A369535 G.f. A(x) satisfies: A( 5*A(x)^5 - 125*A(x)^6 ) = 5*x^5.

1, 5, 100, 2625, 78125, 2502495, 84149950, 2929264125, 104646900625, 3814695312500, 141323209299700, 5305400750059625, 201382516012886250, 7715981043228103125, 298023033141992187500, 11591404812510959303625, 453601322379798576801250, 17846576327002385036896875

Offset

1,2

Links

Paul D. Hanna, Table of n, a(n) for n = 1..300

Formula

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

(1) A( 5*A(x)^5 - 125*A(x)^6 ) = 5*x^5.

(2) A( ( A(5*x^5 - 125*x^6)/5 )^(1/5) ) = x.

Example

G.f.: A(x) = x + 5*x^2 + 100*x^3 + 2625*x^4 + 78125*x^5 + 2502495*x^6 + 84149950*x^7 + 2929264125*x^8 + 104646900625*x^9 + 3814695312500*x^10 + ...
RELATED SERIES.
5*A(x)^5 - 125*A(x)^6 = 5*x^5 - 125*x^10 - 6250*x^15 - 468750*x^20 - 41015625*x^25 - 3896406250*x^30 - 389660156250*x^35 - ...
where A( 5*A(x)^5 - 125*A(x)^6 ) = 5*x^5.
Let B(x) satisfy A(B(x)) = B(A(x)) = x, where
B(x) = x - 5*x^2 - 50*x^3 - 750*x^4 - 13125*x^5 - 249370*x^6 - 4987650*x^7 - 103312125*x^8 - 2195388125*x^9 - 47566768750*x^10 - ...
then B(x)^5 = A(5*x^5 - 125*x^6)/5 which begins
B(x)^5 = x^5 - 25*x^6 + 25*x^10 - 1250*x^11 + 15625*x^12 + 2500*x^15 - 187500*x^16 + 4687500*x^17 - 39062500*x^18 + 328125*x^20 + ...

Prog

(PARI) {a(n) = my(V=[1]); for(i=1, n, V=concat(V, 0); A = x*Ser(V); V[#V] = -polcoeff(subst(G=A, x, 5*A^5 - 125*A^6 ), #V+4)/25); V[n]}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A291314, A291313, A291315, A369531, A369532, A369533, A369534.

Sequence in context: A322715 A208348 A128784 * A078534 A141120 A123668

Adjacent sequences: A369532 A369533 A369534 * A369536 A369537 A369538

Keyword

nonn

Author

Paul D. Hanna, Jan 25 2024

Status

approved