%I #33 Aug 19 2021 10:55:19

%S 0,1,0,0,2,0,0,0,0,3,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,

%T 0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,

%U 0,0,0,0,0,0,0,0,0,0,0,0,0

%N Expansion of Sum_{n>=0} n*q^(n^2).

%C Multiplicative with a(p^(2e)) = p^e, a(p^(2e+1)) = 0. - _Mitch Harris_, Jun 09 2005

%C a(n) is the square root of n if n is square, zero otherwise. - _Carl R. White_, May 23 2009

%H Reinhard Zumkeller, <a href="/A037213/b037213.txt">Table of n, a(n) for n = 0..10000</a>

%F Dirichlet generating function: zeta(2*s-1). - _Franklin T. Adams-Watters_, Sep 11 2005.

%F a(n) = sqrt(n) * floor( cos^2(Pi * sqrt(n)) ). - _Carl R. White_, May 23 2009

%F a(n) = A000196(n) * A010052(n). - _Reinhard Zumkeller_, Jan 27 2010

%F Sum_{k=1..n} a(k) ~ n/2. - _Vaclav Kotesovec_, Aug 19 2021

%t Table[If[IntegerQ[n^(1/2)], n^(1/2), 0], {n, 0, 100}] (* _Geoffrey Critzer_, Feb 21 2015 *)

%o (Haskell)

%o a037213 n = if n == r ^ 2 then r else 0 where r = a000196 n

%o a037213_list = zipWith (*) a010052_list a000196_list

%o -- _Reinhard Zumkeller_, Nov 09 2012

%Y Cf. A000196, A010052.

%K nonn,mult

%O 0,5

%A _N. J. A. Sloane_