The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A321791 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A321791

Table read by descending antidiagonals: T(n,k) is the number of unoriented cycles (bracelets) of length n using up to k available colors.
(history; published version)

#39 by Michel Marcus at Mon Feb 08 02:11:38 EST 2021

STATUS

reviewed

approved

#38 by Joerg Arndt at Sun Feb 07 23:48:24 EST 2021

STATUS

proposed

reviewed

#37 by Petros Hadjicostas at Sun Feb 07 23:12:26 EST 2021

STATUS

editing

proposed

#36 by Petros Hadjicostas at Sun Feb 07 23:12:23 EST 2021

FORMULA

O.g.f. for column k >= 0: Sum_{n>=0} T(n,k)*x^n = 3/4 + (1 + k*x)^2/(4*(1 - k*x^2)) - (1/2) * Sum_{d >= 1} (phi(d)/d) * log(1 - k*x^d). - Petros Hadjicostas, Feb 07 2021

#35 by Petros Hadjicostas at Sun Feb 07 23:01:21 EST 2021

FORMULA

O.g.f. for column k: Sum_{n>=0} T(n,k)*x^n = 13/4 + (1 + k*x)^2/(4*(1 - k*x^2)) - (1/2) * Sum_{d >= 1} (phi(d)/d) * log(1 - k*x^d). - Petros Hadjicostas, Feb 07 2021

#34 by Petros Hadjicostas at Sun Feb 07 23:00:39 EST 2021

FORMULA

O.g.f. for column k: Sum_{n>=0} T(n,k)*x^n = 1/4 + (1 + k*x)^2/(4*(1 - k*x^2)) - (1/2) * Sum_{d >= 1} (phi(d)/d) * log(1 - k*x^d). - Petros Hadjicostas, Feb 07 2021

STATUS

proposed

editing

#33 by Petros Hadjicostas at Sun Feb 07 22:28:10 EST 2021

STATUS

editing

proposed

#32 by Petros Hadjicostas at Sun Feb 07 22:28:06 EST 2021

FORMULA

Linear recurrence for row n: T(n,k) = Sum_{j=0..n} -binomial(j-n-1,j+1) * T(n,k-1-j) for k >= n + 1.

STATUS

proposed

editing

#31 by Petros Hadjicostas at Sun Feb 07 22:04:27 EST 2021

STATUS

editing

proposed

#30 by Petros Hadjicostas at Sun Feb 07 22:04:24 EST 2021

FORMULA

T(n,k) = [n==0] + [n>0] * (k^floor((n+1)/2) + k^ceiling((n+1)/2)) / 4 + (1/2n(2*n)) * Sum_{d|n} phi(d) * k^(n/d)).

STATUS

approved

editing