A291534 Expansion of the series reversion of x/((1 + x)*(1 - x^2)).
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
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 1..1000
- N. J. A. Sloane, Transforms
- Eric Weisstein's World of Mathematics, Series Reversion
- Index entries for reversions of series
Programs
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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 *) -
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
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)
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