OFFSET
2,2
COMMENTS
Self-convolution of A259612.
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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 30 2015
STATUS
approved