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A361056
Expansion of e.g.f. A(x) satisfying A(x) = Sum_{n>=0} (2*A(x)^n + 1)^n * x^n/n!.
7
1, 3, 21, 369, 11025, 465273, 25605585, 1742552325, 141496457985, 13368820514769, 1442273097241809, 175090338669687741, 23642282811004895745, 3517444221383606541849, 572114802197326599160497, 101067684833728895205914757, 19284211878473628720362002689
OFFSET
0,2
LINKS
FORMULA
E.g.f. A(x) = Sum_{n>=0} a(n) * x^n/n! may be defined as follows.
(1) A(x) = Sum_{n>=0} (2*A(x)^n + 1)^n * x^n/n!.
(2) A(x) = Sum_{n>=0} A(x)^(n^2) * exp(x*A(x)^n) * 2^n * x^n/n!.
a(n) = 0 (mod 3) for n > 0.
a(n) = Sum_{k=0..n} A361540(n,k) * 2^(n-k). - Paul D. Hanna, Mar 20 2023
EXAMPLE
E.g.f.: A(x) = 1 + 3*x + 21*x^2/2! + 369*x^3/3! + 11025*x^4/4! + 465273*x^5/5! + 25605585*x^6/6! + 1742552325*x^7/7! + 141496457985*x^8/8! +...
where the e.g.f. satisfies the following series identity:
A(x) = 1 + (2*A(x) + 1)*x + (2*A(x)^2 + 1)^2*x^2/2! + (2*A(x)^3 + 1)^3*x^3/3! + (2*A(x)^4 + 1)^4*x^4/4! + ... + (2*A(x)^n + 1)^n * x^n/n! + ...
and
A(x) = exp(x) + A(x)*exp(x*A(x))*2*x + A(x)^4*exp(x*A(x)^2)*2^2*x^2/2! + A(x)^9*exp(x*A(x)^3)*2^3*x^3/3! + A(x)^16*exp(x*A(x)^4)*2^4*x^4/4! + ... + A(x)^(n^2) * exp(x*A(x)^n) * 2^n * x^n/n! + ...
PROG
(PARI) /* E.g.f.: Sum_{n>=0} (2*A(x)^n + 1)^n * x^n/n! */
{a(n) = my(A = 1); for(i=1, n, A = sum(m=0, n, (2*A^m + 1 +x*O(x^n))^m * x^m/m! )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* E.g.f.: Sum_{n>=0} A(x)^(n^2) * exp(x*A(x)^n) * 2^n * x^n/n! */
{a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, (A +x*O(x^n))^(m^2) * exp(x*A^m +x*O(x^n)) * 2^m * x^m/m! )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 28 2023
STATUS
approved