OFFSET
0,3
COMMENTS
Compare to the identity:
exp(x) = Sum_{n>=0} d^n/dx^n x^(2*n) / (2*n)!.
EXAMPLE
E.g.f.: A(x) = 1 + x + 7*x^2/2! + 33*x^3/3! + 223*x^4/4! + 1753*x^5/5! +...
such that, by definition:
A(x) = 1 + d/dx (x+x^2)^2/2! + d^2/dx^2 (x+x^2)^4/4! + d^3/dx^3 (x+x^2)^6/6! + d^4/dx^4 (x+x^2)^8/8! + d^5/dx^5 (x+x^2)^10/10! +...
Compare to the trivial identity:
exp(x) = 1 + d/dx x^2/2! + d^2/dx^2 x^4/4! + d^3/dx^3 x^6/6! + d^4/dx^4 x^8/8! + d^5/dx^5 x^10/10! +...
PROG
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=1+sum(m=1, n, Dx(m, (x+x^2+x*O(x^n))^(2*m)/(2*m)!)); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 04 2012
STATUS
approved