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A151403
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Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0), ending on the vertical axis and consisting of 2*n steps taken from {(-1, 0), (-1, 1), (1, 0), (1, 1)}.
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19
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1, 4, 32, 320, 3584, 43008, 540672, 7028736, 93716480, 1274544128, 17611882496, 246566354944, 3489862254592, 49855175065600, 717914520944640, 10409760553697280, 151860036312760320, 2227280532587151360, 32823081532863283200, 485781606686376591360, 7217326727911880785920
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OFFSET
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0,2
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COMMENTS
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Number of strings of length 2*n of four different types of balanced parentheses.
The number of strings of length 2*n of t different types of balanced parentheses is given by t^n * A000108(n): there are n opening parentheses in the strings, giving t^n choices for the type (the closing parentheses are chosen to match). (End)
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REFERENCES
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Richard P. Stanley, Catalan Numbers, Cambridge, 2015, p. 106.
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LINKS
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FORMULA
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a(n) = Integral_{x=-2..2} (2*x)^(2*n)*sqrt((2-x)*(2+x)))/(2*Pi) dx. - Peter Luschny, Sep 11 2011
G.f.: 2/(1 + sqrt(1-16*x)) = 1/U(0) where U(k) = 1 - 4*x/U(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 30 2012
a(n) = 4^n*hypergeom([1-n,-n],[2],1). - Peter Luschny, Sep 22 2014
a(n) = 4^n*(2*n)!*[x^(2*n)]hypergeom([],[2],x^2). - Peter Luschny, Jan 31 2015
a(n) ~ 2^(4*n+2)/((2*n+1)*sqrt(Pi*(4*n+5))). - Peter Luschny, Jan 31 2015
D-finite with recurrence: (n+1)*a(n) +8*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Feb 21 2020
Sum_{n>=0} 1/a(n) = 88/75 + 128*arctan(1/sqrt(15)) / (75*sqrt(15)). - Vaclav Kotesovec, Nov 23 2021
Sum_{n>=0} (-1)^n/a(n) = 248/289 - 384*arctanh(1/sqrt(17)) / (289*sqrt(17)). - Amiram Eldar, Jan 25 2022
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MAPLE
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A151403_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := 4*(a[w-1]+add(a[j]*a[w-j-1], j=1..w-1)) od; convert(a, list) end: A151403_list(20); # Peter Luschny, May 19 2011
seq(4^n*(2*n)!*coeff(series(hypergeom([], [2], x^2), x, 2*n+2), x, 2*n), n=0..20); # Peter Luschny, Jan 31 2015
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MATHEMATICA
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aux[i_Integer, j_Integer, n_Integer] := Which[Min[i, j, n] < 0 || Max[i, j] > n, 0, n == 0, KroneckerDelta[i, j, n], True, aux[i, j, n] = aux[-1 + i, -1 + j, -1 + n] + aux[-1 + i, j, -1 + n] + aux[1 + i, -1 + j, -1 + n] + aux[1 + i, j, -1 + n]]; Table[Sum[aux[0, k, 2 n], {k, 0, 2 n}], {n, 0, 25}]
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PROG
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(Sage)
A151403 = lambda n: 4^n*hypergeometric([1-n, -n], [2], 1)
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CROSSREFS
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KEYWORD
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nonn,walk
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AUTHOR
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STATUS
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approved
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