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A073710 Convolution of A073709, which is also the first differences of the unique terms of A073709.

%I A073710 #10 Jun 14 2013 09:45:44
%S A073710 1,2,7,12,35,58,133,208,450,692,1358,2024,3822,5620,10018,14416,25025,
%T A073710 35634,59591,83548,136955,190362,303725,417088,655128,893168,1374632,
%U A073710 1856096,2820456,3784816,5658968,7533120,11144042,14754964,21542374
%N A073710 Convolution of A073709, which is also the first differences of the unique terms of A073709.
%C A073710 First differences consist of duplicated terms: {1, 1, 5, 5, 23, 23, 75, 75, 242, 242, 666, 666, 1798, 1798, ...}; the convolution of these first differences results in: {1, 2, 11, 20, 81, 142, 451, 760, 2143, 3526, 8965, ...}, which in turn has first differences that consist of duplicated terms: {1, 1, 9, 9, 61, 61, 309, 309, ...}.
%H A073710 Reinhard Zumkeller, <a href="/A073710/b073710.txt">Table of n, a(n) for n = 0..10000</a>
%F A073710 Let f(x) = sum_{n=0..inf} A073709(n) x^n, then f(x) satisfies f(x)^2 = sum_{n=0..inf} a(n) x^n, as well as the functional equation f(x^2)^2 = (1 - x)*f(x).
%e A073710 (1 +x +3x^2 +3x^3 +10x^4 +10x^5 +22x^6 +22x^7 +57x^8 +57x^9 +...)^2 = (1 +2x +7x^2 +12x^3 +35x^4 +58x^5 +133x^6 +208x^7 +450x^8 +...) and the first differences of the unique terms {1,3,10,22,57,...} is {1,2,7,12,35,...}.
%t A073710 max = 70; f[x_] := Sum[ a[k]*x^k, {k, 0, max}]; a[0] = a[1] = 1; coes = CoefficientList[ Series[ f[x^2]^2 - (1 - x)*f[x], {x, 0, max}], x]; A073709 = Table[a[k], {k, 0, max}] /. Solve[ Thread[coes == 0]] // First; A073709 // Union // Differences // Prepend[#, 1]&
%o A073710 (Haskell)
%o A073710 a073710 n = a073710_list !! n
%o A073710 a073710_list = conv a073709_list [] where
%o A073710    conv (v:vs) ws = (sum $ zipWith (*) ws' $ reverse ws') : conv vs ws'
%o A073710                     where ws' = v : ws
%o A073710 -- _Reinhard Zumkeller_, Jun 13 2013
%Y A073710 Cf. A073707, A073708, A073709.
%K A073710 easy,nice,nonn
%O A073710 0,2
%A A073710 _Paul D. Hanna_, Aug 05 2002