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A203508 G.f.: Product_{n>=0} (1+a(n)*x^(n+1))^3 = Sum_{n>=0} a(n)*x^n.

%I A203508 #13 Nov 20 2020 09:07:58
%S A203508 1,3,12,64,354,2160,13518,88374,584409,3980736,27291825,190771995,
%T A203508 1339606882,9539905173,68140709607,492072701284,3560322659379,
%U A203508 25984705308156,189940383845883,1398103463338725,10302144982761213,76363018655732307,566463003067056519
%N A203508 G.f.: Product_{n>=0} (1+a(n)*x^(n+1))^3 = Sum_{n>=0} a(n)*x^n.
%H A203508 Alois P. Heinz, <a href="/A203508/b203508.txt">Table of n, a(n) for n = 0..400</a>
%e A203508 G.f.: A(x) = 1 + 3*x + 12*x^2 + 64*x^3 + 354*x^4 + 2160*x^5 + 13518*x^6 +...
%e A203508 where
%e A203508 A(x) = ((1+x)*(1+3*x^2)*(1+12*x^3)*(1+64*x^4)*(1+354*x^5)*...)^3.
%e A203508 Related expansion:
%e A203508 A(x)^(1/3) = 1 + x + 3*x^2 + 15*x^3 + 76*x^4 + 454*x^5 + 2742*x^6 +...
%p A203508 A:= proc(n) option remember; local i, p, q; if n=0 then 1 else
%p A203508       p, q:= A(n-1), 1; for i from 0 to n-1 do q:= convert(
%p A203508         series(q*(1+coeff(p, x, i)*x^(i+1))^3, x, n+1), polynom)
%p A203508       od: q fi
%p A203508     end:
%p A203508 a:= n-> coeff(A(n), x, n):
%p A203508 seq(a(n), n=0..30);  # _Alois P. Heinz_, Aug 01 2013
%t A203508 a[n_] := a[n] = SeriesCoefficient[Product[(1+a[k] x^(k+1))^3, {k, 0, n-1}], {x, 0, n}];
%t A203508 a /@ Range[0, 30] (* _Jean-François Alcover_, Nov 20 2020 *)
%o A203508 (PARI) {a(n) = polcoeff(prod(k=0, n-1, (1+a(k)*x^(k+1)+x*O(x^n)))^3, n)}
%Y A203508 Cf. A032305, A093635.
%K A203508 nonn
%O A203508 0,2
%A A203508 _Paul D. Hanna_, Jan 02 2012