OFFSET
0,3
COMMENTS
Compare to the identity:
exp(x) = Sum_{n>=0} D^(n*(n-1)/2) x^(n*(n+1)/2) / (n*(n+1)/2)!, where operator D^n = d^n/dx^n.
EXAMPLE
E.g.f.: A(x) = 1 + x + 3*x^2/2! + 13*x^3/3! + 103*x^4/4! + 1071*x^5/5! +...
such that, by definition:
A(x) = 1 + (x+x^2) + d/dx (x+x^2)^3/3! + d^3/dx^3 (x+x^2)^6/6! + d^6/dx^6 (x+x^2)^10/10! + d^10/dx^10 (x+x^2)^15/15! +...
Compare to the trivial identity:
exp(x) = 1 + x + d/dx x^3/3! + d^3/dx^3 x^6/6! + d^6/dx^6 x^10/10! + d^10/dx^10 x^15/15! +...
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*(m-1)/2, (x+x^2+x*O(x^n))^(m*(m+1)/2)/(m*(m+1)/2)!)); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 04 2012
STATUS
approved