A352819 G.f. A(x) satisfies: 1 - x = Sum_{n>=0} (x^(3*n) + (-1)^n*A(x))^n.

1, 1, 2, 3, 4, 6, 11, 19, 33, 60, 110, 201, 372, 696, 1305, 2456, 4654, 8855, 16896, 32366, 62217, 119910, 231704, 448879, 871531, 1695541, 3305057, 6454033, 12624041, 24731009, 48520396, 95324187, 187517959, 369329721, 728262059, 1437591201, 2840751293

Offset

1,3

Links

Table of n, a(n) for n=1..37.

Formula

G.f. A(x) satisfies:

(1) 1 - x = Sum_{n>=0} ( x^(3*n) + (-1)^n*A(x) )^n.

(2) 1 - x = Sum_{n>=0} x^(3*n^2) / (1 + (-1)^n*x^(3*n)*A(x))^(n+1).

Example

G.f.: A(x) = x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 11*x^7 + 19*x^8 + 33*x^9 + 60*x^10 + 110*x^11 + 201*x^12 + 372*x^13 + 696*x^14 + ...
where
1 - x = 1 + (x^3 - A(x)) + (x^6 + A(x))^2 + (x^9 - A(x))^3 + (x^12 + A(x))^4 + (x^15 - A(x))^5 + (x^18 + A(x))^6 + ...
Also,
1 - x = 1/(1 + A(x)) + x^3/(1 - x^3*A(x))^2 + x^12/(1 + x^6*A(x))^3 + x^27/(1 - x^9*A(x))^4 + x^48/(1 + x^12*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^(3*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\3), x^(3*m^2)/(1 + (-x)^(3*m)*x*Ser(A))^(m+1) ), #A)); A[n+1]}
for(n=0, 40, print1(a(n), ", "))

Crossrefs

Cf. A317997, A352818, A352820, A352821.

Sequence in context: A295681 A117308 A114412 * A016038 A003099 A061941

Adjacent sequences: A352816 A352817 A352818 * A352820 A352821 A352822

Keyword

nonn

Author

Paul D. Hanna, Apr 05 2022

Status

approved