A173992 Sequence whose Hankel transform is the Somos (4) sequence.

1, 1, 3, 6, 15, 34, 83, 198, 488, 1202, 3015, 7608, 19432, 49994, 129779, 339176, 892600, 2362634, 6288156, 16816232, 45170466, 121812152, 329679487, 895171236, 2437885058, 6657311202, 18224979884, 50006899724, 137502724754

Offset: 0

Author

Paul Barry, Mar 04 2010

Keywords

• nonn
• easy

Comments

Hankel transform is A006720(n+3). In general, the sequence with g.f. ((1-x)/(1-(r+1)*x))*c(x^2*(1-x)/(1-(r+1)*x)) will have a Somos (1,r) Hankel transform.

a(n) is the number of rooted plane 2-trees with integer compositions labeling the leaves (empty labels are allowed), with total size n. The total size is the number of edges in the tree plus the sum of the sizes of the integer compositions labeling the leaves. Example; a(2)=3 because there are two trees that consist of the root and no descendants, hence the root is itself a leaf and it can be labeled by either 2=2 or by 2=1+1, and then there is the tree that consists of the root with two descendants and no labels on the two leaves. - Ricardo Gómez Aíza, Feb 26 2024

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2000
• Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024.

Crossrefs

Cf. A000108, A006720, A056010.

Programs

Magma

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-2*x-Sqrt((1-2*x)*(1-2*x-4*x^2+4*x^3)))/(2*x^2*(1-2*x)))); // G. C. Greubel, Sep 25 2018

Maple

with(LREtools): with(FormalPowerSeries): # requires Maple 2022
ogf:=(1-2*x-sqrt((1-2*x)*(1-2*x-4*x^2+4*x^3)))/(2*x^2*(1-2*x)):
req1:= FindRE(ogf,x,u(n)); inits:= {seq(u(i-1)=[1, 1, 3, 6, 15, 34][i],i=1..6)}:
req2:= subs(n=n-4, MinimalRecurrence(req1,u(n),inits)[1]); # Mathar's recurrence
a:= gfun:-rectoproc({req2} union inits, u(n), remember):
seq(a(n),n=0..28); # Georg Fischer, Nov 03 2022

Mathematica

A173992[n_] := Sum[CatalanNumber[k] Sum[Binomial[k + 1, i] Binomial[n - k - i, n - 2 k - i] (-1)^i Floor[2^(n - 2 k - i)], {i, 0, k + 1}], {k, 0, Floor[n/2]}] (* Eric Rowland, May 15 2017 *)
CoefficientList[Series[(1-2*x -Sqrt[(1-2*x)*(1-2*x-4*x^2+4*x^3)])/(2*x^2* (1-2*x)), {x, 0, 50}], x] (* G. C. Greubel, Sep 25 2018 *)

PARI

a(n) = sum(k=0, n\2, binomial(2*k,k)/(k+1)*sum(i=0, k+1, binomial(k+1,i)*binomial(n-k-i,n-2*k-i)*(-1)^i*2^(n-2*k-i))); \\ Michel Marcus, May 15 2017

PARI

x='x+O('x^50); Vec((1-2*x-((1-2*x)*(1-2*x-4*x^2+4*x^3))^(1/2))/(2*x^2*(1-2*x))) \\ Altug Alkan, Sep 25 2018

Formula

G.f.: ((1-x)/(1-2*x)) * c(x^2*(1-x)/(1-2*x)) = (1-2*x-sqrt((1-2x)*(1-2*x-4*x^2+4*x^3)))/(2*x^2*(1-2*x)), c(x) the g.f. of A000108;

a(n) = Sum_{k=0..floor(n/2), A000108(k)*Sum_{i=0..k+1, C(k+1,i)*C(n-k-i,n-2k-i)*(-1)^i*2^(n-2k-i)}}.

D-finite with recurrence: (n+2)*a(n) -4*(n+1)*a(n-1) +4*a(n-2) +2*(6n-11)*a(n-3) +8*(3-n)*a(n-4)=0. - R. J. Mathar, Nov 17 2011

a(n) ~ sqrt(2-5*c+4*c^2)/(2*c*(1-2*c)*sqrt(Pi*n^3))*(1/c)^n where c=(4+(1+i*sqrt(3))*(1+3*i*sqrt(111))^(1/3)+80/((sqrt(3)+i)^2*(1+3*i*sqrt(111))^(1/3)))/12. - Ricardo Gómez Aíza, Feb 26 2024