A378829 G.f. A(x) satisfies 1 = Sum_{n=-oo..+oo} (A(x)^n - 2*x)^n.

1, -2, 5, -13, 30, -74, 202, -616, 2126, -7828, 29366, -110398, 414214, -1556848, 5892713, -22524354, 86954484, -338421674, 1324660464, -5204326208, 20498580511, -80907096678, 320002290542, -1268500509496, 5040195484362, -20073242195580, 80120884387322, -320442284717582, 1283939790460139

Offset

1,2

Comments

A signed version of A359673.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..303

Formula

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

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

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

(3) 0 = Sum_{n=-oo..+oo} (-1)^n * (A(x)^n - 2*x)^(n-1).

(4) 0 = Sum_{n=-oo..+oo} (-1)^n * A(x)^(2*n) * (A(x)^n + 2*x)^(n+1).

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

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

(7) 0 = Sum_{n=-oo..+oo} (-1)^n * A(x)^(n*(n+1)) / (1 + 2*x*A(x)^n)^(n+1).

Example

G.f.: A(x) = x - 2*x^2 + 5*x^3 - 13*x^4 + 30*x^5 - 74*x^6 + 202*x^7 - 616*x^8 + 2126*x^9 - 7828*x^10 + 29366*x^11 - 110398*x^12 + ...
where 1 = Sum_{n=-oo..+oo} (A(x)^n - 2*x)^n.
SPECIFIC VALUES.
A(t) = 1/6 at t = 0.24134833288352420167420358490093379236139061653959...
  where 1 = Sum_{n=-oo..+oo} (1/6^n - 2*t)^n.
A(t) = 1/7 at t = 0.19473287649699543474178954182484954936895675300220...
  where 1 = Sum_{n=-oo..+oo} (1/7^n - 2*t)^n.
A(t) = 1/8 at t = 0.16330047299490635761734791354706359079698287572429...
  where 1 = Sum_{n=-oo..+oo} (1/8^n - 2*t)^n.
A(t) = exp(-Pi) at t = 0.04720243920412572796492634515550526365563452970121157309...
  where 1 = Sum_{n=-oo..+oo} (exp(-n*Pi) - 2*t)^n,
  also, 1 = Sum_{n=-oo..+oo} exp(-n^2*Pi) / (1 - 2*t*exp(-n*Pi))^n;
  compare to Sum_{n=-oo..+oo} exp(-n^2*Pi) = Pi^(1/4)/gamma(3/4).
A(t) = exp(-2*Pi) at t = 0.001874436990256710694689538031391789940066981740061145959...
  where 1 = Sum_{n=-oo..+oo} (exp(-2*n*Pi) - 2*t)^n,
  also, 1 = Sum_{n=-oo..+oo} exp(-2*n^2*Pi) / (1 - 2*t*exp(-2*n*Pi))^n;
  compare to Sum_{n=-oo..+oo} exp(-2*n^2*Pi) = Pi^(1/4)/gamma(3/4) * sqrt(2+sqrt(2))/2.
A(1/5) = 0.14570268760195709902234365534810153966906514204980...
  where 1 = Sum_{n=-oo..+oo} (A(1/5)^n - 2/5)^n.
A(1/6) = 0.12698642862956730423090954809810167590805619510041...
  where 1 = Sum_{n=-oo..+oo} (A(1/6)^n - 1/3)^n.
A(1/7) = 0.11253270334433369822784652362071431711460474251926...
A(1/8) = 0.10104551587569245791494155789285565556961920656039...
  where 1 = Sum_{n=-oo..+oo} (A(1/8)^n - 1/4)^n.

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=-#V, #V, (A^m - 2*x)^m ), #V-1)/2); V[n+1]}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A355868, A359673.

Sequence in context: A309535 A018012 A359673 * A216684 A065377 A215215

Adjacent sequences: A378826 A378827 A378828 * A378830 A378831 A378832

Keyword

sign,new

Author

Paul D. Hanna, Dec 13 2024

Status

approved