A380062 G.f. A(x) satisfies 1 + 3*A(x) = Sum_{n>=0} (x + 2*A(x)^n)^n.

1, 1, 5, 13, 67, 267, 1333, 6257, 31649, 159205, 824227, 4295599, 22706441, 120952097, 649957785, 3515569465, 19134939879, 104700236227, 575656327477, 3178537378181, 17618587510649, 98000687531045, 546850545803695, 3060334365371927, 17172225926646469, 96593865520419529, 544573760947543421

Offset

1,3

Links

Paul D. Hanna, Table of n, a(n) for n = 1..400

Formula

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

(1) 1 + 3*A(x) = Sum_{n>=0} (x + 2*A(x)^n)^n.

(2) 1 + 3*A(x) = Sum_{n>=0} 2^n * A(x)^(n^2) / (1 - x*A(x)^n)^(n+1).

a(n) ~ c * d^n / n^(3/2), where d = 5.9694701821143914331339... and c = 0.08445223261025391689308... - Vaclav Kotesovec, Jan 22 2025

Example

G.f.: A(x) = x + x^2 + 5*x^3 + 13*x^4 + 67*x^5 + 267*x^6 + 1333*x^7 + 6257*x^8 + 31649*x^9 + 159205*x^10 + 824227*x^11 + 4295599*x^12 + ...
where
1 + 3*A(x) = 1 + (x + 2*A(x)) + (x + 2*A(x)^2)^2 + (x + 2*A(x)^3)^3 + (x + 2*A(x)^4)^4 + (x + 2*A(x)^5)^5 + ...
also
1 + 3*A(x) = 1/(1 - x) + 2*A(x)/(1 - x*A(x))^2 + 2^2*A(x)^4/(1 - x*A(x)^2)^3 + 2^3*A(x)^9/(1 - x*A(x)^3)^4 + 2^4*A(x)^16/(1 - x*A(x)^4)^5 + ...
SPECIFIC VALUES.
A(t) = 3/10 at t = 0.16726377275995072648448117435948148299720377903757...
A(t) = 1/4 at t = 0.16050181723352504846666549277691960802055733336720...
  where 7/4 = Sum_{n>=0} (t + 2/4^n)^n.
A(t) = 1/5 at t = 0.14476857786758172355537218086213390949895147290041...
  where 8/5 = Sum_{n>=0} (t + 2/5^n)^n.
A(t) = 1/6 at t = 0.12943904478896880604661898314490116170232546051386...
  where 3/2 = Sum_{n>=0} (t + 2/6^n)^n.
A(1/6) = 0.29029452204321243394058678922603696260540562320344...
  where 1 + 3*A(1/6) = Sum_{n>=0} (1/6 + 2*A(1/6)^n)^n.
A(1/7) = 0.19534410312734561358652191539351096601493491194423...
A(1/8) = 0.15830595087636851770622457001839792271181094358064...
A(1/9) = 0.13454907242604646240630468598979662673730593892243...

Prog

(PARI) {a(n) = my(V=[0, 1], A=x); for(i=1, n, V = concat(V, 0); A = Ser(V);
V[#V] = polcoef( sum(n=1, #V, (x + 2*A^n)^n ) - 3*A, #V-1) ); V[n+1]}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A380061, A380063, A380064.

Sequence in context: A018678 A149575 A156101 * A372799 A093118 A359690

Adjacent sequences: A380058 A380059 A380061 * A380063 A380064 A380065

Keyword

nonn,changed

Author

Paul D. Hanna, Jan 11 2025

Status

approved