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Let A_n(k) denote the number of different ways to distribute k indistinguishable balls into n constrained urns, with capacities c_1,\cdots, c_n . We consider the normalized \ counting \ functions \ \varphi_n(x) = \gamma_nA_n(|\varrho x|) , where \varphi_n, \varrho_n > 0 are appropriate constants such that supp (\varphi_n) = [0,1] and \int^1_0 \varphi_n (x)dx = 1 . It is shown here that, if (c_n)_{n \in \mathbb N} is asymptotically \ geometric \ with \ weight \ q > \frac{3}{2} , i.e. if q^{–n}c_n converges to some positive real number, then the functions \varphi_n converge to some C^{\infty} -function \varphi on \mathbb R . This function \varphi is the unique solution of the integral equation \varphi(x) = \frac{q}{q–1} \int^{qx}{qx–q+1} \varphi (t)dt satisfying supp \varphi \in [0,1] and \int^1_0 \varphi(t)dt = 1 . Moreover, if q > 2 , it is shown that \varphi is a polynomial on each interval outside a Cantor-like set in the interval [0, 1].