A363185 Expansion of g.f. A(x) satisfying 5 = Sum_{n=-oo..+oo} (-1)^n * x^n * (5*A(x) + x^(2*n-1))^(n+1).

1, 5, 27, 155, 929, 5730, 36083, 230935, 1497739, 9822060, 65021849, 433937545, 2916359840, 19720710150, 134078691289, 915994242780, 6284957607075, 43291450899490, 299248617182754, 2075172105905550, 14432704539830007, 100648564848019045, 703624464015723819

Offset

0,2

Links

Table of n, a(n) for n=0..22.

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.

(1) 5 = Sum_{n=-oo..+oo} (-1)^n * x^n * (5*A(x) + x^(2*n-1))^(n+1).

(2) 5*x = Sum_{n=-oo..+oo} (-1)^n * x^(2*n*(n-1)) / (1 + 5*A(x)*x^(2*n+1))^(n-1).

(3) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^n * (5*A(x) + x^(2*n-1))^n.

(4) A(x) = x / Sum_{n=-oo..+oo} (-1)^n * x^(3*n) * (5*A(x) + x^(2*n-1))^(n-1).

(5) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^(2*n^2) / (1 + 5*A(x)*x^(2*n+1))^n.

a(n) = Sum_{k=0..floor(n/2)} A359670(n-k,n-2*k) * 5^(n-2*k) for n >= 0.

Example

G.f.: A(x) = 1 + 5*x + 27*x^2 + 155*x^3 + 929*x^4 + 5730*x^5 + 36083*x^6 + 230935*x^7 + 1497739*x^8 + 9822060*x^9 + 65021849*x^10 + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(1 - sum(m=-#A, #A, (-1)^m * x^m * (5*Ser(A) + x^(2*m-1))^(m+1) ), #A-1)/5); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A363142, A363182, A363183, A363184.

Cf. A359670.

Sequence in context: A184702 A083326 A083880 * A098409 A351015 A052227

Adjacent sequences: A363182 A363183 A363184 * A363186 A363187 A363188

Keyword

nonn

Author

Paul D. Hanna, May 20 2023

Status

approved