|
%I #32 Aug 11 2023 09:35:55
%S 1,1,0,-3,-7,-4,24,85,99,-215,-1196,-2100,1420,17512,42160,9477,
%T -252073,-815965,-736456,3365813,15248793,22861712,-37036000,
%U -273657748,-575046252,180950476,4658415696,13042693000,6717278152,-73400374512,-275797704864,-321427878811,1012425395135
%N Expansion of the series reversion of x/((1 + x)*(1 - x^2)).
%C Reversion of g.f. for the canonical enumeration of integers (A001057).
%H Seiichi Manyama, <a href="/A291534/b291534.txt">Table of n, a(n) for n = 1..1000</a>
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SeriesReversion.html">Series Reversion</a>
%H <a href="/index/Res#revert">Index entries for reversions of series</a>
%F G.f. A(x) satisfies: A(x)/((1 + A(x))*(1 - A(x)^2)) = x.
%F a(n) = hypergeom([(1 - n)/2, 1 - n/2, -n], [1, 3/2], 1). - _Vladimir Reshetnikov_, Oct 15 2018
%F From _Vladimir Reshetnikov_, Oct 18 2018: (Start)
%F G.f.: 2^(1/3)*(6 - 8*x - 2^(1/3)*t^2)/(6*sqrt(x)*t), where t = (3*sqrt(12 - 39*x + 96*x^2) - (9 + 16*x)*sqrt(x))^(1/3).
%F D-finite with recurrence: 64*n*(n + 1)*(2*n + 1)*a(n) - 4*(n + 1)*(37*n^2 + 134*n + 120)*a(n + 1) + (n + 2)*(55*n^2 + 235*n + 240)*a(n + 2) - 2*(6*n + 21)*(n + 2)*(n + 3)*a(n + 3) = 0. (End)
%F a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * binomial(n,k) * binomial(2*n,n-1-k). - _Seiichi Manyama_, Aug 05 2023
%F From _Seiichi Manyama_, Aug 11 2023: (Start)
%F a(n) = Sum_{k=0..n} (-1)^k * 2^(n-k) * binomial(n,k) * binomial(2*n+k+1,n) / (2*n+k+1).
%F a(n) = (1/n) * Sum_{k=0..n-1} (-2)^k * binomial(n,k) * binomial(3*n-k,n-1-k). (End)
%t Rest[CoefficientList[InverseSeries[Series[x/((1 + x) (1 - x^2)), {x, 0, 33}], x], x]]
%t Table[HypergeometricPFQ[{(1 - n)/2, 1 - n/2, -n}, {1, 3/2}, 1], {n, 1, 33}] (* _Vladimir Reshetnikov_, Oct 15 2018 *)
%o (PARI) a(n) = sum(k=0, n-1, (-1)^k*binomial(n, k)*binomial(2*n, n-1-k))/n; \\ _Seiichi Manyama_, Aug 05 2023
%Y Cf. A001057, A001764, A036765.
%Y Cf. A363982, A364051, A364764.
%K sign
%O 1,4
%A _Ilya Gutkovskiy_, Aug 25 2017
|
