A392209 G.f. A(x) satisfies A( x*A(x)^3 + A(x)^5 ) = A(x)^4.

1, 1, 2, 5, 15, 49, 168, 594, 2150, 7931, 29718, 112814, 432957, 1677050, 6547856, 25742454, 101819101, 404885649, 1617725220, 6491296371, 26147447557, 105691741392, 428578725720, 1742927971200, 7106956875720, 29050379285808, 119015289279194, 488612294241992, 2009889728675210

Offset

1,3

Comments

Different from A063021: first difference is term a(17) = 101819101 != A063021(17) = 101819100.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..1100

Formula

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

(1) A(x)^4 = A( x*A(x)^3 + A(x)^5 ).

(2) A(x)^16 = A( x*A(x)^15 + A(x)^17 + A(x)^20 ).

(3) A(x)^64 = A( x*A(x)^63 + A(x)^65 + A(x)^68 + A(x)^80 ).

(4) A(x)^(4^n) = A( x*A(x)^(4^n-1) + Sum_{k=0..n-1} A(x)^(4^n+4^k) ) for n > 0.

(5) A(x) = x + Sum_{n>=0} A(x)^(4^n+1).

(6) A(x) = Series_Reversion(x - x*Sum_{n>=0} x^(4^n) ).

The radius of convergence r and A(r) satisfy: 1 = Sum_{n>=0} (4^n+1) * A(r)^(4^n) and r = A(r) - Sum_{n>=0} A(r)^(4^n+1), where r = 0.230533258261200156085521786542900412789260200303602699930... and A(r) = 0.421260870588047221340668663556526087333382739055621911819...

a(n) ~ c * d^n / n^(3/2), where d = 4.33776890823698021092287403189070103347406781591211599643306781238... and c = 0.1024483098472250973829268377673544861276998106437743405695... - Vaclav Kotesovec, Jan 27 2026

Example

G.f.: A(x) = x + x^2 + 2*x^3 + 5*x^4 + 15*x^5 + 49*x^6 + 168*x^7 + 594*x^8 + 2150*x^9 + 7931*x^10 + 29718*x^11 + 112814*x^12 + ...
where A( x*A(x)^3 + A(x)^5 ) = A(x)^4.
RELATED SERIES.
(1) A(x)^4 = x^4 + 4*x^5 + 14*x^6 + 48*x^7 + 169*x^8 + 612*x^9 + 2266*x^10 + 8528*x^11 + 32500*x^12 + ...
(2) x*A(x)^3 + A(x)^5 = x^4 + 4*x^5 + 14*x^6 + 48*x^7 + 168*x^8 + 604*x^9 + 2222*x^10 + 8320*x^11 + 31582*x^12 + ...
(3) Let R(x) be the series reversion of A(x), R(A(x)) = x, then
R(x) = x - x^2 - x^5 - x^17 - x^65 - x^257 + ... + -x^(4^n+1) + ...
SPECIFIC VALUES.
A(t) = 2/5 at t = 0.229759828201308159999999986388705323162461461465... where 16/625 = A(t*8/125 + 32/3125) and t = 2/5 - Sum_{n>=0} (2/5)^(4^n+1).
A(t) = 1/3 at t = 0.218106988141249287411848783247915394065888806063... where 1/81 = A(t/27 + 1/243) and t = 1/3 - Sum_{n>=0} 1/3^(4^n+1).
A(t) = 1/4 at t = 0.186523437441792339086532592773437499999265316030... where 1/256 = A(t/64 + 1/1024) and t = 1/4 - Sum_{n>=0} 1/4^(4^n+1).
A(t) = 1/5 at t = 0.159679999998689279999999999999999999999999999631... where 1/625 = A(t/125 + 1/3125) and t = 1/5 - Sum_{n>=0} 1/5^(4^n+1).
A(1/5) = 0.2802960024122365984388546889846792867036674034873...
A(1/6) = 0.2120687471620608889069735993280260247595186049168...
A(1/7) = 0.1729093415279086689022006987660417152259386674281...
A(1/8) = 0.1465421938363728972931556755035998223502665893864...
A(1/9) = 0.1273669762105928346233001079383375534086417143052...
A(1/10) = 0.112725163976012172333598584909084531890886437439...

Prog

(PARI) /* Using series reversion of x - x*Sum_{n>=0} x^(4^n) */
{a(n) = my(A); A = serreverse( x - x*sum(k=0, ceil(log(n)/log(4)), x^(4^k) +x*O(x^n)) ); polcoeff(A, n)}
for(n=1, 35, print1(a(n), ", "))
(PARI) /* Using A(x)^4 = A( x*A(x)^3 + A(x)^5 ) */
{a(n) = my(A=[1], F); for(i=1, n, A = concat(A, 0); F = x*Ser(A);
A[#A] = polcoeff( subst(F, x, x*F^3 + F^5 ) - F^4, #A+3) ); A[n]}
for(n=1, 35, print1(a(n), ", "))

Crossrefs

Cf. A372526, A371713, A371716.

Sequence in context: A369631 A389538 A063021 * A364833 A301832 A272460

Adjacent sequences: A392206 A392207 A392208 * A392210 A392211 A392212

Keyword

nonn

Author

Paul D. Hanna, Jan 24 2026

Status

approved