A245313 G.f. satisfies: A(x) = 1 + x*A(x) + x^2*A(x)^2 + 3*x^3*A(x)*A'(x) + x^4*A(x)*A''(x).

1, 1, 2, 7, 31, 176, 1158, 8919, 76751, 742597, 7865088, 91553100, 1150905332, 15665172108, 227991734414, 3554320236911, 58795765799791, 1033303679424539, 19151079894682674, 374662948814998855, 7691131223011551255, 165785969673935575904, 3734170668741419488552

Offset

0,3

Comments

Do the following limits exist? If so, what are the respective values?

(1) limit a(n)*sqrt(n+1)/(n+1)! ? (Value is near 0.718490 at n=400.)

(2) limit A245313(n)/A245312(n) ? (Value is near 2.721747 at n=400.)

Limit a(n)*sqrt(n+1)/(n+1)! = 0.7189460513696389360211370..., limit A245313(n)/A245312(n) = e. - Vaclav Kotesovec, Jul 20 2014

Links

Paul D. Hanna, Table of n, a(n) for n = 0..400

Formula

G.f. A(x) satisfies:

(1) A(x) = 1/(1-x - x^2*Dx^2(A(x))) where Dx(F(x)) = d/dx x*F(x).

(2) A(x) = G(x*A(x)) where G(x) = A(x/G(x)) is the g.f. of A245312.

(3) A(x) = (1/x)*Series_Reversion(x/G(x)) where G(x) is the g.f. of A245312.

a(n) = [x^n] G(x)^(n+1)/(n+1) for n>=0 where G(x) is the g.f. of A245312.

a(n) = [x^(n+2)] G(x)^(n+1)/(n+1)^3 for n>=0 where G(x) is the g.f. of A245312.

Example

G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 31*x^4 + 176*x^5 + 1158*x^6 +...
where A(x) = 1 / (1-x - x^2*A(x) - 3*x^3*A'(x) - x^4*A''(x)).
Define Dx(A(x)) = d/dx x*A(x) = A(x) + x*A'(x):
Dx(A(x)) = 1 + 2*x + 6*x^2 + 28*x^3 + 155*x^4 + 1056*x^5 + 8106*x^6 +...
so that Dx^2(A(x)) = d/dx x*(d/dx x*A(x)) = A(x) + 3*x*A'(x) + x^2*A''(x):
Dx^2(A(x)) = 1 + 4*x + 18*x^2 + 112*x^3 + 775*x^4 + 6336*x^5 +...
then A(x) = 1/(1-x - x^2*Dx^2(A(x))):
1/A(x) = 1 - x - x^2 - 4*x^3 - 18*x^4 - 112*x^5 - 775*x^6 - 6336*x^7 -...
RELATED SERIES.
The g.f. of A245312 begins:
G(x) = 1 + x + x^2 + 3*x^3 + 10*x^4 + 60*x^5 + 360*x^6 + 2940*x^7 +...
where A(x) = G(x*A(x)) where G(x) = A(x/G(x)) and
a(n) = [x^n] G(x)^(n+1)/(n+1) = [x^(n+2)] G(x)^(n+1)/(n+1)^3 for n>=0.

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*A+x^2*A^2+3*x^3*A*A'+x^4*A*A''+x*O(x^n)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* From A(x) = G(x*A(x)) where G(x) is the g.f. of A245312: */
{a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); m=#A-2; A[#A]=-Vec(Ser(A)^m*(1-m^2*x^2))[#A]/m);
polcoeff(1/x*serreverse(x/Ser(A)), n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A245312, A245311.

Sequence in context: A199675 A352309 A059037 * A306037 A046907 A365561

Adjacent sequences: A245310 A245311 A245312 * A245314 A245315 A245316

Keyword

nonn

Author

Paul D. Hanna, Jul 19 2014

Status

approved