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A073707
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Coefficients of a power series whose convolution consists of only the even-indexed terms of the sequence.
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6
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1, 1, 2, 2, 5, 5, 8, 8, 18, 18, 28, 28, 50, 50, 72, 72, 129, 129, 186, 186, 301, 301, 416, 416, 664, 664, 912, 912, 1368, 1368, 1824, 1824, 2730, 2730, 3636, 3636, 5234, 5234, 6832, 6832, 9788, 9788, 12744, 12744, 17724, 17724, 22704, 22704, 31506, 31506
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OFFSET
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0,3
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LINKS
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FORMULA
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G.f.: A(x) satisfies A(x) = (1+x)*A(x^2)^2, with A(0)=1.
G.f.: A(x) = Product_{n>=0} (1 + x^(2^n))^(2^n).
G.f.: A(x) = (1/(1 - x)) * Product_{n>=0} 1/(1 - x^(2^(n+1)))^(2^n). - Eitan Y. Levine, Jun 24 2023
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EXAMPLE
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(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 + ...).
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MATHEMATICA
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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 *)
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PROG
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(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))
(PARI) {a(n)=polcoeff(prod(k=0, #binary(n), (1+x^(2^k)+x*O(x^n))^(2^k)), n)}
(Haskell)
a073707 n = a073707_list !! n
a073707_list = 1 : f 0 0 [1] where
f x y zs = z : f (x + y) (1 - y) (z:zs) where
z = sum $ zipWith (*) hzs (reverse hzs) where hzs = drop x zs
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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