cp's OEIS Frontend

A199015 G.f.: 1/(1-x) * Product_{n>=1} (1 - x^(2*n))^2/(1 - x^(2*n-1))^2.

%I A199015 #32 Sep 02 2025 14:18:46
%S A199015 1,3,4,6,8,8,11,13,13,15,17,19,20,22,22,24,28,28,30,30,31,35,37,37,39,
%T A199015 41,41,43,45,47,48,52,52,52,54,54,58,60,62,64,64,64,67,69,69,71,75,75,
%U A199015 77,79,79,83,83,83,83,87,90,92,94,94,96,98,98,98,100,102,106,108,108,110,112,112,115,117,117,117,121,121,123
%N A199015 G.f.: 1/(1-x) * Product_{n>=1} (1 - x^(2*n))^2/(1 - x^(2*n-1))^2.
%C A199015 Equals the partial sums of A008441, where A008441(n) is the number of ways of writing n as the sum of 2 triangular numbers.
%H A199015 G. C. Greubel, <a href="/A199015/b199015.txt">Table of n, a(n) for n = 0..1000</a>
%F A199015 Lim_{n->infinity} a(n)/n = Pi/2.
%F A199015 a(n) = Sum_{k=0..n} Sum_{d|4*k+1} (-1)^floor(d/2). - _Michael Somos_ [see A008441]
%F A199015 G.f.: 1/(1-x) * Sum_{n>=0} x^n/(1 - x^(4*n + 1)). - _Michael Somos_ [see A008441]
%F A199015 G.f.: theta_2(sqrt(x))^2/(4*x^(1/4)*(1 - x)), where theta_2() is the Jacobi theta function. - _Ilya Gutkovskiy_, Apr 15 2018
%e A199015 G.f.: A(x) = 1 + 3*x + 4*x^2 + 6*x^3 + 8*x^4 + 8*x^5 + 11*x^6 + 13*x^7 + ...
%e A199015 where the g.f. equals the product:
%e A199015 A(x) = 1/(1-x) * (1-x^2)^2/(1-x)^2 * (1-x^4)^2/(1-x^3)^2 * (1-x^6)^2/(1-x^5)^2 * ...
%e A199015 Illustrate the limit a(n)/n = Pi/2:
%e A199015 a(10)/10 = 1.7, a(10^2)/10^2 = 1.58, a(10^3)/10^3 = 1.574, a(10^4)/10^4 = 1.5704, a(10^5)/10^5 = 1.57086, a(10^6)/10^6 = 1.570784, a(10^7)/10^7 = 1.5707972, ...
%t A199015 CoefficientList[Series[EllipticTheta[2, 0, Sqrt[x]]^2/(4*x^(1/4)*(1 - x)), {x, 0, 50}], x] (* _G. C. Greubel_, Aug 13 2018 *)
%t A199015 a = Compile[{{n, _Integer}}, Block[{c = 0}, Do[ c ++, {i, 0, (1 + Sqrt[1 + 8 n])/4}, {j, 0, (1 + Sqrt[1 + 8(n - i (i + 1))])/2}]; c]]; Array[a, 79, 0] (* _Robert G. Wilson v_, Aug 28 2025 *)
%o A199015 (PARI) {a(n)=sum(k=0,n,sumdiv(4*k+1, d, (-1)^(d\2)))}
%o A199015 (PARI) {a(n)=polcoeff(1/(1-x)*prod(m=1,n\2+1,(1-x^(2*m))/(1-x^(2*m-1)+x*O(x^n)))^2,n)}
%Y A199015 Cf. A008441, A008443.
%K A199015 nonn,changed
%O A199015 0,2
%A A199015 _Paul D. Hanna_, Nov 02 2011