A352821 G.f. A(x) satisfies: 1 - x = Sum_{n>=0} (x^(5*n) + (-1)^n*A(x))^n.

1, 1, 1, 1, 2, 3, 4, 5, 6, 8, 13, 21, 32, 46, 66, 99, 155, 244, 376, 569, 862, 1328, 2077, 3256, 5071, 7853, 12181, 19023, 29882, 47004, 73808, 115757, 181785, 286323, 452111, 714548, 1129185, 1784586, 2823069, 4472449, 7094472, 11261549, 17882350

Offset

1,5

Links

Table of n, a(n) for n=1..43.

Formula

G.f. A(x) satisfies:

(1) 1 - x = Sum_{n>=0} ( x^(5*n) + (-1)^n*A(x) )^n.

(2) 1 - x = Sum_{n>=0} x^(5*n^2) / (1 + (-1)^n*x^(5*n)*A(x))^(n+1).

Example

G.f.: A(x) = x + x^2 + x^3 + x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 5*x^8 + 6*x^9 + 8*x^10 + 13*x^11 + 21*x^12 + 32*x^13 + 46*x^14 + 66*x^15 + ...
where
1 - x = 1 + (x^5 - A(x)) + (x^10 + A(x))^2 + (x^15 - A(x))^3 + (x^20 + A(x))^4 + (x^25 - A(x))^5 + (x^30 + A(x))^6 + ...
Also,
1 - x = 1/(1 + A(x)) + x^5/(1 - x^5*A(x))^2 + x^20/(1 + x^10*A(x))^3 + x^45/(1 - x^15*A(x))^4 + x^80/(1 + x^20*A(x))^5 + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(m=0, #A, (x^(5*m) + (-1)^m*x*Ser(A))^m ), #A)); A[n+1]}
for(n=0, 40, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(m=0, sqrtint(#A\5), x^(5*m^2)/(1 + (-x)^(5*m)*x*Ser(A))^(m+1) ), #A)); A[n+1]}
for(n=0, 40, print1(a(n), ", "))

Crossrefs

Cf. A317997, A352818, A352819, A352820.

Sequence in context: A033071 A306075 A049432 * A102516 A156981 A271949

Adjacent sequences: A352818 A352819 A352820 * A352822 A352823 A352824

Keyword

nonn

Author

Paul D. Hanna, Apr 05 2022

Status

approved