A379195 G.f. A(x) satisfies x = Sum_{n=-oo..+oo} (A(x) - A(x)^n)^(n+1).

1, 1, 1, 2, 5, 10, 21, 56, 148, 359, 906, 2450, 6571, 17338, 46777, 128681, 352859, 967315, 2679764, 7474260, 20860226, 58375826, 164197258, 463322792, 1309547562, 3710517258, 10543567357, 30021808808, 85628123727, 244694423127, 700553813377, 2008780153580, 5768264675938, 16587793685429, 47766704865133

Offset: 1

Author

Paul D. Hanna, Jan 14 2025

Keywords

• nonn

Comments

Compare to the identity 0 = Sum_{n=-oo..+oo} (x - x^(n+1))^n.

Examples

			G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 5*x^5 + 10*x^6 + 21*x^7 + 56*x^8 + 148*x^9 + 359*x^10 + 906*x^11 + 2450*x^12 + 6571*x^13 + 17338*x^14 + ...
where x = Sum_{n=-oo..+oo} (A(x) - A(x)^n)^(n+1).
RELATED SERIES.
F(x) = Sum_{n=-oo..+oo} (x - x^n)^(n+1) = x - x^2 + x^3 - 2*x^4 + 3*x^5 - 3*x^6 + x^7 + x^8 + x^9 - 7*x^10 + 10*x^11 - 6*x^12 + x^13 + x^15 - 8*x^16 + 23*x^17 - 25*x^18 + x^19 + 17*x^20 + x^21 - 32*x^22 + 36*x^23 - 12*x^24 + x^25 + ... + A290003(n)*x^n + ...
where F(A(x)) = x.
SPECIFIC VALUES.
A(t) = 1/2 at t = 0.30725396830704316799197832656390411971168116373389...
  where t = Sum_{n=-oo..+oo} (1/2 - 1/2^n)^(n+1),
  also, t = Sum_{n=-oo..+oo} (2^(n-1) - 1)^(n+1) / 2^(n*(n+1)).
A(t) = 1/3 at t = 0.24338606674563424484910361835257533242309621632065...
  where t = Sum_{n=-oo..+oo} (1/3 - 1/3^n)^(n+1),
  also, t = Sum_{n=-oo..+oo} (3^(n-1) - 1)^(n+1) / 3^(n*(n+1)).
A(t) = 1/4 at t = 0.19758524006807690544490179709803177425355852401229...
  where t = Sum_{n=-oo..+oo} (1/4 - 1/4^n)^(n+1).
A(t) = 1/5 at t = 0.16558333624735433324843855679493132539350188690309...
  where t = Sum_{n=-oo..+oo} (1/5 - 1/5^n)^(n+1).
A(1/4) = 0.34697020435026836163926019675791627488695303305268...
  where 1/4 = Sum_{n=-oo..+oo} (A(1/4) - A(1/4)^n)^(n+1).
A(1/5) = 0.25400492231901630962271637839330240648984255624021...
A(1/6) = 0.20160813481244983396982286666489080077373441727643...
A(1/8) = 0.14327208862930858756346363646363969972815166338945...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..1030

Crossrefs

Cf. A290003.

Programs

PARI

N=40 \\ number of terms
{a(n) = my(R = sum(m=-N-1, N+1, (x - x^m +x^2*O(x^N))^(m+1) ), A=x);
A = serreverse(R); polcoef(A, n)}
for(n=1, N, print1(a(n), ", "))

Formula

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

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

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

From Paul D. Hanna, Jan 25 2025: (Start)

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

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

(End)

a(n) ~ c * d^n / n^(3/2), where d = 3.00914051453408723176675508018... and c = 0.174541635630216521276160108... - Vaclav Kotesovec, Jan 22 2025