A379197 G.f. A(x) satisfies 2 = Sum_{n=-oo..+oo} x^n * (1 + x^n)^n * A(x)^n.

1, 0, 4, 8, 34, 132, 638, 2820, 13168, 62036, 300220, 1464380, 7227616, 35973092, 180572534, 912481936, 4639054638, 23709435920, 121753848044, 627893481912, 3250523102274, 16885954908704, 87997395891936, 459904413345900, 2409993698814872, 12659707993570100, 66651577298786350, 351644960592378116, 1858837753276366880

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..400

Formula

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

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

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

a(n) ~ c * d^n / n^(3/2), where d = 5.57384012479790467653678... and c = 0.36821964478616549325... - Vaclav Kotesovec, Jan 22 2025

Example

G.f.: A(x) = 1 + 4*x^2 + 8*x^3 + 34*x^4 + 132*x^5 + 638*x^6 + 2820*x^7 + 13168*x^8 + 62036*x^9 + 300220*x^10 + 1464380*x^11 + 7227616*x^12 + ...
SPECIFIC VALUES.
A(t) = 3/2 at t = 0.17932110817562092526922034225510656397555988597730...
  where 2 = Sum_{n=-oo..+oo} (3/2)^n * t^n * (1 + t^n)^n.
A(t) = 4/3 at t = 0.17364426414665882921399467667941469013844780878137...
A(t) = 5/4 at t = 0.16521161220996838686500515442487457621181365254065...
A(t) = 6/5 at t = 0.15692802182391781504241199591448307509812700728891...
  where 2 = Sum_{n=-oo..+oo} (6/5)^n * t^n * (1 + t^n)^n.
A(1/6) = 1.2610831429301963024065444169902163365622216552672...
  where 2 = Sum_{n=-oo..+oo} 1/6^n * (1 + 1/6^n)^n * A(1/6)^n.
A(1/7) = 1.1433278204701872784987607646709591793469859419026...
A(1/8) = 1.0962135625311591216691131663240630319942784156181...
A(1/9) = 1.0702210207780083759033558223635819530998710547013...
A(1/10) = 1.0538928783555246970697587839922150753996404838066...
  where 2 = Sum_{n=-oo..+oo} 1/10^n * (1 + 1/10^n)^n * A(1/10)^n.

Prog

(PARI) {a(n) = my(V=[1], A=1); for(i=1, n, V=concat(V, 0); A=Ser(V);
V[#V] = polcoef( sum(m=-#V, #V, x^m * (1 + x^m +x*O(x^#V))^m * A^m), #V-1)); V[n+1]}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Sequence in context: A149107 A197157 A324410 * A218628 A149108 A149109

Adjacent sequences: A379194 A379195 A379196 * A379198 A379199 A379200

Keyword

nonn

Author

Paul D. Hanna, Jan 15 2025

Status

approved