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A173992
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Sequence whose Hankel transform is the Somos (4) sequence.
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3
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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
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OFFSET
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0,3
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COMMENTS
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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
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LINKS
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FORMULA
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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
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MAPLE
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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):
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MATHEMATICA
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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 *)
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PROG
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(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
(Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-2*x-Sqrt((1-2x)*(1-2*x-4*x^2+4*x^3)))/(2*x^2*(1-2*x)))); // G. C. Greubel, Sep 25 2018
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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