login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A200074 G.f. satisfies: A(x) = (1 + x*A(x)^2)*(1 + x^2*A(x)). 10
1, 1, 3, 9, 30, 108, 406, 1577, 6280, 25499, 105169, 439388, 1855636, 7908909, 33975250, 146954693, 639460707, 2797384235, 12295494109, 54272825103, 240480529815, 1069257987503, 4769306203838, 21334400243252, 95687482105807, 430217846136134, 1938651904470374, 8754225470415889 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

More generally, for fixed parameters p, q, r, and s, if F(x) satisfies:

F(x) = exp( Sum_{n>=1} x^(n*r)*F(x)^(n*p)/n * [Sum_{k=0..n} C(n,k)^2 * x^(k*s)*F(x)^(k*q)] ),

then F(x) = (1 + x^r*F(x)^(p+1))*(1 + x^(r+s)*F(x)^(p+q+1)).

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..450

FORMULA

G.f. satisfies:

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

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

(3) A(x) = x / Series_Reversion( x*G(x) ) where G(x) is the g.f. of A199876.

(4) A(x) = G(x/A(x)) where G(x) = A(x*G(x)) is the g.f. of A199876.

Recurrence: (n+1)*(n+2)*(1241*n^4 - 10636*n^3 + 25417*n^2 - 7382*n - 17136)*a(n) =  - 18*(n+1)*(443*n^3 - 3889*n^2 + 9734*n - 5712)*a(n-1) + 4*(6205*n^6 - 53180*n^5 + 115741*n^4 + 64762*n^3 - 370103*n^2 + 246727*n - 25704)*a(n-2) + 6*(2482*n^6 - 24995*n^5 + 76519*n^4 - 36347*n^3 - 185471*n^2 + 293092*n - 140400)*a(n-3) + 2*(4964*n^6 - 57436*n^5 + 228617*n^4 - 276802*n^3 - 361447*n^2 + 956696*n - 320496)*a(n-4) - 6*(2482*n^6 - 32441*n^5 + 140587*n^4 - 173153*n^3 - 266705*n^2 + 677518*n - 291840)*a(n-5) + 12*(n-4)*(2*n - 11)*(11*n^2 + 73*n - 748)*a(n-6) + 2*(n-5)*(2*n - 13)*(1241*n^4 - 5672*n^3 + 955*n^2 + 16508*n - 8496)*a(n-7). - Vaclav Kotesovec, Aug 18 2013

a(n) ~ c*d^n/n^(3/2), where d = 4.770539985405... is the root of the equation -4 + 12*d^2 - 8*d^3 - 12*d^4 - 20*d^5 + d^7 = 0 and c = 0.612892860188927397373456... - Vaclav Kotesovec, Aug 18 2013

EXAMPLE

G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 30*x^4 + 108*x^5 + 406*x^6 + 1577*x^7 +...

Related expansions:

A(x)^2 = 1 + 2*x + 7*x^2 + 24*x^3 + 87*x^4 + 330*x^5 + 1289*x^6 +...

A(x)^3 = 1 + 3*x + 12*x^2 + 46*x^3 + 180*x^4 + 720*x^5 + 2928*x^6 +...

where A(x) = 1 + x*A(x)^2 + x^2*A(x) + x^3*A(x)^3.

The logarithm of the g.f. A = A(x) equals the series:

log(A(x)) = (A + x)*x + (A^2 + 2^2*x*A + x^2)*x^2/2 +

(A^3 + 3^2*x*A^2 + 3^2*x^2*A + x^3)*x^3/3 +

(A^4 + 4^2*x*A^3 + 6^2*x^2*A^2 + 4^2*x^3*A + x^4)*x^4/4 +

(A^5 + 5^2*x*A^4 + 10^2*x^2*A^3 + 10^2*x^3*A^2 + 5^2*x^4*A + x^5)*x^5/5 +

(A^6 + 6^2*x*A^5 + 15^2*x^2*A^4 + 20^2*x^3*A^3 + 15^2*x^4*A^2 + 6^2*x^5*A + x^6)*x^6/6 +...

more explicitly,

log(A(x)) = x + 5*x^2/2 + 19*x^3/3 + 77*x^4/4 + 331*x^5/5 + 1445*x^6/6 + 6392*x^7/7 + 28565*x^8/8 +...

MAPLE

a:= n-> coeff(series(RootOf(A=(1+x*A^2)*(1+x^2*A), A), x, n+1), x, n):

seq(a(n), n=0..30);  # Alois P. Heinz, May 16 2012

MATHEMATICA

m = 28; A[_] = 0;

Do[A[x_] = (1 + x A[x]^2)(1 + x^2 A[x]) + O[x]^m, {m}];

CoefficientList[A[x], x] (* Jean-Fran├žois Alcover, Oct 02 2019 *)

PROG

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

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

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, (1-x/A)^(2*m+1)*sum(j=0, n, binomial(m+j, j)^2*x^j/A^j)*x^m*A^m/m))); polcoeff(A, n, x)}

CROSSREFS

Cf. A199876, A199874, A199877, A198951, A198953, A198957, A192415, A198888, A200075, A004148, A036765.

Sequence in context: A053022 A128725 A099783 * A032125 A246472 A091699

Adjacent sequences:  A200071 A200072 A200073 * A200075 A200076 A200077

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Nov 13 2011

STATUS

approved

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 July 14 23:49 EDT 2020. Contains 335762 sequences. (Running on oeis4.)