A305625 Number of chiral pairs of rows of n colors with exactly 5 different colors.

0, 0, 0, 0, 60, 900, 8400, 63000, 417000, 2551440, 14802900, 82763100, 450501660, 2404493700, 12645952200, 65771370000, 339164682000, 1737485315640, 8855354531100, 44952362878500, 227475739300260, 1148269299919500, 5785013208282000, 29100046926951000, 146201097996135000, 733811769167043840, 3680292427100043300, 18446421887430345900, 92412024657725026860, 462780012983867889300, 2316780309783100387800

Offset

1,5

Comments

If the row is achiral, i.e., the same as its reverse, we ignore it. If different from its reverse, we count it and its reverse as a chiral pair.

Links

Table of n, a(n) for n=1..31.

Formula

a(n) = (k!/2) * (S2(n,k) - S2(ceiling(n/2),k)), with k=5 colors used and where S2(n,k) is the Stirling subset number A008277.

a(n) = (A001118(n) - A056456(n)) / 2.

a(n) = A001118(n) - A056312(n) = A056312(n) - A056456(n).

G.f.: -(k!/2) * (x^(2k-1) + x^(2k)) / Product_{j=1..k} (1 - j*x^2) + (k!/2) * x^k / Product_{j=1..k} (1 - j*x) with k=5 colors used.

Example

For a(5) = 60, the chiral pairs are the 5! = 120 permutations of ABCDE, each paired with its reverse.

Mathematica

k=5; Table[(k!/2) (StirlingS2[n, k] - StirlingS2[Ceiling[n/2], k]), {n, 1, 40}]

Prog

(PARI) a(n) = 60*(stirling(n, 5, 2) - stirling(ceil(n/2), 5, 2)); \\ Altug Alkan, Sep 26 2018

Crossrefs

Fifth column of A305622.

A056456(n) is number of achiral rows of n colors with exactly 5 different colors.

Cf. A001118, A056312.

Sequence in context: A259539 A138898 A229368 * A056321 A056312 A268967

Adjacent sequences: A305622 A305623 A305624 * A305626 A305627 A305628

Keyword

nonn,easy

Author

Robert A. Russell, Jun 06 2018

Status

approved