A382320 G.f. A(x) satisfies A(x) = x + ( Sum_{n>=1} A(x^n) )^2.

1, 1, 4, 14, 52, 195, 774, 3140, 13118, 55861, 241988, 1062411, 4718380, 21156811, 95652842, 435553638, 1995707806, 9194770161, 42570402238, 197957907525, 924157498638, 4329762257151, 20351029400480, 95938011359954, 453492517932696, 2148971058064469, 10206782449568402, 48581518322215785

Offset

1,3

Comments

Moebius transform of A382321.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..1000

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

(1) A(x) = x + ( Sum_{n>=1} A(x^n) )^2.

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

(3) a(n) = Sum_{d|n} mu(n/d) * A382321(d), where mu(n) = A008683(n).

a(n) = Sum_{k=1..n-1} A382321(k) * A382321(n-k) for n>1. - Seiichi Manyama, Jan 31 2026

Example

G.f.: A(x) = x + x^2 + 4*x^3 + 14*x^4 + 52*x^5 + 195*x^6 + 774*x^7 + 3140*x^8 + 13118*x^9 + 55861*x^10 + 241988*x^11 + 1062411*x^12 + ...
where
A(x) = x + (A(x) + A(x^2) + A(x^3) + A(x^4) + A(x^5) + ...)^2.
Also,
A(x) = x + (1*x/(1-x) + 1*x^2/(1-x^2) + 4*x^3/(1-x^3) + 14*x^4/(1-x^4) + 52*x^5/(1-x^5) + ... + a(n)*x^n/(1-x^n) + ...)^2.
RELATED SERIES.
Let B(x) be the g.f. of A382321, then
B(x) = sqrt(A(x) - x) = A(x) + A(x^2) + A(x^3) + A(x^4) + A(x^5) + ...
also,
B(x) = 1*x/(1-x) + 1*x^2/(1-x^2) + 4*x^3/(1-x^3) + 14*x^4/(1-x^4) + 52*x^5/(1-x^5) + ... + a(n)*x^n/(1-x^n) + ...
where
B(x) = x + 2*x^2 + 5*x^3 + 16*x^4 + 53*x^5 + 201*x^6 + 775*x^7 + 3156*x^8 + 13123*x^9 + 55915*x^10 + ... + A382321(n)*x^n + ...
SPECIFIC VALUES.
A(t) = 1/3 at t = 0.18953359270108621177863789682602463898350...
A(t) = 1/4 at t = 0.16868191745255221372158901026603005319647546231828...
A(t) = 1/5 at t = 0.14869000629919786864454042949377631100682323978543...
A(t) = 1/6 at t = 0.13170763103610823029989547694942003850417668554436...
A(t) = 1/8 at t = 0.10603568604310227135468587801069505665797192687138...
A(t) = 1/10 at t = 0.08820689026962139114391178645419956379788657573350...
A(1/6) = 0.24416775678766113280027523896358848592770995258866...
A(1/7) = 0.18785458600287838621718435407412736595937048585958...
A(1/8) = 0.15495563809965924498768163961764534025883482146155...
A(1/9) = 0.13262497998255262729422999557604551803999178259100...
A(1/10) = 0.11624684613671210668252975279854687090990941727560...

Prog

(PARI) {a(n) = my(A=x +x*O(x^n), V=[1]); for(i=1, n, V=Vec(A); A = x + sum(k=1, n-1, V[k]*x^k/(1-x^k +x*O(x^n)) )^2 ); polcoef(A, n)}
for(n=1, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=[0, 1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoef(-Ser(A) + x + sum(k=1, #A, subst(Ser(A), x, x^k))^2, #A-1) ); A[n+1]}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A382321, A008683.

Sequence in context: A369577 A099486 A047033 * A017947 A394668 A244898

Adjacent sequences: A382317 A382318 A382319 * A382321 A382322 A382323

Keyword

nonn

Author

Paul D. Hanna, Apr 09 2025

Status

approved