A359712 a(n) = coefficient of x^n in A(x) such that 2 = Sum_{n=-oo..+oo} (-x)^n * (2*A(x) + x^(n-1))^(n+1).

1, 4, 20, 106, 586, 3356, 19728, 118382, 722208, 4466050, 27931600, 176371300, 1122867012, 7199842666, 46454345844, 301384205640, 1964899532794, 12866563846920, 84585757496444, 558060746899684, 3693810227983576, 24521903234307786, 163234951757526400

Offset

0,2

Links

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

Formula

G.f. A(x) = Sum_{n>=0} a(n) * x^n may be described as follows.

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

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

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

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

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

From Paul D. Hanna, May 12 2023: (Start)

(6) 2 = Sum_{n=-oo..+oo} (-1)^n * x^(3*n+1) * (2*A(x) + x^n)^n.

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

(8) 2*x = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 + 2*A(x)*x^(n+1))^(n+1).

(9) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (2*A(x) + x^n)^(n+1).

(10) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 2*A(x)*x^n)^n.

(11) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 2*A(x)*x^(n+1))^n. (End)

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

Example

G.f.: A(x) = 1 + 4*x + 20*x^2 + 106*x^3 + 586*x^4 + 3356*x^5 + 19728*x^6 + 118382*x^7 + 722208*x^8 + 4466050*x^9 + 27931600*x^10 +  ...

Prog

(PARI) {a(n) = my(A=1, y=2); for(i=1, n,
A = 1/sum(m=-#A, #A, (-1)^m * (x*y*A + x^m + x*O(x^n) )^m ) );
polcoeff( A, n, x)}
for(n=0, 25, print1( a(n), ", "))
(PARI) {a(n) = my(A=[1], y=2); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(-y + sum(n=-#A, #A, (-1)^n * x^n * (y*Ser(A) + x^(n-1))^(n+1) )/(-y), #A-1, x) ); A[n+1]}
for(n=0, 25, print1( a(n), ", "))

Crossrefs

Cf. A359670, A359711, A359713, A363104, A363105, A361778.

Sequence in context: A135159 A190724 A243585 * A365226 A263965 A265084

Adjacent sequences: A359709 A359710 A359711 * A359713 A359714 A359715

Keyword

nonn

Author

Paul D. Hanna, Jan 17 2023

Status

approved