OFFSET
0,2
COMMENTS
a(n) is the number of Green's H classes in the semigroup of n X n matrices over GF(2) (cf. A359313). - Geoffrey Critzer, Jun 20 2023
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..60
Wikipedia, Green's relations
FORMULA
a(n) ~ c * 2^(n^2/2), where c = 18.0796893855819714431... if n is even and c = 18.02126069886312898683... if n is odd. - Vaclav Kotesovec, Jun 23 2014
Sum_{n>=0} a(n)*x^n/A005329(n)^2 = E(x)^2 where E(x) = Sum_{n>=0} x^n/A005329(n)^2. - Geoffrey Critzer, Jun 20 2023_
EXAMPLE
G.f.: A(x) = 1 + 2*x + 11*x^2 + 100*x^3 + 1677*x^4 + 49974*x^5 + 2801567*x^6 + ...
Related integer series:
A(x)^(1/2) = 1 + x + 5*x^2 + 45*x^3 + 781*x^4 + 23981*x^5 + 1371885*x^6 + 145101805*x^7 + 29560055405*x^8 + ... + A243951(n)*x^n + ...
A022166, the triangle of q-binomial coefficients for q=2, begins:
1;
1, 1;
1, 3, 1;
1, 7, 7, 1;
1, 15, 35, 15, 1;
1, 31, 155, 155, 31, 1;
1, 63, 651, 1395, 651, 63, 1;
1, 127, 2667, 11811, 11811, 2667, 127, 1; ...
from which we can illustrate the initial terms of this sequence:
a(0) = 1^2 = 1;
a(1) = 1^2 + 1^2 = 2;
a(2) = 1^2 + 3^2 + 1^2 = 11;
a(3) = 1^2 + 7^2 + 7^2 + 1^2 = 100;
a(4) = 1^2 + 15^2 + 35^2 + 15^2 + 1^2 = 1677;
a(5) = 1^2 + 31^2 + 155^2 + 155^2 + 31^2 + 1^2 = 49974;
a(6) = 1^2 + 63^2 + 651^2 + 1395^2 + 651^2 + 63^2 + 1^2 = 2801567; ...
MATHEMATICA
a[n_] := Sum[QBinomial[n, k, 2]^2, {k, 0, n}]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Apr 09 2016 *)
PROG
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 21 2014
STATUS
approved