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A118195 Self-convolution square-root of A118191, where A118191 is column 0 of the matrix square of triangle A118190 with A118190(n,k) = (5^k)^(n-k). 2

%I

%S 1,1,3,23,411,15771,1353045,252512065,106798723795,99080638950595,

%T 208993838938550873,968425792397232696773,10208662119796586878979989,

%U 236472963735267887311598074949,12462692176683507314938059670486683

%N Self-convolution square-root of A118191, where A118191 is column 0 of the matrix square of triangle A118190 with A118190(n,k) = (5^k)^(n-k).

%C In general, sqrt( Sum_{n>=0} x^n/(1 - q^n*x) ) is an integer series whenever q == 1 (mod 4).

%H G. C. Greubel, <a href="/A118195/b118195.txt">Table of n, a(n) for n = 0..75</a>

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

%e A(x) = 1 + x + 3*x^2 + 23*x^3 + 411*x^4 + 15771*x^5 + ...

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

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

%t With[{m = 30}, CoefficientList[Series[Sqrt[Sum[x^j/(1 - 5^j*x), {j, 0, m + 2}]], {x, 0, m}], x]] (* _G. C. Greubel_, Jun 30 2021 *)

%o (PARI) a(n)=polcoeff(sqrt(sum(k=0,n,sum(j=0, k, (5^j)^(k-j) )*x^k+x*O(x^n))),n)

%o (MAGMA)

%o m:=30;

%o R<x>:=PowerSeriesRing(Rationals(), m);

%o Coefficients(R!( Sqrt( (&+[x^j/(1-5^j*x): j in [0..m+2]]) ) )); // _G. C. Greubel_, Jun 30 2021

%o (Sage)

%o m=30;

%o def A118195_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P( sqrt(sum( x^j/(1-5^j*x) for j in (0..m+2))) ).list()

%o A118195_list(m) # _G. C. Greubel_, Jun 30 2021

%Y Cf. A118190, A118191.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Apr 15 2006

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Last modified January 20 15:42 EST 2022. Contains 350472 sequences. (Running on oeis4.)