A378584 G.f. A(x) satisfies 1 + 4*A(x) = Sum_{n=-oo..+oo} (2*x + A(x)^n)^n.

1, 2, 5, 13, 46, 170, 666, 2648, 10944, 46296, 199828, 873616, 3863994, 17255100, 77710773, 352486026, 1608824138, 7383210030, 34048297506, 157700809314, 733283992543, 3421740870626, 16018406425836, 75208586777464, 354067037861768, 1671014124932980, 7904396487947504, 37469496390403704

Offset

1,2

Links

Paul D. Hanna, Table of n, a(n) for n = 1..380

Formula

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

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

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

a(n) ~ c * d^n / n^(3/2), where d = 5.00995688191010718208... and c = 0.138126946089315438... - Vaclav Kotesovec, Jan 22 2025

Example

G.f: A(x) = x + 2*x^2 + 5*x^3 + 13*x^4 + 46*x^5 + 170*x^6 + 666*x^7 + 2648*x^8 + 10944*x^9 + 46296*x^10 + 199828*x^11 + 873616*x^12 + ...
where 1 + 4*A(x) = Sum_{n=-oo..+oo} (2*x + A(x)^n)^n.
SPECIFIC VALUES.
A(t) = 1/3 at t = 0.1831955345652769491608413185059304103972018821511469491...
  where 7/3 = Sum_{n=-oo..+oo} (2*t + 1/3^n)^n,
  also, 7/3 = Sum_{n=-oo..+oo} (1/3)^(n^2) / (1 + 2*t/3^n)^n.
A(t) = 1/4 at t = 0.1588173428481951756613569181105876136100886668873381763...
  where 2 = Sum_{n=-oo..+oo} (2*t + 1/4^n)^n,
  also, 2 = Sum_{n=-oo..+oo} (1/4)^(n^2) / (1 + 2*t/4^n)^n.
A(1/6) = 0.272948495724025612756651865489...
A(1/7) = 0.20982707763947888749753415117948193411561266617062...
A(1/8) = 0.17189390051289055911618566206048257447724135382709...
A(1/10) = 0.1270422272370836396738203332383911172335941544364...

Prog

(PARI) {a(n) = my(V=[0, 1], A=x); for(i=1, n, V=concat(V, 0); A = Ser(V);
V[#V] = (1/2)*polcoef( sum(n=-#V, #V, (2*x + A^n)^n ) - 4*A, #V-1) ); V[n+1]}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A378583.

Sequence in context: A149875 A221546 A085632 * A111563 A079573 A360513

Adjacent sequences: A378581 A378582 A378583 * A378585 A378586 A378587

Keyword

nonn

Author

Paul D. Hanna, Jan 10 2025

Status

approved