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A291534
Expansion of the series reversion of x/((1 + x)*(1 - x^2)).
10
1, 1, 0, -3, -7, -4, 24, 85, 99, -215, -1196, -2100, 1420, 17512, 42160, 9477, -252073, -815965, -736456, 3365813, 15248793, 22861712, -37036000, -273657748, -575046252, 180950476, 4658415696, 13042693000, 6717278152, -73400374512, -275797704864, -321427878811, 1012425395135
OFFSET
1,4
COMMENTS
Reversion of g.f. for the canonical enumeration of integers (A001057).
LINKS
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Series Reversion
FORMULA
G.f. A(x) satisfies: A(x)/((1 + A(x))*(1 - A(x)^2)) = x.
a(n) = hypergeom([(1 - n)/2, 1 - n/2, -n], [1, 3/2], 1). - Vladimir Reshetnikov, Oct 15 2018
From Vladimir Reshetnikov, Oct 18 2018: (Start)
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).
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)
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
From Seiichi Manyama, Aug 11 2023: (Start)
a(n) = Sum_{k=0..n} (-1)^k * 2^(n-k) * binomial(n,k) * binomial(2*n+k+1,n) / (2*n+k+1).
a(n) = (1/n) * Sum_{k=0..n-1} (-2)^k * binomial(n,k) * binomial(3*n-k,n-1-k). (End)
MATHEMATICA
Rest[CoefficientList[InverseSeries[Series[x/((1 + x) (1 - x^2)), {x, 0, 33}], x], x]]
Table[HypergeometricPFQ[{(1 - n)/2, 1 - n/2, -n}, {1, 3/2}, 1], {n, 1, 33}] (* Vladimir Reshetnikov, Oct 15 2018 *)
PROG
(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
CROSSREFS
KEYWORD
sign
AUTHOR
Ilya Gutkovskiy, Aug 25 2017
STATUS
approved