A377573 Cogrowth sequence for the 14-element dihedral group D7 = .

1, 0, 1, 0, 3, 0, 10, 1, 35, 9, 126, 55, 462, 286, 1717, 1365, 6451, 6188, 24463, 27132, 93518, 116281, 360031, 490337, 1394582, 2043275, 5430530, 8439210, 21242341, 34621041, 83411715, 141290436, 328589491, 574274008, 1297937234, 2326683921, 5138431851

Offset: 0

Author

Sean A. Irvine, Nov 01 2024

Keywords

• nonn
• easy

Comments

Taking the overlay of the two generating functions in the bisections A072844 and A072266, shows that a(n) = A094052(n-1), n>0. - R. J. Mathar, Nov 05 2024

Examples

			a(4) = 3 corresponds to the TTTT = TSTS = STST = 1. Note: TSTS = (TSTS)(TT) = T(STST)T = TT = 1.
a(9) = 9 corresponds to the words SSSSSSSTT = SSSSSSTTS = SSSSSTTSS = SSSSTTSSS = SSSTTSSSS = SSTTSSSSS = STTSSSSSS = TTSSSSSSS = TSSSSSSST = 1.
		

Links

• Sean A. Irvine, Table of n, a(n) for n = 0..1000
• H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 134.
• Haggai Liu, Enumerative Properties of Cogrowth Series on Free Products of Finite Groups, ACA 2021 Session on Algorithmic Combinatorics, 2021.
• Sean A. Irvine, Java program (github)

Crossrefs

Bisections: A072266, A072844.

Cf. A052964 (D5), A007583 (D6), A007582 (D8).

Formula

G.f.: F_7(x) where F_n(x) = 1/2 + (1/(2*n)) * Sum_{j=0..n-1} 1 / (1 - 2*cos(2*Pi*j/n)*x).