A216359 G.f. satisfies: A(x) = (1 + x*A(x)^2) * (1 + x/A(x)).

1, 2, 3, 13, 32, 147, 445, 2067, 7019, 32590, 119209, 551551, 2125429, 9795863, 39221165, 180177403, 742575760, 3403131833, 14342166121, 65626369612, 281459129188, 1286834885967, 5596229192396, 25580269950635, 112492633046446, 514323765191879, 2282371511598955

Offset: 0

Author

Paul D. Hanna, Sep 04 2012

Keywords

• nonn
• easy

Comments

The radius of convergence of g.f. A(x) is r = 0.209619875959405379599013693... with A(r) = 2.36951367232829409921688546894691317519410... where y=A(r) satisfies y^7 - 2*y^6 - 4*y^4 + 4*y^3 + 4*y - 2 = 0.

Examples

			G.f.: A(x) = 1 + 2*x + 3*x^2 + 13*x^3 + 32*x^4 + 147*x^5 + 445*x^6 +...
Related expansions.
A(x)^2 = 1 + 4*x + 10*x^2 + 38*x^3 + 125*x^4 + 500*x^5 + 1839*x^6 +...
A(x)^3 = 1 + 6*x + 21*x^2 + 83*x^3 + 315*x^4 + 1269*x^5 + 5061*x^6 +...
where A(x) = (1-x^2)*A(x)^2 - x*A(x)^3 - x.
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + 1/A(x)^3)*x*A(x) + (1 + 2^2/A(x)^3 + 1/A(x)^6)*x^2*A(x)^2/2 +
(1 + 3^2/A(x)^3 + 3^2/A(x)^6 + 1/A(x)^9)*x^3*A(x)^3/3 +
(1 + 4^2/A(x)^3 + 6^2/A(x)^6 + 4^2/A(x)^9 + 1/A(x)^12)*x^4*A(x)^4/4 +
(1 + 5^2/A(x)^3 + 10^2/A(x)^6 + 10^2/A(x)^9 + 5^2/A(x)^12 + 1/A(x)^15)*x^5*A(x)^5/5 +...
		

Links

• Robert Israel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A073157, A198953, A215623, A215624, A364331, A364336.

Programs

Maple

S:= series(RootOf(x+y+x^2*y^2-y^2+x*y^3, y, 1), x, 41):
seq(coeff(S,x,j),j=0..40); # Robert Israel, Jul 10 2017

Mathematica

nmax=20; aa=ConstantArray[0,nmax]; aa[[1]]=2; Do[AGF=1+Sum[aa[[n]]*x^n,{n,1,j-1}]+koef*x^j; sol=Solve[Coefficient[(1-x^2)*AGF^2 - x*AGF^3 - x - AGF,x,j]==0,koef][[1]];aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{1,aa}] (* Vaclav Kotesovec, Sep 18 2013 *)

PARI

{a(n)=local(A=1+x); for(i=1, n, A=(1 + x*A^2)*(1 + x/(A+x*O(x^n)))); polcoeff(A, n)}

PARI

{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2/A^(3*j))*x^m*A^m/m))); polcoeff(A, n)}
for(n=0, 31, print1(a(n), ", "))

Formula

G.f. satisfies:

A(x) = exp( Sum_{n>=1} x^n*A(x)^n/n * Sum_{k=0..n} C(n,k)^2 / A(x)^(3*k) ).

The formal inverse of the g.f. A(x) is (sqrt(1 - 2*x^3 + 4*x^4 + x^6) - (1+x^3))/(2*x^2).

Recurrence: n*(n+1)*(1241*n^5 - 21306*n^4 + 135203*n^3 - 381522*n^2 + 435524*n - 104880)*a(n) = 6*n*(1201*n^4 - 19476*n^3 + 114613*n^2 - 287442*n + 255364)*a(n-1) + 2*(12410*n^7 - 237880*n^6 + 1771109*n^5 - 6388366*n^4 + 11032829*n^3 - 6363274*n^2 - 3856020*n + 4157712)*a(n-2) + 6*(2482*n^7 - 51299*n^6 + 419427*n^5 - 1705769*n^4 + 3477465*n^3 - 2797370*n^2 - 637684*n + 1410288)*a(n-3) + 2*(4964*n^7 - 110044*n^6 + 983093*n^5 - 4442260*n^4 + 10160177*n^3 - 8790970*n^2 - 4722180*n + 9233280)*a(n-4) - 6*(2482*n^7 - 58745*n^6 + 553921*n^5 - 2617109*n^4 + 6255337*n^3 - 6022682*n^2 - 1392300*n + 4289616)*a(n-5) + 60*(n-7)*(2*n - 11)*(n^3 - 40*n^2 + 280*n - 552)*a(n-6) + 2*(n-8)*(2*n - 13)*(1241*n^5 - 15101*n^4 + 62389*n^3 - 91339*n^2 - 930*n + 64260)*a(n-7). - Vaclav Kotesovec, Sep 18 2013

a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 4.77053998540509708... is the root of the equation -4 + 12*d^2 - 8*d^3 - 12*d^4 - 20*d^5 + d^7 = 0 and c = 1.27852844884923435863262213680985089152... - Vaclav Kotesovec, Sep 18 2013

In closed form, c = (-4 + (1 + sqrt(1+8/d^2))*d^2) * sqrt((d^3*(1 + sqrt(1+8/d^2) + (4*(4 + d^2*(-3-sqrt(1+8/d^2) + d*(4+d))))/d^6)) / (1 + 1/64*(1 + sqrt(1+8/d^2)-4/d^2)^3*d^3)) / (32*d). - Vaclav Kotesovec, Aug 18 2014

From Peter Bala, Sep 10 2024: (Start)

For n not of the form 3*m + 1, we conjecture that a(n) = Sum_{k = 0..n} binomial(-n+3*k+1, k)*binomial(-n+3*k+1, n-k)/(-n+3*k+1).

Define a sequence operator R: {u(n): n >= 0} -> {v(n): n >= 0} by Sum_{n >= 0} v(n)*x^n = (1/x) * series_reversion(x/Sum_{n >= 0} u(n)*x^n). Then R({a(n)}) = A364336, R^2({a(n)}) = A215623 and R^3({a(n)}) = A364331. Cf. A073157. (End)