A259268 G.f. satisfies: A(x) = Series_Reversion( x - 3*A(x)*B(x)^2 ), where B'(x) = 2*A(x).

1, 3, 60, 1848, 70722, 3112692, 151785933, 8028464085, 454496257269, 27288493264020, 1726435358946081, 114532053780495258, 7937351775801976725, 572926949672305285692, 42967912938983806922313, 3341457670910162832959766, 268988340093587305693448265

Offset

1,2

Comments

Conjecture: (2*n-1) | a(n) for n>=1 and therefore all the terms of this sequence are integers.

Links

Table of n, a(n) for n=1..17.

Formula

Let B(x) = Integral 2*A(x) dx, then g.f. A(x) satisfies:

(1) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) 3^n * A(x)^n * B(x)^(2*n) / n!.

(2) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) 3^n * A(x)^n * B(x)^(2*n) / (n!*x) ).

a(n)/(2*n-1) = A259269(n).

Example

G.f.: A(x) = x + 3*x^5 + 60*x^9 + 1848*x^13 + 70722*x^17 + 3112692*x^21 +...
Let B(x) = Integral 2*A(x) dx, an integer series that begins:
B(x) = x^2 + x^6 + 12*x^10 + 264*x^14 + 7858*x^18 + 282972*x^22 + 11675841*x^26 +...+ A259269(n)*x^(4*n+2) +...
B(x)^2 = x^4 + 2*x^8 + 25*x^12 + 552*x^16 + 16388*x^20 + 587996*x^24 + 24175914*x^28 +...
then A(x - 3*A(x)*B(x)^2) = x, where
A(x)*B(x)^2 = x^5 + 5*x^9 + 91*x^13 + 2595*x^17 + 93962*x^21 + 3970616*x^25 + 187722645*x^29 +...
Series_Reversion(A(x)) = x - 3*x^5 - 15*x^9 - 273*x^13 - 7785*x^17 - 281886*x^21 - 11911848*x^25 +...
Also,
A(x) = x + 3*A(x)*B(x)^2 + [d/dx 9*A(x)^2*B(x)^4]/2! + [d^2/dx^2 27*A(x)^3*B(x)^6]/3! + [d^3/dx^3 81*A(x)^4*B(x)^8]/4! + [d^4/dx^4 243*A(x)^5*B(x)^10]/5! +...
Logarithmic series:
log(A(x)/x) = 2*A(x)*B(x)^2/x + [d/dx 9*A(x)^2*B(x)^4/x]/2! + [d^2/dx^2 27*A(x)^3*B(x)^6/x]/3! + [d^3/dx^3 81*A(x)^4*B(x)^8/x]/4! + [d^4/dx^4 243*A(x)^5*B(x)^10/x]/5! +...

Prog

(PARI) {a(n)=local(A=x, G=x^2); for(i=0, n, G=intformal(2*A);  A = serreverse(x - 3*A*G^2 +O(x^(4*n)))); polcoeff(A, 4*n-3)}
for(n=1, 25, print1(a(n), ", "))
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x*O(x^n), B=x^2); for(i=1, n, B=intformal(2*A);  A = x + sum(m=1, n+1, Dx(m-1, 3^m*A^m*B^(2*m)/m!)) +O(x^(4*n))); polcoeff(A, 4*n-3)}
for(n=1, 25, print1(a(n), ", "))
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x*O(x^n), B=x^2); for(i=1, n, B=intformal(2*A); A = x*exp(sum(m=1, n, Dx(m-1, 3^m*A^m*B^(2*m)/(m!*x))) +O(x^(4*n)))); polcoeff(A, 4*n-3)}
for(n=1, 25, print1(a(n), ", "))

Crossrefs

Cf. A259269, A259270, A259271.

Sequence in context: A248707 A219870 A326283 * A268964 A361536 A081854

Adjacent sequences: A259265 A259266 A259267 * A259269 A259270 A259271

Keyword

nonn

Author

Paul D. Hanna, Jun 30 2015

Status

approved