A352817 G.f. A(x) satisfies: 1 - x = Sum_{n>=0} x^n * (x^(2*n) + (-1)^n*A(x))^n.

1, 1, 2, 3, 4, 8, 15, 27, 54, 107, 211, 429, 870, 1776, 3669, 7600, 15832, 33183, 69766, 147267, 312000, 662801, 1412103, 3016367, 6457814, 13856491, 29792374, 64174954, 138483001, 299324140, 647967785, 1404737352, 3049490131, 6628499431, 14425441436

Offset

0,3

Links

Table of n, a(n) for n=0..34.

Formula

G.f. A(x) satisfies:

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

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

Example

G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 8*x^5 + 15*x^6 + 27*x^7 + 54*x^8 + 107*x^9 + 211*x^10 + 429*x^11 + 870*x^12 + ...
where
1 - x = 1 + x*(x^2 - A(x)) + x^2*(x^4 + A(x))^2 + x^3*(x^6 - A(x))^3 + x^4*(x^8 + A(x))^4 + x^5*(x^10 - A(x))^5 + x^6*(x^12 + A(x))^6 + ...
Also,
1 - x = 1/(1 + x*A(x)) + x^3/(1 - x^3*A(x))^2 + x^10/(1 + x^5*A(x))^3 + x^21/(1 - x^7*A(x))^4 + x^36/(1 + x^9*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^m * (x^(2*m) + (-1)^m*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\2), x^((2*m+1)*m)/(1 + (-1)^m*x^(2*m+1)*Ser(A))^(m+1) ), #A)); A[n+1]}
for(n=0, 40, print1(a(n), ", "))

Crossrefs

Cf. A352816, A352818.

Sequence in context: A122774 A274166 A364595 * A189740 A118841 A126294

Adjacent sequences: A352814 A352815 A352816 * A352818 A352819 A352820

Keyword

nonn

Author

Paul D. Hanna, Apr 05 2022

Status

approved