A367385 Expansion of e.g.f. A(x) satisfying A(x/A(x)) = exp(x*A(x)).

1, 1, 5, 61, 1329, 43841, 1987153, 116322249, 8430315169, 733890562273, 75025552012641, 8851196086238969, 1188516164483406289, 179619377095898214801, 30271231938826215582001, 5645050489627807288153321, 1157185379272549414363693377, 259281400277115714365664526529

Offset: 0

Author

Paul D. Hanna, Dec 22 2023

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 61*x^3/3! + 1329*x^4/4! + 43841*x^5/5! + 1987153*x^6/6! + 116322249*x^7/7! + 8430315169*x^8/8! + 733890562273*x^9/9! + ...
where A(x/A(x)) = exp(x*A(x)) and
exp(x*A(x)) = 1 + x + 3*x^2/2! + 22*x^3/3! + 329*x^4/4! + 8396*x^5/5! + 318577*x^6/6! + 16388086*x^7/7! + 1075939601*x^8/8! + 86549687704*x^9/9! + ...
Also,
A(x) = exp(x*B(x)^2) where B(x) = A(x*B(x)) begins
B(x) = 1 + x + 7*x^2/2! + 112*x^3/3! + 2989*x^4/4! + 115136*x^5/5! + 5899159*x^6/6! + 381657928*x^7/7! + 30082660633*x^8/8! + 2814548348224*x^9/9! + ...
B(x)^2 = 1 + 2*x + 16*x^2/2! + 266*x^3/3! + 7168*x^4/4! + 275842*x^5/5! + 14058520*x^6/6! + 903187826*x^7/7! + 70653972896*x^8/8! + 6560662418306*x^9/9! + ...
Further,
A(x/C(x)^2) = exp(x) where C(x) = A(x/C(x)) begins
C(x) = 1 + x + 3*x^2/2! + 28*x^3/3! + 533*x^4/4! + 16096*x^5/5! + 680827*x^6/6! + 37544368*x^7/7! + 2577391273*x^8/8! + 213306280480*x^9/9! + ...
C(x)^2 = 1 + 2*x + 8*x^2/2! + 74*x^3/3! + 1344*x^4/4! + 39202*x^5/5! + 1618456*x^6/6! + 87693090*x^7/7! + 5940234656*x^8/8! + 486479747906*x^9/9! + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A144681.

Programs

PARI

{a(n) = my(A=1+x); for(i=0,n, A = exp( (1/x)*serreverse( x/(A + x*O(x^n)) )^2 )); n!*polcoeff(A,n)}
for(n=0,20, print1(a(n),", "))

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.

(1) A(x/A(x)) = exp(x*A(x)).

(2) A(x) = exp(x*B(x)^2) where B(x) = A(x*B(x)) = (1/x)*Series_Reversion(x/A(x)).

(3) A(x/C(x)^2) = exp(x) where C(x) = A(x/C(x)) = x / Series_Reversion(x*A(x)).