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A162326
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Let a(0) = a(1) = 1, and n*a(n) = 2*(-7+5*n)*a(n-1) + 9*(2-n)*a(n-2) for n >= 2.
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14
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1, 1, 3, 13, 71, 441, 2955, 20805, 151695, 1135345, 8671763, 67320573, 529626839, 4213228969, 33833367963, 273892683573, 2232832964895, 18314495896545, 151037687326755, 1251606057754605, 10416531069771111, 87029307323766681
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OFFSET
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0,3
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COMMENTS
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Let y = y(x) be implicitly defined by g(x,y(x)) = 0, with dg/dy not identically zero. For n >= 1, the sequence a(n) is the number of terms in the expansion of the divided difference [x0,...,xn]y in terms of bivariate divided differences of g.
(1 + 3*x + 13*x^2 + 71*x^3 + ...) = (1 + 4*x + 20*x^2 + 116*x^3 + ...) * 1/(1 + x + 4*x^2 + 20*x^3 + 116*x^4 + ...); where A082298 = (1, 4, 20, 116, 740, ...). - Gary W. Adamson, Nov 17 2011
a(n+1) is the number of Schröder paths of semilength n in which the (2,0)-steps come in 3 colors and with no peaks at level 1. - José Luis Ramírez Ramírez, Mar 31 2013
Define an infinite triangle by T(n,0)=1 and the other cells by T(n,k) = Sum_{c=0..k-1} T(n,c) + Sum_{r=k..n-1} T(r,k), the sum of the cells to the left and above a cell. The column k=1 contains A000079, the column k=2 essentially A001792. Then T(n,n)=a(n) on the diagonal. - J. M. Bergot, May 22 2013
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LINKS
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FORMULA
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Let E = N x N \ {(0,0), (0,1)} be a set of pairs of natural numbers. The number of terms a(n) is the coefficient of x^n*y^{n-1} of the generating function 1 - log(1 - Sum_{(s,t) in E} x^s*y^{s+t-1}) = 1 + Sum_{q >= 1} (Sum_{(s,t) in E} x^s*y^{s+t-1})^q / q.
a(n) = 2F1(1/2,1-n;2;-8), where 2F1 is the Gauss hypergeometric series.
G.f.: (5 - sqrt( (1-9*x)/(1-x) ))/4.
Quadratic recurrence relation: a(n) = 1 + 2*Sum_{m=1..n-1} a(m)*a(n-m).
(End)
a(n) = Sum_{k=0..n} (binomial(n,k)*2^(n-k-1)*binomial(2*n-2*k-2,n-k-1))/n, n>0, a(0)=1. - Vladimir Kruchinin, Mar 13 2016
a(n+1) = Sum_{k = 0..n} 2^k*C(n,k)*Catalan(k).
a(n+1) = (2/Pi) * Integral_{x = -1..1} (1 + 8*x^2)^n*sqrt(1 - x^2) dx.
O.g.f.: 1 + x/(1 - x)*c(2*x/(1-x)), where c(x) is the o.g.f. for A000108. (End)
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EXAMPLE
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Write [0...n]y for [x0,...,xn]y and [0...s,0...t]g for [x0,...,xs;y0,...,yt]g.
For n = 1 one finds 1 term, [01]y = -[01;1]g/[0;01]g.
For n = 2 one finds 3 terms, [012]y = -[012;2]g/[0;02]g + ([01;12]g[12;2]g)/([0;02]g[1;12]g) - ([0;012]g[01;1]g[12;2]g)/([0;02]g[0;01]g[1;12]g).
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MATHEMATICA
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CoefficientList[Series[(5-Sqrt[(1-9*x)/(1-x)])/4, {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)
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PROG
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(Python)
L = [1, 1]
for n in range(2, 22):
L.append( ((-14 + 10*n)*L[-1] + (18-9*n)*L[-2])//n )
print(L)
(PARI) a(n) = if(n<2, 1, (2*(-7+5*n)*a(n-1) + 9*(2-n)*a(n-2))/n);
(PARI) my(x='x+O('x^20)); Vec((5-sqrt((1-9*x)/(1-x)))/4) \\ G. C. Greubel, Feb 07 2019
(Maxima)
a(n):=if n=0 then 1 else sum(binomial(n, k)*2^(n-k-1)*binomial(2*n-2*k-2, n-k-1), k, 0, n)/n; /* Vladimir Kruchinin, Mar 13 2016 */
(Magma) m:=20; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (5-Sqrt((1-9*x)/(1-x)))/4 )); // G. C. Greubel, Feb 07 2019
(Magma) a:=[1, 3]; for n in [3..21] do Append(~a, (2*(-7+5*n)*a[n-1] + 9*(2-n)*a[n-2]) div n); end for ; [1] cat a; // Marius A. Burtea, Jan 20 2020
(Sage) ((5-sqrt((1-9*x)/(1-x)))/4).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Feb 07 2019
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CROSSREFS
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Cf. A172003, which is a generalization to bivariate implicit functions.
Cf. A003262, which is the analogous sequence for implicit derivatives, and A172004 for its generalization to bivariate implicit functions.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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