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A063020
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Reversion of y - y^2 - y^3 + y^4.
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2
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0, 1, 1, 3, 9, 32, 119, 466, 1881, 7788, 32868, 140907, 611871, 2685732, 11896906, 53115412, 238767737, 1079780412, 4909067468, 22424085244, 102865595140, 473678981820, 2188774576575, 10145798119530, 47165267330415
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Seems to be the inverse of A007858. Can someone prove this?
a(n+1) counts paths from (0,0) to (n,n) which do not go above the line y=x, using steps (1,0) and (2k,1), where k ranges over the nonnegative integers. For example, the 9 paths from (0,0) to (3,3) are the 5 Catalan paths, as well as DNEN, DENN, EDNN and ENDN. Here E=(1,0), N=(0,1), D=(2,1). - Brian Drake (bdrake(AT)brandeis.edu), Sep 20 2007
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REFERENCES
| Drake, Brian, Limits of areas under lattice paths. Discrete Math. 309 (2009), no. 12, 3936-3953.
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LINKS
| Index entries for reversions of series
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FORMULA
| a(n)=sum(k=0..n-1,(sum(j=0..k, binomial(j,n-3*k+2*j-1)*(-1)^(j-k)*binomial(k,j)))*binomial(n+k-1,n-1))/n. [From Vladimir Kruchinin, Oct 11 2011]
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MAPLE
| A:= series(RootOf(_Z-_Z^2-_Z^3+_Z^4-x), x, 21): seq(coeff(A, x, i), i=0..20); - Brian Drake (bdrake(AT)brandeis.edu), Sep 20 2007
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MATHEMATICA
| CoefficientList[InverseSeries[Series[y - y^2 - y^3 + y^4, {y, 0, 30}], x], x]
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PROG
| (Maxima)
a(n):=sum((sum(binomial(j, n-3*k+2*j-1)*(-1)^(j-k)*binomial(k, j), j, 0, k))*binomial(n+k-1, n-1), k, 0, n-1)/n; [From Vladimir Kruchinin, Oct 11 2011]
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CROSSREFS
| Cf. A007848.
Cf. A052709, A064641.
Sequence in context: A122452 A192206 A091841 * A104184 A193621 A183425
Adjacent sequences: A063017 A063018 A063019 * A063021 A063022 A063023
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KEYWORD
| nonn,easy
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AUTHOR
| Olivier Gerard (olivier.gerard(AT)gmail.com), Jul 05 2001.
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