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

%I A243950 #31 Aug 03 2025 21:05:06
%S A243950 1,2,11,100,1677,49974,2801567,293257480,59426801521,23154622451162,
%T A243950 17786849024835651,26694462878992491180,79786045619298591331605,
%U A243950 469805503062346255040726910,5538428985758278544518994721255,129179377104085570277109465712798800,6048537751321912538368011648067930447545
%N A243950 Sum of the squares of q-binomial coefficients for q=2 in row n of triangle A022166, for n >= 0.
%C A243950 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
%H A243950 Paul D. Hanna, <a href="/A243950/b243950.txt">Table of n, a(n) for n = 0..60</a>
%H A243950 Wikipedia, <a href="http://en.wikipedia.org/wiki/Green%27s_relations">Green's relations</a>
%F A243950 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
%F A243950 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
%e A243950 G.f.: A(x) = 1 + 2*x + 11*x^2 + 100*x^3 + 1677*x^4 + 49974*x^5 + 2801567*x^6 + ...
%e A243950 Related integer series:
%e A243950 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 + ...
%e A243950 A022166, the triangle of q-binomial coefficients for q=2, begins:
%e A243950   1;
%e A243950   1,   1;
%e A243950   1,   3,    1;
%e A243950   1,   7,    7,     1;
%e A243950   1,  15,   35,    15,     1;
%e A243950   1,  31,  155,   155,    31,    1;
%e A243950   1,  63,  651,  1395,   651,   63,   1;
%e A243950   1, 127, 2667, 11811, 11811, 2667, 127, 1; ...
%e A243950 from which we can illustrate the initial terms of this sequence:
%e A243950   a(0) = 1^2 = 1;
%e A243950   a(1) = 1^2 + 1^2 = 2;
%e A243950   a(2) = 1^2 + 3^2 + 1^2 = 11;
%e A243950   a(3) = 1^2 + 7^2 + 7^2 + 1^2 = 100;
%e A243950   a(4) = 1^2 + 15^2 + 35^2 + 15^2 + 1^2 = 1677;
%e A243950   a(5) = 1^2 + 31^2 + 155^2 + 155^2 + 31^2 + 1^2 = 49974;
%e A243950   a(6) = 1^2 + 63^2 + 651^2 + 1395^2 + 651^2 + 63^2 + 1^2 = 2801567; ...
%t A243950 a[n_] := Sum[QBinomial[n, k, 2]^2, {k, 0, n}]; Table[a[n], {n, 0, 16}] (* _Jean-François Alcover_, Apr 09 2016 *)
%o A243950 (PARI) {A022166(n, k)=polcoeff(x^k/prod(j=0, k, 1-2^j*x+x*O(x^n)), n)}
%o A243950 {a(n)=sum(k=0,n,A022166(n, k)^2)}
%o A243950 for(n=0,20,print1(a(n),", "))
%Y A243950 Cf. A022166, A243951, A359313.
%K A243950 nonn
%O A243950 0,2
%A A243950 _Paul D. Hanna_, Jun 21 2014