A380677 G.f. A(x) satisfies 1 = Sum_{n=-oo..+oo} x^(2*n) * (x^n - A(x))^(3*n+1).

1, 2, 8, 36, 198, 1128, 6837, 42690, 273960, 1792650, 11922735, 80342746, 547403208, 3764568202, 26097746670, 182183863242, 1279566641040, 9035527984360, 64109825254786, 456834687004440, 3267926616628182, 23458797921291994, 168936073477132102, 1220121029135864026, 8835737467337361482

Offset

0,2

Links

Paul D. Hanna, Table of n, a(n) for n = 0..350

Formula

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

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

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

Example

G.f.: A(x) = 1 + 2*x + 8*x^2 + 36*x^3 + 198*x^4 + 1128*x^5 + 6837*x^6 + 42690*x^7 + 273960*x^8 + 1792650*x^9 + 11922735*x^10 + ...
SPECIFIC VALUES.
A(t) = 7/4 at t = 0.12654949614445186746403892264694555335923498557738...
  where 1 = Sum_{n=-oo..+oo} t^(2*n) * (t^n - 7/4)^(3*n+1).
A(t) = 5/3 at t = 0.12374694612565134762563311753154796236873902596812...
A(t) = 3/2 at t = 0.11392195456863186572686610752037791827642247932473...
A(t) = 4/3 at t = 0.09535917714046949923896929084305426642940930464927...
A(t) = 5/4 at t = 0.08098320583796566321668508295130093344916245020730...
A(1/8) = 1.69987163237671043867918157348979527169465395859405...
  where 1 = Sum_{n=-oo..+oo} (1/8)^(2*n) * ((1/8)^n - A(1/8))^(3*n+1).
A(1/9) = 1.46724009425513930419976858432180568713155056224164...
A(1/10) = 1.3665270076239843695076027726524469708778850053524...
A(1/11) = 1.3048130783240200786482939740924774873262324649207...
A(1/12) = 1.2620494023042372384830602119971826992309809007730...
A(1/14) = 1.2057100150678855865365454675611764497376238367914...
A(1/16) = 1.1698113057379453133949062841882391284824341375308...

Prog

(PARI) {a(n) = my(V=[1]); for(i=1, n, V = concat(V, 0); A = Ser(V);
V[#V] = polcoef(-1 + sum(n=-#V, #V, x^(2*n) * (x^n - A)^(3*n+1) ), #V-1) ); H=A; V[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A380067, A355866, A380676.

Sequence in context: A052582 A367920 A222468 * A374162 A020021 A213076

Adjacent sequences: A380674 A380675 A380676 * A380678 A380679 A380680

Keyword

nonn

Author

Paul D. Hanna, Feb 02 2025

Status

approved