A353739 E.g.f. A(x) satisfies: 1 = Sum_{n>=0} 2^n * (exp(n*x) - A(x))^n / n!.

1, 1, 3, 39, 983, 36575, 1820855, 114756943, 8825761159, 806801625375, 85941554164631, 10503158853530607, 1454442091007070247, 225879929852989030463, 39006369984778846202935, 7435376490890285530440719, 1554727341153441175044808647, 354663953592562868624029236063

Offset

0,3

Comments

More generally, the following sums are equal:

(1) Sum_{n>=0} (q^n + p)^n * r^n/n!,

(2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;

here, q = exp(x) with p = -A(x), r = 2.

Links

Table of n, a(n) for n=0..17.

Formula

E.g.f. A(x) satisfies:

(1) 1 = Sum_{n>=0} 2^n * (exp(n*x) - A(x))^n / n!.

(2) 1 = Sum_{n>=0} 2^n * exp(n^2*x - 2*A(x)*exp(n*x)) / n!.

Example

E.g.f: A(x) = 1 + x + 3*x^2/2! + 39*x^3/3! + 983*x^4/4! + 36575*x^5/5! + 1820855*x^6/6! + 114756943*x^7/7! + 8825761159*x^8/8! + 806801625375*x^9/9! + ...
such that
1 = 1 + 2*(exp(x) - A(x)) + 2^2*(exp(2*x) - A(x))^2/2! + 2^3*(exp(3*x) - A(x))^3/3! + 2^4*(exp(4*x) - A(x))^4/4! + 2^5*(exp(5*x) - A(x))^5/5! + 2^6*(exp(6*x) - A(x))^6/6! + ...
also
1 = exp(-2*A(x)) + 2*exp(x - 2*A(x)*exp(x)) + 2^2*exp(4*x - 2*A(x)*exp(2*x))/2! + 2^3*exp(9*x - 2*A(x)*exp(3*x))/3! + 2^4*exp(16*x - 2*A(x)*exp(4*x))/4! + 2^5*exp(25*x - 2*A(x)*exp(5*x))/5! + ...
Related series.
log(A(x)) = x + 2*x^2/2! + 32*x^3/3! + 830*x^4/4! + 31384*x^5/5! + 1580150*x^6/6! + 100484768*x^7/7! + 7786595710*x^8/8! + 716464194216*x^9/9! + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(m=0, #A, (exp(m*x +x*O(x^#A)) - Ser(A))^m * 2^m/m! ), #A-1)/2; ); H=A; n!*A[n+1]}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A326095, A353740.

Sequence in context: A276964 A378232 A367596 * A274573 A278750 A082159

Adjacent sequences: A353736 A353737 A353738 * A353740 A353741 A353742

Keyword

nonn

Author

Paul D. Hanna, May 06 2022

Status

approved