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A118191 Row sums of triangle A118190: a(n) = Sum_{k=0..n} 5^(k*(n-k)) for n>=0.

%I A118191 #10 Aug 20 2025 09:57:23
%S A118191 1,2,7,52,877,32502,2740627,507843752,214111484377,198376465625002,
%T A118191 418186492923828127,1937270172119160156252,20419262349796295263671877,
%U A118191 472966350615029335022460937502,24925857360591180741786959228515627
%N A118191 Row sums of triangle A118190: a(n) = Sum_{k=0..n} 5^(k*(n-k)) for n>=0.
%C A118191 Self-convolution of A118195; in general, sqrt(Sum_{n>=0} x^n/(1-q^n*x)) is an integer series whenever q == 1 (mod 4). Also equals column 0 of the matrix square of triangle A118190, where [A118190^2](n,k) = a(n-k)*5^(k*(n-k)) for n>=k>=0.
%H A118191 G. C. Greubel, <a href="/A118191/b118191.txt">Table of n, a(n) for n = 0..70</a>
%F A118191 G.f.: A(x) = Sum_{n>=0} x^n/(1-5^n*x).
%F A118191 a(n) ~ c * 5^(n^2/4), where c = EllipticTheta[3, 0, 1/5] (in Mathematica) = JacobiTheta3(0,1/5) (in Maple) = 1.40320102401310720671088653743895... if n is even and c = EllipticTheta[2, 0, 1/5] = JacobiTheta2(0,1/5) = 1.39106543858832939481476315485543... if n is odd. - _Vaclav Kotesovec_, Aug 20 2025
%e A118191 A(x) = 1/(1-x) + x/(1-5*x) + x^2/(1-25*x) + x^3/(1-125*x) + ...
%e A118191   = 1 + 2*x + 7*x^2 + 52*x^3 + 877*x^4 + 32502*x^5 + ...
%t A118191 Table[Sum[5^(k*(n-k)), {k,0,n}], {n,0,30}] (* _G. C. Greubel_, Jun 29 2021 *)
%o A118191 (PARI) a(n)=sum(k=0, n, (5^k)^(n-k))
%o A118191 (Magma) [(&+[5^(k*(n-k)): k in [0..n]]): n in [0..30]]; // _G. C. Greubel_, Jun 29 2021
%o A118191 (Sage) [sum(5^(k*(n-k)) for k in (0..n)) for n in (0..30)] # _G. C. Greubel_, Jun 29 2021
%Y A118191 Cf. A118190 (triangle), A118192 (antidiagonal sums), A118195 (A(x)^(1/2)).
%K A118191 nonn
%O A118191 0,2
%A A118191 _Paul D. Hanna_, Apr 15 2006