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A073712 Self-convolution of A073711.

%I A073712 #31 Jul 19 2016 10:50:00
%S A073712 1,2,3,6,7,12,16,26,31,42,59,72,104,116,184,186,303,282,497,406,783,
%T A073712 612,1224,840,1856,1232,2784,1656,4136,2376,6008,3138,8735,4362,12345,
%U A073712 5754,17693,7756,24432,10170,34471,13302,46771,17688,65144,22296,87008
%N A073712 Self-convolution of A073711.
%C A073712 The g.f. G(x) of A073711 satisfies: G(x) = G(x^2) + x*G(x^2)^2.
%C A073712 The terms of this sequence found at odd-indexed positions are equal to twice that of A194279, which equals the self-convolution cube of A073711.
%H A073712 Reinhard Zumkeller, <a href="/A073712/b073712.txt">Table of n, a(n) for n = 0..10000</a>
%F A073712 a(n) = A073711(2*n+1) for n>=0.
%F A073712 a(2*n+1) = 2*A194279(n) for n>=0, where A194279 equals the self-convolution cube of A073711.
%t A073712 nmax = 46; max = 2*nmax+1; f[x_] := Sum[ a[k]*x^k, {k, 0, max}]; a[0] = a[1] = a[2] = 1; coes = CoefficientList[ Series[ f[x] - f[x^2] - x*f[x^2]^2, {x, 0, max}], x]; sol = Solve[ Thread[ coes == 0]] // First; Table[ a[2*n+1], {n, 0, nmax}] /. sol (* _Jean-François Alcover_, Mar 06 2013 *)
%o A073712 (Haskell)
%o A073712 a073712 n = a073712_list !! n
%o A073712 a073712_list = map (g a073711_list) [1..] where
%o A073712 g xs k = sum $ zipWith (*) xs $ reverse $ take k xs
%o A073712 -- _Reinhard Zumkeller_, Dec 20 2012
%o A073712 (PARI) a(n)=local(A=1); for(i=0,#binary(n), A=subst(A,x,x^2+x*O(x^n))+x*subst(A,x,x^2+x*O(x^n))^2);polcoeff(A^2,n)
%o A073712 for(n=0,65,print1(a(n),", ")) \\ _Paul D. Hanna_, Dec 21 2012
%Y A073712 Cf. A073711, A194279.
%K A073712 easy,nice,nonn,look
%O A073712 0,2
%A A073712 _Paul D. Hanna_, Aug 05 2002
%E A073712 Name changed and entry revised by _Paul D. Hanna_, Dec 21 2012