login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A178685 G.f.: A(x) = x/Series_Reversion(G(x)) where x*A(G(x)) = G(x) is the g.f. of A000699. 2

%I

%S 1,1,3,17,143,1569,20911,324673,5720319,112365569,2430174335,

%T 57327084033,1464259637503,40255921184257,1185312426797823,

%U 37219582236845057,1241715255479353343,43867539372405948417,1636223622007966705663,64261630682976366944257

%N G.f.: A(x) = x/Series_Reversion(G(x)) where x*A(G(x)) = G(x) is the g.f. of A000699.

%C The limit of (a(n+1)/a(n))/n = 2. - _Paul D. Hanna_, Aug 08 2014

%C The g.f. of related sequence A000699, where A000699(n) is the number of irreducible diagrams with 2n nodes, satisfies: G(x) = x + x^2*[d/dx G(x)^2/x].

%H Paul D. Hanna, <a href="/A178685/b178685.txt">Table of n, a(n) for n = 0..300</a>

%F G.f. satisfies:

%F (1) [x^(n+1)] A(x)^(n+2) = 2(n+1)*[x^n] A(x)^(n+2).

%F (2) A(x) = 1 + x*A(x) * (A(x) + x*A'(x)) / (A(x) - x*A'(x)). - _Paul D. Hanna_, Aug 08 2014

%e G.f.: A(x) = 1 + x + 3*x^2 + 17*x^3 + 143*x^4 + 1569*x^5 + 20911*x^6 +...

%e A(x) satisfies: A(G(x)) = G(x)/x where G(x) is the g.f. of A000699:

%e G(x) = x + x^2 + 4*x^3 + 27*x^4 + 248*x^5 + 2830*x^6 + 38232*x^7 +...

%e which in turn satisfies: G(x) = x + x^2*[d/dx G(x)^2/x].

%e ...

%e The table of coefficients in the initial powers of g.f. A(x) begins:

%e A^1: [1, 1, 3, 17, 143, 1569, 20911, 324673, 5720319, ...];

%e A^2: [1, 2, 7, 40, 329, 3526, 46107, 705444, 12289245, ...];

%e A^3: [1, 3, 12, 70, 567, 5949, 76350, 1150920, 19818801, ...];

%e A^4: [1, 4, 18, 108, 867, 8928, 112524, 1671000, 28436085, ...];

%e A^5: [1, 5, 25, 155, 1240, 12566, 155650, 2277050, 38285275, ...];

%e A^6: [1, 6, 33, 212, 1698, 16980, 206902, 2982096, 49529871, ...];

%e A^7: [1, 7, 42, 280, 2254, 22302, 267624, 3801036, 62355195, ...];

%e A^8: [1, 8, 52, 360, 2922, 28680, 339348, 4750872, 76971171, ...];

%e ...

%e In the above table, notice that the main diagonal:

%e [1, 2, 12, 108, 1240, 16980, 267624, 4750872, 93615408, ...]

%e is related to the secondary diagonal in the following way:

%e [2*1, 4*3, 6*18, 8*155, 10*1698, 12*22302, 14*339348, 16*5850963, ...].

%o (PARI) {a(n)=local(A=[1,1]);for(i=2,n,A=concat(A,0);A[#A]=(2*(#A-1)*Vec(Ser(A)^(#A))[#A-1]-Vec(Ser(A)^(#A))[#A])/(#A));A[n+1]}

%o for(n=0,25,print1(a(n),", "))

%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A = 1 + x*A*(A + x*A')/(A - x*A' +x*O(x^n))); polcoeff(A,n)}

%o for(n=0,25,print1(a(n),", "))

%Y Cf. A000699.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 31 2010

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified February 20 22:51 EST 2019. Contains 320362 sequences. (Running on oeis4.)