A379203 G.f. A(x) satisfies 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 3)^(n+1).

1, 5, 34, 290, 2820, 29629, 327301, 3744868, 43981858, 527126689, 6420981368, 79260797860, 989306411413, 12464737320229, 158320378037652, 2025016002188169, 26060398562711196, 337197048402240367, 4384067953773647268, 57245716462267462224, 750403639664344374239, 9871281245683966836462

Offset

1,2

Comments

Related identity: Sum_{n=-oo..+oo} x^n*(y - x^n)^n = 0, which holds formally for all y.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..300

Formula

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

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

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

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

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

(5) A(B(x)) = x where B(x) = 1/( Sum_{n=-oo..+oo} x^n * (x^n + 3)^(n+1) ).

Example

G.f.: A(x) = x + 5*x^2 + 34*x^3 + 290*x^4 + 2820*x^5 + 29629*x^6 + 327301*x^7 + 3744868*x^8 + 43981858*x^9 + 527126689*x^10 + ...
SPECIFIC VALUES.
A(t) = 1/7 at t = 0.069769772400266469707360138034033927488705716660080...
  where t = 1/Sum_{n=-oo..+oo} (1 + 3*7^(n-1))^n / 7^(n^2-1).
A(t) = 1/8 at t = 0.067295105779482404156544832668824160420208234924667...
  where t = 1/Sum_{n=-oo..+oo} (1 + 3*8^(n-1))^n / 8^(n^2-1).
A(t) = 1/9 at t = 0.064327556053208007320009998534415581932268509899202...
  where t = 1/Sum_{n=-oo..+oo} (1 + 3*9^(n-1))^n / 9^(n^2-1).
A(t) = 1/10 at t = 0.06126924119589872239866986020862532219839002819792...
  where t = 1/Sum_{n=-oo..+oo} (1 + 3*10^(n-1))^n / 10^(n^2-1).
A(1/15) = 0.12166176397390884847529063617720403039492284665035...
A(1/16) = 0.10420546336336096378642246758350885785023968035181...
A(1/20) = 0.07053009254165709187694647754531300907207762301254...

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=-#A, #A, A^m*(A^m + 3)^(m+1) ), #V-3); ); polcoef(A, n)}
for(n=1, 40, print1(a(n), ", "))

Crossrefs

Cf. A379200, A379199, A166952, A378264, A379202, A379204, A379205.

Sequence in context: A284864 A208677 A380619 * A259906 A379991 A355258

Adjacent sequences: A379200 A379201 A379202 * A379204 A379205 A379206

Keyword

nonn

Author

Paul D. Hanna, Dec 20 2024

Status

approved