A375181 Inverse binomial transform of A376277.

1, 1, 2, 3, 6, 12, 28, 70, 189, 533, 1551, 4599, 13807, 41784, 127144, 388373, 1189661, 3651910, 11228851, 34571914, 106555917, 328713138, 1014797705, 3134841053, 9689148780, 29961083746, 92683964271, 286816872102, 887849075464, 2749110140301, 8514323952447

Offset: 0

Author

Thomas Scheuerle, Sep 23 2024

Keywords

• nonn
• less

Comments

The Hankel sequence transform gives {1, 1, 1, 1, 1, ...}.

The Hankel sequence transform starting with the second term gives {1, -3, -2, 1, 3, 2, -1, -3, -2, ...}.

The periodic pattern in the continued fractions of the generating function starts after some prefix, this causes high complexity in the generating function and makes a nice combinatorial interpretation less likely, therefore the keyword "less" was considered.

Crossrefs

Cf. A059738, A376277.

Programs

PARI

my(N=30, x='x+O('x^N)); Vec((2*x^2-sqrt(-3*x^2-2*x+1)+3*x-1)/(-5*x^3+sqrt(-3*x^2-2*x+1)*(x^2+x-1)+4*x-1))

Formula

G.f.: 1/(1-1*x/(1-1*x/(1+1*x/(1+1*x/(1-3*x/(1-(1/3)*x/(1-(2/3)*x/(1-(3/2)*x/(1+(1/2)*x/(...)))))))))), a continued fraction expansion. The coefficients of x are {-1, -1, 1, 1, -3, -(1/3), -(2/3), -(3/2), (1/2), 2, -3, ...}. The numerators will repeat {1, 2, 3} the denominators {1, 1, 2, 2, 3, 3} the sign repeats {-,-,-,-,+,+}.

G.f.: (2*x^2 - sqrt(-3*x^2 - 2*x + 1) + 3*x - 1)/(-5*x^3 + sqrt(-3*x^2 - 2*x + 1)*(x^2 + x - 1) + 4*x - 1)

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*A376277(k).

D-finite with recurrence (-n+3)*a(n) +(7*n-25)*a(n-1) +2*(-6*n+25)*a(n-2) +10*(-n+4)*a(n-3) +2*(16*n-73)*a(n-4) +(n-5)*a(n-5) +21*(-n+5)*a(n-6)=0. - R. J. Mathar, Oct 24 2024