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#16 by N. J. A. Sloane at Thu Jan 10 22:54:16 EST 2019
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#15 by Peter Luschny at Thu Jan 10 13:56:24 EST 2019
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#14 by Peter Luschny at Thu Jan 10 13:50:35 EST 2019
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| EXAMPLE
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Generated from Fibonacci polynomials (A011973) and coefficients of odd powers of 1/(1-x):
coefficients of odd powers of 1/(1-x):
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| MATHEMATICA
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ShiftedReversion[ser_, n_, sgn_] := CoefficientList[(sgn/x)InverseSeries[Series[x sgn ser, {x, , 0, , n}]], x];
Jacobsthal := (2x^2 - 1)/((x + 1)(2x - 1)); (* with aA001045(0) = 1 *)
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| PROG
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(PARI) {a(n)=if(n==0, 1, sum(k=0, (n-1)\2, binomial(n-k-1, k)*binomial(n, 2*k)*2^k/(/ (2*k+1)))}
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#13 by Peter Luschny at Thu Jan 10 13:45:09 EST 2019
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| MATHEMATICA
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ShiftedReversion[ser_, n_, sgn_] := CoefficientList[(sgn/x)InverseSeries[Series[x sgn ser, {x, 0, n}]], x];
Jacobsthal := (2x^2 - 1)/((x + 1)(2x - 1)); (* with a(0) = 1 *)
ShiftedReversion[Jacobsthal, 27, -1] (* Peter Luschny, Jan 10 2019 *)
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| STATUS
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approved
editing
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#12 by Alois P. Heinz at Fri Sep 20 05:55:53 EDT 2013
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#11 by Vincenzo Librandi at Fri Sep 20 01:13:15 EDT 2013
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#10 by Vincenzo Librandi at Fri Sep 20 01:13:07 EDT 2013
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| LINKS
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Vincenzo Librandi, <a href="/A101786/b101786.txt">Table of n, a(n) for n = 0..1000</a>
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| STATUS
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approved
editing
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#9 by Paul D. Hanna at Thu Sep 19 20:49:01 EDT 2013
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#8 by Paul D. Hanna at Thu Sep 19 20:48:58 EDT 2013
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| FORMULA
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a(n) = Sum_{k=0..[(n-1)/2]} C(n-k-1, k)*C(n, 2*k)*2^k/(2*k+1) for n>0, with a(0)=1. G.f.: A(x) = [ series reversion of x*(1-2*x^2)/(1+x-2*x^2) ]/x.
G.f.: A(x) = (1/x)*Series_Reversion( x*(1 - 2*x^2)/(1+x - 2*x^2) ).
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| STATUS
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approved
editing
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#7 by Bruno Berselli at Tue Sep 17 10:24:32 EDT 2013
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