A326262 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( 1/(1-x)^(2*n) - A(x) )^n.

1, 2, 7, 80, 1742, 51842, 1902589, 82219592, 4071164749, 226803165574, 14029472009781, 953926536359084, 70723894649169937, 5679305945331227594, 491179287055641264989, 45527108214667404725616, 4503148842172835722939285, 473502491643614888369261116, 52748299277043902326373361722, 6206479798643382507763241117360, 769187266152748793100664986340382, 100156538984193022704291755068539370

Offset

0,2

Links

Table of n, a(n) for n=0..21.

Formula

G.f. A(x) satisfies:

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

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

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

Example

G.f.: A(x) = 1 + 2*x + 7*x^2 + 80*x^3 + 1742*x^4 + 51842*x^5 + 1902589*x^6 + 82219592*x^7 + 4071164749*x^8 + 226803165574*x^9 + 14029472009781*x^10 + ...
such that
1 = 1  +  (1/(1-x)^2 - A(x))  +  (1/(1-x)^4 - A(x))^2  +  (1/(1-x)^6 - A(x))^3  +  (1/(1-x)^8 - A(x))^4  +  (1/(1-x)^10 - A(x))^5  +  (1/(1-x)^12 - A(x))^6  +  (1/(1-x)^14 - A(x))^7  + ...
Also,
1 = 1/(1 + A(x))  +  (1-x)^2/((1-x)^2 + A(x))^2  +  (1-x)^4/((1-x)^4 + A(x))^3  +  (1-x)^6/((1-x)^6  +  A(x))^4 + (1-x)^8/((1-x)^8 + A(x))^5  +  (1-x)^10/((1-x)^10 + A(x))^6  +  (1-x)^12/((1-x)^12 + 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)^(-2*m) - Ser(A))^m ) )[#A] ); H=A; A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A304639, A326263, A326264, A326265.

Cf. A321602.

Sequence in context: A128293 A354426 A307291 * A071409 A232041 A135756

Adjacent sequences: A326259 A326260 A326261 * A326263 A326264 A326265

Keyword

nonn

Author

Paul D. Hanna, Jun 20 2019

Status

approved