A352820 G.f. A(x) satisfies: 1 - x = Sum_{n>=0} (x^(4*n) + (-1)^n*A(x))^n.

1, 1, 1, 2, 3, 4, 5, 7, 12, 20, 31, 48, 78, 131, 217, 354, 581, 971, 1634, 2739, 4580, 7699, 13027, 22092, 37449, 63551, 108176, 184637, 315530, 539625, 924125, 1585371, 2723675, 4683890, 8062277, 13892645, 23966392, 41384842, 71522034, 123706840, 214148865

Offset

1,4

Links

Table of n, a(n) for n=1..41.

Formula

G.f. A(x) satisfies:

(1) 1 - x = Sum_{n>=0} ( x^(4*n) + (-1)^n*A(x) )^n.

(2) 1 - x = Sum_{n>=0} x^(4*n^2) / (1 + (-1)^n*x^(4*n)*A(x))^(n+1).

Example

G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 5*x^7 + 7*x^8 + 12*x^9 + 20*x^10 + 31*x^11 + 48*x^12 + 78*x^13 + 131*x^14 + ...
where
1 - x = 1 + (x^4 - A(x)) + (x^8 + A(x))^2 + (x^12 - A(x))^3 + (x^16 + A(x))^4 + (x^20 - A(x))^5 + (x^24 + A(x))^6 + ...
Also,
1 - x = 1/(1 + A(x)) + x^4/(1 - x^4*A(x))^2 + x^16/(1 + x^8*A(x))^3 + x^36/(1 - x^12*A(x))^4 + x^64/(1 + x^16*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^(4*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\4), x^(4*m^2)/(1 + (-x^4)^m*x*Ser(A))^(m+1) ), #A)); A[n+1]}
for(n=0, 40, print1(a(n), ", "))

Crossrefs

Cf. A317997, A352818, A352819, A352821.

Sequence in context: A060437 A133428 A090703 * A137357 A100676 A193769

Adjacent sequences: A352817 A352818 A352819 * A352821 A352822 A352823

Keyword

nonn

Author

Paul D. Hanna, Apr 05 2022

Status

approved