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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
 1, 1, 3, 23, 411, 15771, 1353045, 252512065, 106798723795, 99080638950595, 208993838938550873, 968425792397232696773, 10208662119796586878979989, 236472963735267887311598074949, 12462692176683507314938059670486683 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS In general, sqrt( Sum_{n>=0} x^n/(1 - q^n*x) ) is an integer series whenever q == 1 (mod 4). LINKS G. C. Greubel, Table of n, a(n) for n = 0..75 FORMULA G.f.: A(x) = sqrt( Sum_{n>=0} x^n/(1-5^n*x) ). EXAMPLE 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) + ... 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 *) PROG (PARI) a(n)=polcoeff(sqrt(sum(k=0, n, sum(j=0, k, (5^j)^(k-j) )*x^k+x*O(x^n))), n) (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 (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 CROSSREFS Cf. A118190, A118191. Sequence in context: A178315 A210910 A280654 * A055326 A271851 A133338 Adjacent sequences:  A118192 A118193 A118194 * A118196 A118197 A118198 KEYWORD nonn AUTHOR Paul D. Hanna, Apr 15 2006 STATUS approved

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Last modified November 28 16:34 EST 2021. Contains 349413 sequences. (Running on oeis4.)