A326263 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( 1/(1-x)^(3*n) - A(x) )^n.

1, 3, 15, 262, 8616, 384873, 21181421, 1372455324, 101895990777, 8511828635054, 789539638329648, 80506096148928303, 8951189588697000825, 1078020157296224938479, 139830500253903232730304, 19438947952499889395212003, 2883820412306778479104733811, 454810046719340404484233328331, 75993667094400965507408118716882, 13411571696501962452150617362998648, 2493074269436929464139674369969509811

Offset

0,2

Links

Table of n, a(n) for n=0..20.

Formula

G.f. A(x) satisfies:

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

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

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

Example

G.f.: A(x) = 1 + 3*x + 15*x^2 + 262*x^3 + 8616*x^4 + 384873*x^5 + 21181421*x^6 + 1372455324*x^7 + 101895990777*x^8 + 8511828635054*x^9 + 789539638329648*x^10 + ...
such that
1 = 1  +  (1/(1-x)^3 - A(x))  +  (1/(1-x)^6 - A(x))^2  +  (1/(1-x)^9 - A(x))^3  +  (1/(1-x)^12 - A(x))^4  +  (1/(1-x)^15 - A(x))^5  +  (1/(1-x)^18 - A(x))^6  +  (1/(1-x)^21 - A(x))^7  + ...
Also,
1 = 1/(1 + A(x))  +  (1-x)^3/((1-x)^3 + A(x))^2  +  (1-x)^6/((1-x)^6 + A(x))^3  +  (1-x)^9/((1-x)^9  +  A(x))^4 + (1-x)^12/((1-x)^12 + A(x))^5  +  (1-x)^15/((1-x)^15 + A(x))^6  +  (1-x)^18/((1-x)^18 + A(x))^7 + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ((1-x)^(-3*m) - Ser(A))^m ) )[#A] ); H=A; A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A304639, A326262, A326264, A326265.

Cf. A321603.

Sequence in context: A050474 A051179 A122591 * A120607 A013352 A013354

Adjacent sequences: A326260 A326261 A326262 * A326264 A326265 A326266

Keyword

nonn

Author

Paul D. Hanna, Jun 20 2019

Status

approved