A378582 G.f. Sum_{n=-oo..+oo} (x^n - x)^(n+1).

2, -1, 1, -1, 4, -3, -1, -1, 13, -1, -7, -10, 10, -1, 22, -1, 10, -23, -25, -1, 43, -1, 50, -36, 14, -1, -19, -26, 16, -55, 78, -1, 201, -1, -129, -78, 20, -108, 211, -1, 22, -105, 9, -1, 349, -1, 274, -430, 26, -1, -421, -50, 568, -171, 441, -1, 769, -661, -238, -210, 32, -1, 1291, -1, 34, -591, -897, -1288, 1765, -1

Offset

0,1

Comments

Compare g.f. to: Sum_{n=-oo..+oo} (x - x^(n+1))^n = 0.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..8200

Formula

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

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

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

Example

G.f.: A(x) = 2 - x + x^2 - x^3 + 4*x^4 - 3*x^5 - x^6 - x^7 + 13*x^8 - x^9 - 7*x^10 - 10*x^11 + 10*x^12 - x^13 + 22*x^14 - x^15 + 10*x^16 - 23*x^17 - 25*x^18 - x^19 + 43*x^20 - x^21 + 50*x^22 - 36*x^23 + 14*x^24 - x^25 + ...
RELATED SERIES.
F(x) = x + 4*x^2 + 16*x^3 + 255*x^4 + 4344*x^5 + 49104*x^6 + 543744*x^7 + 8203012*x^8 + 130252849*x^9 + 1857148424*x^10 + 26419178032*x^11 + 406394717168*x^12 + ... + A379764(n)*x^n + ...
where A(4*F(x)) = 2 - 4*x + 4*x^4 - 4*x^9 + 4*x^16 - 4*x^25 + 4*x^36 - 4*x^49 + 4*x^64 - 4*x^81 + 4*x^100 + ...
which equals 2*theta_4(x).
SPECIFIC VALUES.
A local minimum of A(x) is at x = z, A'(z) = 0,
  where z = 0.397529435491742842870725714009076671931564550115616181...
  and A(z) = 1.76925395689645126935316774753841505734121715127456863...
A(t) = 5 at t = 0.83799627848215104988844491211534885329390252098950...
  where 5 = Sum_{n=-oo..+oo} (t^n - t)^(n+1).
A(t) = 4 at t = 0.81576458148282505480367740238923856698108048584006...
A(t) = 3 at t = 0.77374138107025616474622853242292423166707448715221...
A(t) = 2 at t = 0.63388217567664405459819983625203116490495970722052...
  where 2 = Sum_{n=-oo..+oo} (t^n - t)^(n+1).
A(5/6) = 4.7450460128227344201120209017241324429858113440230...
A(4/5) = 3.5316007525899668447063164855200492187769453202119...
A(3/4) = 2.6807156006575496013515132361048335609447924890231...
A(2/3) = 2.1104745199027122018154499017357730364485251172261...
A(3/5) = 1.9196428858805646516374432938412152594386473418308...
A(1/2) = 1.7984818587175528236453909117209424070005516950258...
  where A(1/2) = Sum_{n=-oo..+oo} (1/2^n - 1/2)^(n+1).
A(2/5) = 1.7692678774574642220664568097843035516621774732422...
A(1/3) = 1.7777271141981820071010562139080023372566201778866...
  where A(1/3) = Sum_{n=-oo..+oo} (1/3^n - 1/3)^(n+1).
A(1/4) = 1.8094512892069907186358957280354026387421254606042...
A(1/5) = 1.8373950901417623688604868970319735279603621700924...
A(1/6) = 1.8591645951522733561268876665948564129719912163274...

Prog

(PARI) {a(n) = my(A = sum(m=-n-1, n+1, (x^m - x +x*O(x^n))^(m+1))); polcoef(A, n)}
for(n=0, 70, print1(a(n), ", "))

Crossrefs

Cf. A379764, A290003.

Sequence in context: A322020 A294895 A285328 * A321030 A373514 A290529

Adjacent sequences: A378579 A378580 A378581 * A378583 A378584 A378585

Keyword

sign

Author

Paul D. Hanna, Jan 13 2025

Status

approved