A259611 G.f. A(x) satisfies: A'(x)/2 = Series_Reversion( x - x*A'(x) - A(x) ).

1, 2, 13, 128, 1592, 23148, 377584, 6739594, 129507154, 2648996432, 57205691948, 1296296380644, 30678145489088, 755447291969728, 19299230525813281, 510253174406747552, 13933861551059205804, 392343665999527604536, 11374928331862130417384, 339140678016090755172248

Offset

2,2

Comments

Self-convolution of A259612.

Links

Table of n, a(n) for n=2..21.

Formula

G.f. A(x) satisfies:

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

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

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

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

a(n) = 2*A259610(n-1)/n for n>=2.

Example

G.f. A(x) = x^2 + 2*x^3 + 13*x^4 + 128*x^5 + 1592*x^6 + 23148*x^7 + 377584*x^8 +...
such that
A'(x - x*A'(x) - A(x)) = 2*x.
RELATED SERIES.
A'(x)/2 = x + 3*x^2 + 26*x^3 + 320*x^4 + 4776*x^5 + 81018*x^6 + 1510336*x^7 + 30328173*x^8 +...+ A259610(n)*x^n +...
and
sqrt(A(x)) = x + x^2 + 6*x^3 + 58*x^4 + 720*x^5 + 10506*x^6 + 172284*x^7 + 3092717*x^8 +...+ A259612(n)*x^n +...

Prog

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

Crossrefs

Cf. A259610, A259612.

Sequence in context: A036078 A290219 A057065 * A351299 A215624 A247365

Adjacent sequences: A259608 A259609 A259610 * A259612 A259613 A259614

Keyword

nonn

Author

Paul D. Hanna, Jun 30 2015

Status

approved