A326264 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( 1/(1-x)^(4*n) - A(x) )^n.

1, 4, 26, 612, 26919, 1603680, 117660064, 10162944112, 1005838347950, 112009295740916, 13850874442895434, 1882848486231714788, 279100448753985866813, 44813411860476850508720, 7749809454081027489860264, 1436399220794697421878462832, 284111046278259235057207651469, 59740768193467931633275499487660, 13308884562229489858971683010469182, 3131623636896229572958776700673759164

Offset

0,2

Links

Table of n, a(n) for n=0..19.

Formula

G.f. A(x) satisfies:

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

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

(3) 1 = Sum_{n>=0} (1-x)^(4*n) / ( (1-x)^(4*n) + A(x) )^(n+1).

Example

G.f.: A(x) = 1 + 4*x + 26*x^2 + 612*x^3 + 26919*x^4 + 1603680*x^5 + 117660064*x^6 + 10162944112*x^7 + 1005838347950*x^8 + 112009295740916*x^9 + 13850874442895434*x^10 + ...
such that
1 = 1  +  (1/(1-x)^4 - A(x))  +  (1/(1-x)^8 - A(x))^2  +  (1/(1-x)^12 - A(x))^3  +  (1/(1-x)^16 - A(x))^4  +  (1/(1-x)^20 - A(x))^5  +  (1/(1-x)^24 - A(x))^6  +  (1/(1-x)^28 - A(x))^7  + ...
Also,
1 = 1/(1 + A(x))  +  (1-x)^4/((1-x)^4 + A(x))^2  +  (1-x)^8/((1-x)^8 + A(x))^3  +  (1-x)^12/((1-x)^12  +  A(x))^4 + (1-x)^16/((1-x)^16 + A(x))^5  +  (1-x)^20/((1-x)^20 + A(x))^6  +  (1-x)^24/((1-x)^24 + 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)^(-4*m) - Ser(A))^m ) )[#A] ); H=A; A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A304639, A326262, A326263, A326265.

Cf. A321604.

Sequence in context: A194926 A167147 A322395 * A132488 A320626 A144990

Adjacent sequences: A326261 A326262 A326263 * A326265 A326266 A326267

Keyword

nonn

Author

Paul D. Hanna, Jun 20 2019

Status

approved