A380061 G.f. A(x) satisfies 1 + 2*A(x) = Sum_{n>=0} (x + A(x)^n)^n.

1, 1, 3, 6, 22, 64, 238, 813, 3064, 11276, 43290, 165781, 647916, 2541826, 10085272, 40216458, 161537304, 651906163, 2644581776, 10771863433, 44052178217, 180771472434, 744215912074, 3072704927332, 12720703058636, 52791694969046, 219588568135410, 915306832688642, 3822734157002821, 15994557980983589

Offset

1,3

Links

Paul D. Hanna, Table of n, a(n) for n = 1..370

Formula

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

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

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

a(n) ~ c * d^n / n^(3/2), where d = 4.404603275448413351115... and c = 0.124757791288661710... - Vaclav Kotesovec, Jan 22 2025

Example

G.f.: A(x) = x + x^2 + 3*x^3 + 6*x^4 + 22*x^5 + 64*x^6 + 238*x^7 + 813*x^8 + 3064*x^9 + 11276*x^10 + 43290*x^11 + 165781*x^12 + ...
where
1 + 2*A(x) = 1 + (x + A(x)) + (x + A(x)^2)^2 + (x + A(x)^3)^3 + (x + A(x)^4)^4 + (x + A(x)^5)^5 + ...
also
1 + 2*A(x) = 1/(1 - x) + A(x)/(1 - x*A(x))^2 + A(x)^4/(1 - x*A(x)^2)^3 + A(x)^9/(1 - x*A(x)^3)^4 + A(x)^16/(1 - x*A(x)^4)^5 + ...
SPECIFIC VALUES.
A(t) = 1/3 at t = 0.21113418444205264671945193671032382558997406426968...
  where 5/3 = Sum_{n>=0} (t + 1/3^n)^n.
A(t) = 1/4 at t = 0.18142430617769577223861981972629677110643259922912...
  where 3/2 = Sum_{n>=0} (t + 1/4^n)^n.
A(t) = 1/5 at t = 0.15629861486046332792061966616353733814216002304152...
  where 7/5 = Sum_{n>=0} (t + 1/5^n)^n.
A(t) = 1/6 at t = 0.13647025270389097347232067790887467697059927206469...
  where 4/3 = Sum_{n>=0} (t + 1/6^n)^n.
A(1/5) = 0.29687312646844966972432360640338759607076917682394...
  where 1 + 2*A(1/5) = Sum_{n>=0} (1/5 + A(1/5)^n)^n.
A(1/6) = 0.21933159677831824800451769369205221017201954379946...
A(1/7) = 0.17695875419420417559520731471081672048402598324828...
A(1/8) = 0.14906980560864112152590985614625198492432883898111...
A(1/10) = 0.1139208273120548952749666258536995401947215571004...

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 + A^n)^n ) - 2*A, #V-1) ); V[n+1]}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A380062, A380063, A380064.

Sequence in context: A148640 A148641 A148642 * A049415 A208939 A209067

Adjacent sequences: A380058 A380059 A380060 * A380062 A380063 A380064

Keyword

nonn

Author

Paul D. Hanna, Jan 11 2025

Status

approved