A370718 Expansion of g.f. A(x) satisfying A(x) = exp(x*A(x) + L(x)), where L'(x) = is the least nonnegative integer series such that A(x) is an integer series with A'(0) = 1.

1, 1, 2, 4, 9, 20, 47, 113, 279, 702, 1793, 4637, 12123, 31983, 85042, 227665, 613124, 1659927, 4515112, 12333189, 33816577, 93041508, 256792871, 710774480, 1972519207, 5487331792, 15299316997, 42744746059, 119654728359, 335549390828, 942564726188, 2651841948281, 7471773621129

Offset

0,3

Comments

The g.f. is motivated by the following identities involving the Catalan function C(x) = 1 + x*C(x)^2 (A000108) and the Motzkin function M(x) = 1 + x*M(x) + x^2*M(x)^2 (A001006):

(1) C(x)^2 = exp( x*C(x)^2 + Integral C(x)^2 dx ),

(2) C(x) = exp( x*C(x) + Integral x*C(x)^4/(1 - x^2*C(x)^4) dx ),

(3) M(x) = exp( x*M(x) + Integral x*M(x)/(1 - x^2*M(x)) dx ).

Links

Paul D. Hanna, Table of n, a(n) for n = 0..3000

Formula

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

(1) A(x) = exp(x*A(x) + L(x)), where L(x) = Sum_{n>=1} A370719(n)*x^n/n with A370719(1) = 0.

(2) [x^n] A'(x)/A(x) = (n+1)*a(n) + A370719(n+1), where 0 <= A370719(n+1) <= n, for n >= 0.

a(n) ~ c * d^n / n^(3/2), where d = 2.95096506072791643602456246927765372318152877881296246187793662... and c = 1.29175249398142508103898586184438451356153263383148632000580381... - Vaclav Kotesovec, Mar 02 2024

Let r be the radius of convergence, then A(r) = 1/r = exp(1 + L(r)) and L(r) = log(1/r) - 1 where L(x) is specified in formula (1). Note that r = 1/d where d is given by Vaclav Kotesovec's formula. Explicitly, r = 0.338872192459415... and L(r) = 0.082132256083160093570046274341567930991471748011061606257615544... - Paul D. Hanna, Mar 04 2024

Example

G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 20*x^5 + 47*x^6 + 113*x^7 + 279*x^8 + 702*x^9 + 1793*x^10 + 4637*x^11 + 12123*x^12 + ...
where A(x) = exp(x*A(x) + L(x)) and
L(x) = x^2/2 + x^3/3 + 3*x^4/4 + x^5/5 + 3*x^6/6 + x^7/7 + 3*x^8/8 + 7*x^9/9 + 8*x^10/10 + x^11/11 + 7*x^12/12 + ... + A370719(n)*x^n/n + ...
RELATED SERIES.
log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 19*x^4/4 + 46*x^5/5 + 123*x^6/6 + 330*x^7/7 + 907*x^8/8 + 2518*x^9/9 + 7028*x^10/10 + ...
where log(A(x)) = x*A(x) + L(x).
The logarithmic derivative of A(x) begins
A'(x)/A(x) = 1 + 3*x + 7*x^2 + 19*x^3 + 46*x^4 + 123*x^5 + 330*x^6 + 907*x^7 + 2518*x^8 + 7028*x^9 + 19724*x^10 + ...
and the derivative of x*A(x) starts as
(x*A(x))' = 1 + 2*x + 6*x^2 + 16*x^3 + 45*x^4 + 120*x^5 + 329*x^6 + 904*x^7 + 2511*x^8 + 7020*x^9 + 19723*x^10 + ...
the difference A'(x)/A(x) - (x*A(x))' equals
L'(x) = x + x^2 + 3*x^3 + x^4 + 3*x^5 + x^6 + 3*x^7 + 7*x^8 + 8*x^9 + x^10 + ...

Prog

(PARI) {a(n) = my(A=[1], L=[0], F); for(i=1, n, A = concat(A, 0);
L = concat(L, 0); F = exp( x*Ser(A) + sum(n=1, #L, L[n]*x^n/n) );
L[#L] = (#L - (#L)*polcoeff(F, #L))%(#L);
A = Vec(F +O(x^#A)); ); A[n+1]}
for(n=0, 40, print1(a(n), ", "))

Crossrefs

Cf. A370719.

Sequence in context: A014267 A089405 A091500 * A318799 A318852 A052329

Adjacent sequences: A370715 A370716 A370717 * A370719 A370720 A370721

Keyword

nonn

Author

Paul D. Hanna, Mar 01 2024

Status

approved