A378579 G.f. A(x) satisfies x = Sum_{n>=1} ((1 + A(x)^n)^n - 1).

1, -2, 5, -15, 54, -226, 1041, -5045, 25090, -126674, 646764, -3335207, 17359589, -91138625, 482237135, -2569446532, 13774698084, -74245779493, 402105384051, -2187066640025, 11941274232967, -65425584835537, 359598131529024, -1982178299221646, 10955208670488609, -60696056311093958, 337040131916813474

Offset

1,2

Links

Paul D. Hanna, Table of n, a(n) for n = 1..500

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

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

(2) x = Sum_{n>=1} A(x)^(n^2) / (1 - A(x)^n)^(n+1), from formula by Seiichi Manyama in A318636.

(3) x = Sum_{n>=1} A(x)^n * Sum_{d|n} binomial(n/d,d), from formula by Ridouane Oudra in A318636.

(4) A(x) = Series_Reversion(G(x)), where G(x) = Sum_{n>=1} ((1 + x^n)^n - 1) is the g.f. of A318636.

Example

G.f.: A(x) = x - 2*x^2 + 5*x^3 - 15*x^4 + 54*x^5 - 226*x^6 + 1041*x^7 - 5045*x^8 + 25090*x^9 - 126674*x^10 + 646764*x^11 - 3335207*x^12 + ...
where
x = (1 + A(x))-1 + (1 + A(x)^2)^2-1 + (1 + A(x)^3)^3-1 + (1 + A(x)^4)^4-1 + ...
The expansions of (1 + A(x)^n)^n - 1 begin:
n=1: x - 2*x^2 + 5*x^3 - 15*x^4 + 54*x^5 - 226*x^6 + 1041*x^7 - 5045*x^8 + ...
n=2: 2*x^2 - 8*x^3 + 29*x^4 - 108*x^5 + 430*x^6 - 1848*x^7 + 8484*x^8 + ...
n=3: 3*x^3 - 18*x^4 + 81*x^5 - 336*x^6 + 1395*x^7 - 6048*x^8 + ...
n=4: 4*x^4 - 32*x^5 + 176*x^6 - 848*x^7 + 3934*x^8 - 18416*x^9 + ...
n=5: 5*x^5 - 50*x^6 + 325*x^7 - 1775*x^8 + 9000*x^9 + ...
n=6: 6*x^6 - 72*x^7 + 540*x^8 - 3300*x^9 + 18234*x^10 + ...
n=7: 7*x^7 - 98*x^8 + 833*x^9 - 5635*x^10 + 33761*x^11 + ...
n=8: 8*x^8 - 128*x^9 + 1216*x^10 - 9024*x^11 + 58336*x^12 + ...
...
the sum of which equals x.
SPECIFIC VALUES.
A(t) = 1/8 at t = 0.16352126551257248889045664875683784263524590236453...
  where t = Sum_{n>=1} ((1 + 1/8^n)^n - 1),
  also, t = Sum_{n>=1} (1/8)^(n^2) / (1 - 1/8^n)^(n+1).
A(t) = 1/9 at t = 0.14078320572038685935740333771629838603314392626246...
  where t = Sum_{n>=1} ((1 + 1/9^n)^n - 1),
  also, t = Sum_{n>=1} (1/9)^(n^2) / (1 - 1/9^n)^(n+1).
A(t) = 1/10 at t = 0.12355985214267974666409476695653610216564400778886...
  where t = Sum_{n>=1} ((1 + 1/10^n)^n - 1).
A(t) = -1/4 at t = -0.15526284433046589758223569590356891892154738705096...
A(t) = -1/5 at t = -0.13708093574671812870578995929148440226274633630611...
A(1/6) = 0.12685609485901293251324636636937755144064758593774...
  where 1/6 = Sum_{n>=1} ((1 + A(1/6)^n)^n - 1).
A(1/7) = 0.11241354571385669088090100601380487815275189296537...
A(1/8) = 0.10094983523585092678474357142194212014408583724977...
A(1/9) = 0.09162346270443389626958872306814641680247795571686...
A(-1/6) = -0.29924902046763454720023815313494776169729752567409...
A(-1/7) = -0.21417128882821263382592721321354392301151580309678...
A(-1/8) = -0.17350785733170913051439226143409362909355792177797...

Prog

(PARI) {a(n) = my(A = serreverse( sum(m=1, n, (1 + x^m +x*O(x^n))^m - 1) ));
polcoef(A, n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A318636.

Sequence in context: A185040 A369599 A208237 * A321958 A107112 A193318

Adjacent sequences: A378576 A378577 A378578 * A378580 A378581 A378582

Keyword

sign,new

Author

Paul D. Hanna, Jan 08 2025

Status

approved