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).

1, 1, 3, 23, 411, 15771, 1353045, 252512065, 106798723795, 99080638950595, 208993838938550873, 968425792397232696773, 10208662119796586878979989, 236472963735267887311598074949, 12462692176683507314938059670486683

Offset: 0

Author

Paul D. Hanna, Apr 15 2006

Keywords

• nonn

Comments

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

Examples

			A(x) = 1 + x + 3*x^2 + 23*x^3 + 411*x^4 + 15771*x^5 + ...
A(x)^2 = 1 + 2*x + 7*x^2 + 52*x^3 + 877*x^4 + 32502*x^5 + ...
= 1/(1-x) + x/(1-5x) + x^2/(1-25x) + x^3/(1-125x) + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..75

Crossrefs

Cf. A118190, A118191.

Programs

Magma

m:=30;
R:=PowerSeriesRing(Rationals(), m);
Coefficients(R!( Sqrt( (&+[x^j/(1-5^j*x): j in [0..m+2]]) ) )); // G. C. Greubel, Jun 30 2021

Mathematica

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 *)

PARI

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

Sage

m=30;
def A118195_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( sqrt(sum( x^j/(1-5^j*x) for j in (0..m+2))) ).list()
A118195_list(m) # G. C. Greubel, Jun 30 2021

Formula

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

a(n) ~ A118191(n)/2. - Vaclav Kotesovec, Aug 20 2025