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A122992
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Expansion of g.f.: A(x) = Product_{n>=0} ( 1 + x*(1+x)^n )^( 1/2^(n+1) ).
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2
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1, 1, 1, 1, 0, -2, -5, -3, 27, 145, 382, -24, -6796, -44972, -167234, -105302, 4182671, 41042943, 232150003, 618910867, -4104725087, -76739338173, -670022786184, -3614373261686, -3033843119112, 208905541624840, 3094995814651910, 27593428414596086
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OFFSET
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0,6
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LINKS
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FORMULA
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G.f. exp( Sum_{n>=1} log(1 + x*(1+x)^n) / 2^(n+1) ). - Paul D. Hanna, Sep 11 2018
G.f. exp( Sum_{n>=1} -(-x)^n / (n*(2 - (1+x)^n)) ). - Paul D. Hanna, Sep 11 2018
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EXAMPLE
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G.f.: A(x) = 1 + x + x^2 + x^3 - 2*x^5 - 5*x^6 - 3*x^7 + 27*x^8 + 145*x^9 + 382*x^10 - 24*x^11 - 6796*x^12 + ...
such that
A(x) = (1+x)^(1/2) * (1 + x*(1+x))^(1/4) * (1 + x*(1+x)^2)^(1/8) * (1 + x*(1+x)^3)^(1/16) * ...
RELATED SERIES.
log(A(x)) = x + x^2/2 + x^3/3 - 3*x^4/4 - 9*x^5/5 - 17*x^6/6 + 15*x^7/7 + 237*x^8/8 + 1045*x^9/9 + 2321*x^10/10 - 4509*x^11/11 - 80913*x^12/12 - 493921*x^13/13 - 1703911*x^14/14 + 934221*x^15/15 + 68486893*x^16/16 + 625660487*x^17/17 + 3435765859*x^18/18 + 7342805551*x^19/19 - 94385950863*x^20/20 + ...
which equals Sum_{n>=1} log(1 + x*(1+x)^n) / 2^(n+1).
Also,
log(A(x)) = x/(2 - (1+x)) - x^2/(2*(2 - (1+x)^2)) + x^3/(3*(2 - (1+x)^3)) - x^4/(4*(2 - (1+x)^4)) + x^5/(5*(2 - (1+x)^5)) - x^6/(6*(2 - (1+x)^6)) +- ...
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MATHEMATICA
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terms = 28;
gf = Exp[Sum[2^(-1-n)*Log[1+x*(1+x)^n] + O[x]^terms, {n, 0, 5 terms}]];
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PROG
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(PARI) {a(n)=round(polcoeff(prod(i=0, 6*n+10, (1+x*(1+x)^i +x*O(x^n))^(1/2^(i+1))), n))}
(PARI) {a(n)=local(A); if(n<0, 0, A=1+O(x); for(k=1, n, A=truncate(A)+x*O(x^k); A+=substvec(A, [x, y], [x*(1+x*y), y/(1+x*y+O(x^k))]) -A^2/(1+x)); subst(polcoeff(A, n), y, 1))} /* Michael Somos, Oct 21 2006 */
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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