A352816 G.f. A(x) satisfies: 1 - x = Sum_{n>=0} x^n * (x^n + (-1)^n*A(x))^n.

1, 2, 3, 7, 16, 40, 105, 281, 770, 2147, 6080, 17431, 50459, 147265, 432821, 1279978, 3806154, 11373668, 34136769, 102863360, 311065038, 943740309, 2871717550, 8762164066, 26801981013, 82172139800, 252470206882, 777243596670, 2397210487402, 7406340364950

Offset

0,2

Links

Table of n, a(n) for n=0..29.

Formula

G.f. A(x) satisfies:

(1) 1 - x = Sum_{n>=0} x^n * ( x^n + (-1)^n*A(x) )^n.

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

Example

G.f.: A(x) = 1 + 2*x + 3*x^2 + 7*x^3 + 16*x^4 + 41*x^5 + 107*x^6 + 286*x^7 + 782*x^8 + 2179*x^9 + 6172*x^10 + 17702*x^11 + 51272*x^12 + ...
where
1 - x = 1 + x*(x - A(x)) + x^2*(x^2 + A(x))^2 + x^3*(x^3 - A(x))^3 + x^4*(x^4 + A(x))^4 + x^5*(x^5 - A(x))^5 + x^6*(x^6 + A(x))^6 + ...
Also,
1 - x = 1/(1 + x*A(x)) + x^2/(1 - x^2*A(x))^2 + x^6/(1 + x^3*A(x))^3 + x^12/(1 - x^4*A(x))^4 + x^20/(1 + x^5*A(x))^5 + ...
Specific values.
A(x) = 2 at x = 0.2515345800906597549860933755999098710430442043412...
A(x) = 3 at x = 0.3019496486281905495940467534583642873678976347045...

Prog

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

Crossrefs

Cf. A352817, A317997.

Sequence in context: A334398 A027118 A114582 * A107387 A091487 A247332

Adjacent sequences: A352813 A352814 A352815 * A352817 A352818 A352819

Keyword

nonn

Author

Paul D. Hanna, Apr 05 2022

Status

approved