A259608 G.f. A(x) satisfies: A(x) = Series_Reversion( x - x^2*A(x) - x*Integral 2*A(x) dx ).

1, 2, 15, 164, 2190, 33384, 561659, 10226376, 198975366, 4101249990, 88985266436, 2022670569000, 47986654728506, 1184722493746988, 30364559922967455, 806313807163378768, 22146014022165507644, 628220131284285896472, 18382404744008384580629, 554214116675011187495440

Offset

1,2

Links

Table of n, a(n) for n=1..20.

Formula

G.f. A(x) satisfies:

(1) A(x) = Series_Reversion( x - Sum_{n>=1} (n+1)/n * a(n) * x^(2*n+1) ).

(2) A(x) = x + Sum_{n>=1} (n+1)/n * a(n) * A(x)^(2*n+1).

Let B(x) = Integral 2*A(x) dx, then

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

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

a(n)/n = A259609(n) for n>=1.

Example

G.f.: A(x) = x + 2*x^3 + 15*x^5 + 164*x^7 + 2190*x^9 + 33384*x^11 +...
Let B(x) = Integral 2*A(x) dx
B(x) = x^2 + x^4 + 5*x^6 + 41*x^8 + 438*x^10 + 5564*x^12 + 80237*x^14 + 1278297*x^16 + 22108374*x^18 +...+ A259609(n)*x^(2*n) +...
such that A(x - x^2*A(x) - x*B(x)) = x.
Also,
A(x) = x + (x*A(x) + B(x))*x + [d/dx (x*A(x) + B(x))^2*x^2]/2! + [d^2/dx^2 (x*A(x) + B(x))^3*x^3]/3! + [d^3/dx^3 (x*A(x) + B(x))^4*x^4]/4! + [d^4/dx^4 (x*A(x) + B(x))^5*x^5]/5! +...
Logarithmic series:
log(A(x)/x) = (x*A(x) + B(x)) + [d/dx (2*x*A(x) + B(x))^2*x]/2! + [d^2/dx^2 (2*x*A(x) + B(x))^3*x^2]/3! + [d^3/dx^3 (2*x*A(x) + B(x))^4*x^3]/4! + [d^4/dx^4 (2*x*A(x) + B(x))^5*x^4]/5! +...

Prog

(PARI) {a(n)=local(A=x); for(i=0, n, A = serreverse(x - x^2*A - x*intformal(2*A) +x*O(x^(2*n)))); polcoeff(A, 2*n-1)}
for(n=1, 25, print1(a(n), ", "))

Crossrefs

Cf. A259609.

Sequence in context: A204679 A363564 A364331 * A317278 A140809 A153852

Adjacent sequences: A259605 A259606 A259607 * A259609 A259610 A259611

Keyword

nonn

Author

Paul D. Hanna, Jun 30 2015

Status

approved