Moment sequence of the trace of a random matrix in G=USp(2)=SU(2). If X=tr(A) is a random variable (A distributed according to the Haar measure on G) then a(n) = E[X^n]. - Andrew V. Sutherland, Feb 29 2008
Number of distinct proper binary trees with n nodes. - Chris R. Sims (chris.r.sims(AT)gmail.com), Jun 30 2010
-a(n-1), with a(-1):=0, n>=0, is the Z-sequence for the Riordan array A049310 (Chebyshev S). For the definition see that triangle. - Wolfdieter Lang, Nov 04 2011
A signed version is generated by evaluating polynomials in A126216 that are essentially the face polynomials of the associahedra. This entry's sequence is related to an inversion relation on p. 34 of Mizera, related to Feynman diagrams. - Tom Copeland, Dec 09 2019
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REFERENCES
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Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Ch. 49, Hemisphere Publishing Corp., 1987.
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LINKS
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Kiran S. Kedlaya and Andrew V. Sutherland, HyperellipticCurves, L-Polynomials, and Random Matrices. In: Arithmetic, Geometry, Cryptography, and Coding Theory: International Conference, November 5-9, 2007, CIRM, Marseilles, France. (Contemporary Mathematics; v.487)
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FORMULA
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G.f.: (1 - sqrt(1 - 4*x^2)) / (2*x^2) = 1/(1-x^2/(1-x^2/(1-x^2/(1-x^2/(1-...(continued fraction). - Philippe Deléham, Nov 24 2009
E.g.f.: I_1(2x)/x Where I_n(x) is the modified Bessel function. - Benjamin Phillabaum, Mar 07 2011
Apart from the first term the e.g.f. is given by x*HyperGeom([1/2],[3/2,2], x^2). - Benjamin Phillabaum, Mar 07 2011
a(n) = Integral_{x=-2..2} x^n*sqrt((2-x)*(2+x)))/(2*Pi). - Peter Luschny, Sep 11 2011
E.g.f.: E(0)/(1-x) where E(k) = 1-x/(1-x/(x-(k+1)*(k+2)/E(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 05 2013
G.f.: 3/2- sqrt(1-4*x^2)/2 = 1/x^2 + R(0)/x^2, where R(k) = 2*k-1 - x^2*(2*k-1)*(2*k+1)/R(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 28 2013 (warning: this is not the g.f. of this sequence, R. J. Mathar, Sep 23 2021)
G.f.: 1/Q(0), where Q(k) = 2*k+1 + x^2*(1-4*(k+1)^2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 09 2014
a(n) = n!*[x^n]hypergeom([],[2],x^2). - Peter Luschny, Jan 31 2015
a(n) = 2^n*hypergeom([3/2,-n],[3],2). - Peter Luschny, Feb 03 2015
a(n) = ((-1)^n+1)*2^(2*floor(n/2)-1)*Gamma(floor(n/2)+1/2)/(sqrt(Pi)* Gamma(floor(n/2)+2)). - Ilya Gutkovskiy, Jul 23 2016
D-finite with recurrence (n+2)*a(n) +4*(-n+1)*a(n-2)=0. - R. J. Mathar, Mar 21 2021
a(n) = 2^n * Sum_{k = 0..n} (-2)^(-k)*binomial(n, k)*Catalan(k+1).
G.f.: 1/(1 + 2*x) * c(x/(1 + 2*x))^2 = 1/(1 - 2*x) * c(-x/(1 - 2*x))^2 = c(x^2), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)
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EXAMPLE
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G.f. = 1 + x^2 + 2*x^4 + 5*x^6 + 14*x^8 + 42*x^10 + 132*x^12 + 429*x^14 + ...
The a(0) = 1 through a(8) = 14 ordered binary rooted trees with n + 1 nodes (ranked by A358375):
o . (oo) . ((oo)o) . (((oo)o)o) . ((((oo)o)o)o)
(o(oo)) ((o(oo))o) (((o(oo))o)o)
((oo)(oo)) (((oo)(oo))o)
(o((oo)o)) (((oo)o)(oo))
(o(o(oo))) ((o((oo)o))o)
((o(o(oo)))o)
((o(oo))(oo))
((oo)((oo)o))
((oo)(o(oo)))
(o(((oo)o)o))
(o((o(oo))o))
(o((oo)(oo)))
(o(o((oo)o)))
(o(o(o(oo))))
(End)
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MAPLE
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with(combstruct): grammar := { BB = Sequence(Prod(a, BB, b)), a = Atom, b = Atom }: seq(count([BB, grammar], size=n), n=0..47); # Zerinvary Lajos, Apr 25 2007
BB := {E=Prod(Z, Z), S=Union(Epsilon, Prod(S, S, E))}: ZL:=[S, BB, unlabeled]: seq(count(ZL, size=n), n=0..45); # Zerinvary Lajos, Apr 22 2007
BB := [T, {T=Prod(Z, Z, Z, F, F), F=Sequence(B), B=Prod(F, Z, Z)}, unlabeled]: seq(count(BB, size=n+1), n=0..45); # valid for n> 0. # Zerinvary Lajos, Apr 22 2007
seq(n!*coeff(series(hypergeom([], [2], x^2), x, n+2), x, n), n=0..45); # Peter Luschny, Jan 31 2015
# Using function CompInv from A357588.
CompInv(48, n -> ifelse(irem(n, 2) = 0, 0, (-1)^iquo(n-1, 2))); # Peter Luschny, Oct 07 2022
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MATHEMATICA
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a[n_?EvenQ] := CatalanNumber[n/2]; a[n_] = 0; Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Sep 10 2012 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ BesselI[ 1, 2 x] / x, {x, 0, n}]]; (* Michael Somos, Mar 19 2014 *)
bot[n_]:=If[n==1, {{}}, Join@@Table[Tuples[bot/@c], {c, Table[{k, n-k-1}, {k, n-1}]}]];
Table[Length[bot[n]], {n, 10}] (* Gus Wiseman, Nov 14 2022 *)
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PROG
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(Sage)
D = [0]*(n+2); D[1] = 1
b = True; h = 2; R = []
for i in range(2*n-1) :
if b :
for k in range(h, 0, -1) : D[k] -= D[k-1]
h += 1; R.append(abs(D[1]))
else :
for k in range(1, h, 1) : D[k] += D[k+1]
b = not b
return R
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CROSSREFS
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These trees (ordered binary rooted) are ranked by A358375.
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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