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A118191 Row sums of triangle A118190: a(n) = Sum_{k=0..n} 5^(k*(n-k)) for n>=0. 4

%I

%S 1,2,7,52,877,32502,2740627,507843752,214111484377,198376465625002,

%T 418186492923828127,1937270172119160156252,20419262349796295263671877,

%U 472966350615029335022460937502,24925857360591180741786959228515627

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

%C 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 G. C. Greubel, <a href="/A118191/b118191.txt">Table of n, a(n) for n = 0..70</a>

%F G.f.: A(x) = Sum_{n>=0} x^n/(1-5^n*x).

%e A(x) = 1/(1-x) + x/(1-5*x) + x^2/(1-25*x) + x^3/(1-125*x) + ...

%e = 1 + 2*x + 7*x^2 + 52*x^3 + 877*x^4 + 32502*x^5 + ...

%t Table[Sum[5^(k*(n-k)), {k,0,n}], {n,0,30}] (* _G. C. Greubel_, Jun 29 2021 *)

%o (PARI) a(n)=sum(k=0, n, (5^k)^(n-k))

%o (MAGMA) [(&+[5^(k*(n-k)): k in [0..n]]): n in [0..30]]; // _G. C. Greubel_, Jun 29 2021

%o (Sage) [sum(5^(k*(n-k)) for k in (0..n)) for n in (0..30)] # _G. C. Greubel_, Jun 29 2021

%Y Cf. A118190 (triangle), A118192 (antidiagonal sums), A118195 (A(x)^(1/2)).

%K nonn

%O 0,2

%A _Paul D. Hanna_, Apr 15 2006

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Last modified November 28 19:19 EST 2021. Contains 349415 sequences. (Running on oeis4.)