A324617 G.f. A(x) satisfies: 1 + x = Sum_{n>=0} x^n*(A(x)^n + 1)^n/(1 + x*A(x)^n)^(n+1).

1, -1, 1, -6, 10, -27, 28, -107, 502, -1996, -1015, 39035, -76739, -1078632, 7222569, 9644362, -337421969, 1171731119, 9909483512, -109536156966, 74836320374, 5651749289781, -37674051339344, -117589711277053, 3186640549115616, -12979461559921647, -138543759546567508, 1942263572054253138, -3322718404632175707, -132968516893238601191, 1307791482651889603081, 1344751233503556511150

Offset

0,4

Links

Paul D. Hanna, Table of n, a(n) for n = 0..200

Formula

G.f. A(x) satisfies:

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

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

Example

G.f.: A(x) = 1 - x + x^2 - 6*x^3 + 10*x^4 - 27*x^5 + 28*x^6 - 107*x^7 + 502*x^8 - 1996*x^9 - 1015*x^10 + 39035*x^11 - 76739*x^12 - 1078632*x^13 + ...
such that
1 + x = 1/(1 + x*A(x)) + x*(A(x) + 1)/(1 + x*A(x))^2 + x^2*(A(x)^2 + 1)^2 / (1 + x*A(x)^2)^3 + x^3*(A(x)^3 + 1)^3/(1 + x*A(x)^3)^4 + x^4*(A(x)^4 + 1)^4 / (1 + x*A(x)^4)^5 + x^5*(A(x)^5 + 1)^5 / (1 + x*A(x)^5)^6 + ...
also
1 + x = 1/(1 - x*A(x)) + x*(A(x) - 1)/(1 - x*A(x))^2 + x^2*(A(x)^2 - 1)^2 / (1 - x*A(x)^2)^3 + x^3*(A(x)^3 - 1)^3/(1 - x*A(x)^3)^4 + x^4*(A(x)^4 - 1)^4 / (1 - x*A(x)^4)^5 + x^5*(A(x)^5 - 1)^5 / (1 - x*A(x)^5)^6 + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0);
A[#A] = -polcoeff( sum(n=0, #A+1, x^n*(Ser(A)^n + 1)^n/(1 + x*Ser(A)^n)^(n+1) ), #A)); polcoeff(Ser(A), n)}
for(n=0, 40, print1(a(n), ", "))

Crossrefs

Sequence in context: A077621 A298736 A336845 * A359145 A287989 A081394

Adjacent sequences: A324614 A324615 A324616 * A324618 A324619 A324620

Keyword

sign

Author

Paul D. Hanna, Mar 12 2019

Status

approved