OFFSET
0,1
COMMENTS
This is just one row of a double sequence a(n,m) for n = 0,1,2, ... and m = 0,1,2,...: a(n,m) = 2^(n+m+1)*(Sum_{r=0..m} (2^(-r) * binomial(n, n-r)* binomial(m, r))) - 1, with 0 <= m <= n and a(0,0)=1.
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
V. Murali, FSRG, Rhodes University.
V. Murali and B. B. Makamba, Fuzzy subgroups of finite Abelian groups, Far East Journal of Mathematical Sciences (FJMS), Vol. 14, No. 1 (2004), pp. 113-125.
V. Murali and B. B. Makamba, Counting the fuzzy subgroups of an Abelian group of order p^n q^m, Fuzzy Sets And Systems, Vol. 144, No.3 (2004), pp. 459-470.
Index entries for linear recurrences with constant coefficients, signature (7,-18,20,-8).
FORMULA
a(n) = (2^n)*(n^2 + 7n + 8) - 1 for n=0..14.
G.f.: (12*x^2 - 18*x + 7) / ((x-1)*(2*x-1)^3). - Colin Barker, Jan 15 2015
EXAMPLE
a(3) = 303. A fuzzy subgroup is simply a chain of subgroups in the lattice of subgroups. Counting of chains in the lattice of subgroups of Z_{p^3} + Z_2 gives us a(3) = 303. The two papers cited describe the counting process using fuzzy subgroup concept.
MATHEMATICA
LinearRecurrence[{7, -18, 20, -8}, {7, 31, 103, 303}, 30] (* Harvey P. Dale, Dec 31 2015 *)
PROG
(PARI) Vec((12*x^2-18*x+7)/((x-1)*(2*x-1)^3) + O(x^100)) \\ Colin Barker, Jan 15 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Venkat Murali (v.murali(AT)ru.ac.za), May 25 2005
EXTENSIONS
Corrected by T. D. Noe, Nov 08 2006
STATUS
approved