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A133656
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Number of below-diagonal paths from (0,0) to (n,n) using steps (1,0), (0,1) and (2k-1,1), k a positive integer.
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2
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1, 2, 6, 23, 99, 456, 2199, 10962, 56033, 292094, 1546885, 8299058, 45010492, 246377362, 1359339710, 7551689783, 42206697209, 237156951618, 1338917298708, 7591380528489, 43207023511013, 246773061257046, 1413889039642479
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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
| Brian Drake, Table of n, a(n) for n = 1..50
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FORMULA
| G.f. g(x) satisfies g(x) = 1 + x*g(x)^2+x*g(x)/(1-x^2*g(x)^2).
a(n) = sum(k=0..n, binomial(n+k,n)*sum(j=0..k+n+1, binomial(j,-n-3*k+2*j-2) *(-1)^(n+k-j+1)*binomial(n+k+1,j)))/(n+1). - Vladimir Kruchinin, Oct 11 2011
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EXAMPLE
| a(4) = 99 since there are 90 Schroeder paths (A006318) from (0,0) to (4,4) plus DNNEN, DNENN, DENNN, DdNN, DNdN, DNNd, EDNNN, ENDNN and dDNN, where E=(1,0), N=(0,1), D=(3,1) and d=(1,1).
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MAPLE
| A:=series(RootOf(1+_Z*(x-1)+_Z^2*(x-x^2)+_Z^3*x^2-_Z^4*x^3), x, 21): seq(coeff(A, x, i), i=0..20);
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PROG
| (Maxima) a(n):=sum(binomial(n+k, n)*sum(binomial(j, -n-3*k+2*j-2)*(-1)^(n+k-j+1) *binomial(n+k+1, j), j, 0, k+n+1), k, 0, n)/(n+1); [From Vladimir Kruchinin, Oct 11 2011]
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CROSSREFS
| Cf. A006318, A064641, A052709, A063020. Row sums of A201080.
Sequence in context: A150298 A196018 A009449 * A078487 A193038 A120346
Adjacent sequences: A133653 A133654 A133655 * A133657 A133658 A133659
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KEYWORD
| easy,nonn
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AUTHOR
| Brian Drake (bdrake(AT)brandeis.edu), Sep 20 2007
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