A377257 G.f. A(x) satisfies A(x)^5 = A(x^5) + 5*A(x^6).

1, 1, -2, 6, -21, 80, -320, 1327, -5641, 24452, -107626, 479595, -2158961, 9801964, -44825612, 206274835, -954379090, 4436725739, -20712537332, 97058595375, -456348299572, 2152172842439, -10177774161826, 48252146708494, -229284953354357, 1091812174888210, -5209089231759676, 24897363431677317

Offset

1,3

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) = ( A(x^5) + 5*A(x^6) )^(1/5).

(2) A(x) = ( (A(x^25) + 5*A(x^30))^(1/5) + 5*(A(x^30) + 5*A(x^36))^(1/5) )^(1/5).

Example

G.f.: A(x) = x + x^2 - 2*x^3 + 6*x^4 - 21*x^5 + 80*x^6 - 320*x^7 + 1327*x^8 - 5641*x^9 + 24452*x^10 - 107626*x^11 + 479595*x^12 + ...
A(x)^5 = x^5 + 5*x^6 + x^10 + 5*x^12 - 2*x^15 - 10*x^18 + 6*x^20 + 30*x^24 - 21*x^25 - 25*x^30 - 320*x^35 + 400*x^36 + 1327*x^40 + ...
where A(x)^5 = A(x^5) + 5*A(x^6).
SPECIFIC VALUES.
A(t) = 1/5 at t = 0.17625092697531094932792874275629659329509743342882...
A(t) = 1/6 at t = 0.14909464281682869249948621914173923623608956025540...
A(t) = 1/8 at t = 0.11420184542314071056231183654632387325643195167188...
A(t) = 1/10 at t = 0.0926681481249947691070754347999703575997310652178...
A(1/5) = 0.22974848743734887045086411262913643095786932783070...
A(1/6) = 0.18814989214838108394438827324387704988563438585036...
A(1/7) = 0.15911895687908896565504803748643966782898349461855...
A(1/8) = 0.13774709372620054863291445090474626459620181135902...
A(1/10) = 0.1084473289424864366298051397819700935054955647083...

Prog

(PARI) {a(n) = my(A, V=[0, 1]); for(i=1, n+1, V=concat(V, 0); A=Ser(V);
V[#V] = polcoef( (subst(A, x, x^5) + 5*subst(A, x, x^6) - A^5 )/5, #V+3) ); polcoef(A, n)}
for(n=1, 50, print1(a(n), ", "))

Crossrefs

Cf. A377255, A377256, A377258.

Sequence in context: A196345 A352703 A352704 * A257561 A150203 A106228

Adjacent sequences: A377254 A377255 A377256 * A377258 A377259 A377260

Keyword

sign

Author

Paul D. Hanna, Nov 29 2024

Status

approved