A385911 If n = 2^b*3^c, then a(n) = (-1)^b * binomial(b+c, b), else a(n) = 0, for n >= 1.

1, -1, 1, 1, 0, -2, 0, -1, 1, 0, 0, 3, 0, 0, 0, 1, 0, -3, 0, 0, 0, 0, 0, -4, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -10, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -6, 0

Offset

1,6

Comments

See comment by David Wasserman in related sequence A061984.

Triangle A385910 has g.f. A(x,y) where A(x,y) = A(x^3 + 3*x*y*A(x,y)^3, y) / A(x^2 + 2*x*y*A(x,y)^2, y).

a(n) = A385910(n+1, 1) for n >= 1.

Links

Table of n, a(n) for n=1..97.

Formula

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

(1) A(x) = Sum_{n>=0} (-1)^n * Sum_{k>=0} binomial(n+k,k) * x^(2^n*3^k).

(2) If n = 2^b*3^c, then a(n) = (-1)^b * binomial(b+c, b) else a(n) = 0, for n >= 1.

(3) A(x) = x - A(x^2) + A(x^3). - Paul D. Hanna, Jan 05 2026

Example

G.f. A(x) = x - x^2 + x^3 + x^4 - 2*x^6 - x^8 + x^9 + 3*x^12 + x^16 - 3*x^18 - 4*x^24 + x^27 - x^32 + 6*x^36 + 5*x^48 - 4*x^54 + x^64 - 10*x^72 + x^81 - 6*x^96 + ...
where A(x) equals the sum of the following series
A(x) = (x + x^3 + x^9 + x^27 + ... + x^(3^k) + ...)
  - (x^2 + 2*x^6 + 3*x^18 + 4*x^54 + ... + (k+1)*x^(2*3^k) + ...)
  + (x^4 + 3*x^12 + 6*x^36 + 10*x^108 + ... + C(2+k,k)*x^(2^2*3^k) + ...)
  - (x^8 + 4*x^24 + 10*x^72 + 20*x^216 + ... + C(3+k,k)*x^(2^3*3^k) + ...)
  + (x^16 + 5*x^48 + 15*x^144 + 35*x^432 + ... + C(4+k,k)*x^(2^4*3^k) + ...)
  + ... + (-1)^n * Sum_{k>=0} binomial(n+k,k) * x^(2^n*3^k) + ...
which implies A(x) = x - A(x^2) + A(x^3).

Prog

(PARI) {a(n) = my(p2, p3); if(n<1, 0, p2 = valuation(n, 2); p3 = valuation(n, 3);
if(n/(2^p2*3^p3)>1, 0, (-1)^p2 * binomial(p2 + p3, p2) ))}
for(n=1, 100, print1(a(n), ", "))

Crossrefs

Cf. A385910, A061984.

Sequence in context: A066620 A219023 A025427 * A348536 A245963 A382138

Adjacent sequences: A385908 A385909 A385910 * A385912 A385913 A385914

Keyword

sign

Author

Paul D. Hanna, Jul 12 2025

Status

approved