A379204 G.f. A(x) satisfies 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 4)^(n+1).

1, 6, 52, 572, 7154, 96444, 1365480, 20015404, 301104656, 4622137698, 72110068424, 1140008607808, 18223311950352, 294049155429240, 4783093039542544, 78348659072215696, 1291254702576739650, 21396346604365855060, 356250789435149406252, 5957201829333106382128, 100003077199160840926640

Offset

1,2

Comments

Related identity: Sum_{n=-oo..+oo} x^n*(y - x^n)^n = 0, which holds formally for all y.

Conjecture: a(n) is even for n > 1.

Conjecture: a(n) == 2 (mod 4) iff n = (k-1)^2 + 1 for some k > 1.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..200

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

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

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

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

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

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

Example

G.f.: A(x) = x + 6*x^2 + 52*x^3 + 572*x^4 + 7154*x^5 + 96444*x^6 + 1365480*x^7 + 20015404*x^8 + 301104656*x^9 + 4622137698*x^10 + ...

Prog

(PARI) {a(n) = my(V=[0, 1], A); for(i=1, n, V=concat(V, 0); A = Ser(V);
V[#V] = polcoef( sum(m=-#A, #A, A^m*(A^m + 4)^(m+1) ), #V-3); ); polcoef(A, n)}
for(n=1, 40, print1(a(n), ", "))

Crossrefs

Cf. A379200, A379199, A166952, A378264, A379202, A379203, A379205.

Sequence in context: A365755 A232821 A355162 * A127133 A243249 A075756

Adjacent sequences: A379201 A379202 A379203 * A379205 A379206 A379207

Keyword

nonn

Author

Paul D. Hanna, Dec 20 2024

Status

approved