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%I #29 Jun 25 2023 02:07:57
%S 1,1,2,2,5,5,8,8,18,18,28,28,50,50,72,72,129,129,186,186,301,301,416,
%T 416,664,664,912,912,1368,1368,1824,1824,2730,2730,3636,3636,5234,
%U 5234,6832,6832,9788,9788,12744,12744,17724,17724,22704,22704,31506,31506
%N Coefficients of a power series whose convolution consists of only the even-indexed terms of the sequence.
%H Reinhard Zumkeller (confirmed by Paul D. Hanna), <a href="/A073707/b073707.txt">Table of n, a(n) for n = 0..10000</a>
%F G.f.: A(x) satisfies A(x) = (1+x)*A(x^2)^2, with A(0)=1.
%F G.f.: A(x) = Product_{n>=0} (1 + x^(2^n))^(2^n).
%F G.f.: A(x) = (1/(1 - x)) * Product_{n>=0} 1/(1 - x^(2^(n+1)))^(2^n). - _Eitan Y. Levine_, Jun 24 2023
%e (1 + x + 2x^2 + 2x^3 + 5x^4 + 5x^5 + 8x^6 + 8x^7 + 28x^8 + 28x^9 + ...)^2 = (1 + 2x + 5x^2 + 8x^3 + 18x^4 + 28x^5 + 50x^6 + 72x^7 + 129x^8 + ...).
%t nmax = 49; CoefficientList[ Series[ Product[ (1+x^(2^n))^(2^n), {n, 0, Log[nmax]/Log[2]}], {x, 0, nmax}], x] (* _Jean-François Alcover_, Jan 04 2013, from 2nd formula, modified by _Vaclav Kotesovec_, Oct 23 2020 *)
%o (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 (PARI) {a(n)=polcoeff(prod(k=0,#binary(n),(1+x^(2^k)+x*O(x^n))^(2^k)),n)}
%o (Haskell)
%o a073707 n = a073707_list !! n
%o a073707_list = 1 : f 0 0 [1] where
%o f x y zs = z : f (x + y) (1 - y) (z:zs) where
%o z = sum $ zipWith (*) hzs (reverse hzs) where hzs = drop x zs
%o -- _Reinhard Zumkeller_, Dec 21 2011
%Y Cf. A073708, A073709.
%K easy,nonn
%O 0,3
%A _Paul D. Hanna_, Aug 04 2002
%E Definition corrected by _Paul D. Hanna_, Feb 25 2010
%E Data corrected for n > 45 by _Reinhard Zumkeller_, Dec 21 2011
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