A243950 Sum of the squares of q-binomial coefficients for q=2 in row n of triangle A022166, for n >= 0.

1, 2, 11, 100, 1677, 49974, 2801567, 293257480, 59426801521, 23154622451162, 17786849024835651, 26694462878992491180, 79786045619298591331605, 469805503062346255040726910, 5538428985758278544518994721255, 129179377104085570277109465712798800, 6048537751321912538368011648067930447545

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

(PARI) {A022166(n, k)=polcoeff(x^k/prod(j=0, k, 1-2^j*x+x*O(x^n)), n)}
{a(n)=sum(k=0, n, A022166(n, k)^2)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A022166, A243951, A359313.

Sequence in context: A205806 A220433 A318007 * A056732 A157715 A001271

Adjacent sequences: A243947 A243948 A243949 * A243951 A243952 A243953

Keyword

nonn

Author

Paul D. Hanna, Jun 21 2014

Status

approved