%I #32 Jul 26 2024 10:21:28

%S 0,0,0,1,5,13,27,47,75,112,156,214,278,358,444,552,660,796,930,1099,

%T 1259,1457,1649,1885,2101,2377,2623,2933,3221,3569,3879,4279,4623,

%U 5056,5452,5926,6334,6878,7328,7892,8404,9018,9540,10228,10788,11504,12142,12898

%N Number of 2 X 2 matrices M of positive integers such that permanent(M) < n.

%C For a guide to related sequences, see A211795.

%H Robert Israel, <a href="/A212151/b212151.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) + A212240(n) = n^4.

%F a(n) = Sum_{k=1..n-1} Sum_{i=1..n-1} d(k) * floor((n-k-1)/i), where d(k) is the number of divisors of k (A000005). - _Wesley Ivan Hurt_, Nov 16 2017

%F G.f.: (x/(1-x))*(Sum_{i>=1} x^i/(1-x^i))^2. - _Robert Israel_, Nov 16 2017

%F from _Ridouane Oudra_, Oct 10 2023: (Start)

%F a(n) = Sum_{i=1..n-1} Sum_{j=1..n-1} tau(i*j)*floor((n-1)/(i+j)) ;

%F a(n) = Sum_{i=1..n-1} Sum_{j=1..i-1} tau(j)*tau(i-j) ;

%F a(n+2) = Sum_{i=1..n} A055507(i). (End)

%p N:= 100: # to get a(0)..a(N)

%p g:= z*(1-z)^(-1)*add(z^i/(1-z^i),i=1..N-2)^2:

%p S:=series(g,z,N+1):

%p seq(coeff(S,z,n),n=0..N); # _Robert Israel_, Nov 16 2017

%t t = Compile[{{n, _Integer}}, Module[{s = 0},

%t (Do[If[w*x + y*z < n, s = s + 1],

%t {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];

%t Map[t[#] &, Range[0, 40]] (* A212151 *)

%t (* _Peter J. C. Moses_, Apr 13 2012 *)

%o (Python)

%o from sympy import divisor_count

%o def A212151(n): return sum((sum(divisor_count(i+1)*divisor_count(j-i) for i in range(j>>1))<<1)+(divisor_count(j+1>>1)**2 if j&1 else 0) for j in range(1,n-1)) # _Chai Wah Wu_, Jul 26 2024

%Y Cf. A000005, A211795, A212240.

%Y Cf. A055507.

%K nonn,easy

%O 0,5

%A _Clark Kimberling_, May 07 2012