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A073708 Generating function A(x) satisfies A(x) = (1+x)^2*A(x^2)^2, with A(0)=1.

%I A073708 #19 Jul 01 2025 19:23:54
%S A073708 1,2,5,8,18,28,50,72,129,186,301,416,664,912,1368,1824,2730,3636,5234,
%T A073708 6832,9788,12744,17724,22704,31506,40308,54730,69152,93592,118032,
%U A073708 156888,195744,259625,323506,423021,522536,681642,840748,1083402,1326056,1705665
%N A073708 Generating function A(x) satisfies A(x) = (1+x)^2*A(x^2)^2, with A(0)=1.
%H A073708 Paul D. Hanna, <a href="/A073708/b073708.txt">Table of n, a(n) for n = 0..1000</a>
%F A073708 Equals the self-convolution of A073707.
%e A073708 G.f.: A(x) = 1 + 2*x + 5*x^2 + 8*x^3 + 18*x^4 + 28*x^5 + 50*x^6 +...
%e A073708 where A(x)^2 = 1 + 4*x + 14*x^2 + 36*x^3 + 93*x^4 + 208*x^5 + 456*x^6 +...
%e A073708 This sequence equals the self-convolution of A073707, which begins:
%e A073708 [1, 1, 2, 2, 5, 5, 8, 8, 18, 18, 28, 28, 50, 50, ...].
%e A073708 The first differences of this sequence result in A073709:
%e A073708 [1, 1, 3, 3, 10, 10, 22, 22, 57, 57, 115, 115, ...];
%e A073708 the self-convolution of A073709 yields A073710:
%e A073708 [1, 2, 7, 12, 35, 58, 133, 208, ...],
%e A073708 which in turn equals the first differences of the unique terms of A073709.
%t A073708 A073708list[n_] := Module[{m = 1, A = 1}, While[m <= n, m = 2 m; A = ((1 + x)*(A /. x -> x^2))^2] + O[x]^m; CoefficientList[A, x][[1 ;; n]]]; A073708list[50] (* _Jean-François Alcover_, Apr 21 2016, adapted from PARI *)
%o A073708 (PARI) a(n)=local(A,m); if(n<0,0,m=1; A=1+O(x); while(m<=n,m*=2; A=((1+x)*subst(A,x,x^2))^2); polcoeff(A,n))
%o A073708 (Haskell)
%o A073708 a073708 n = a073708_list !! n
%o A073708 a073708_list = conv a073707_list [] where
%o A073708    conv (v:vs) ws = (sum $ zipWith (*) ws' $ reverse ws') : conv vs ws'
%o A073708                     where ws' = v : ws
%o A073708 -- _Reinhard Zumkeller_, Jun 13 2013
%Y A073708 Cf. A073709, A073710. A073707(2n)=a(n).
%K A073708 easy,nonn
%O A073708 0,2
%A A073708 _Paul D. Hanna_, Aug 04 2002
%E A073708 Edited by _Michael Somos_, May 03 2003
%E A073708 Edited by _Paul D. Hanna_, Jan 04 2013