A215973 a(0) = 1, for n > 0: a(n) = Sum_{k=0..n-1} a(k) * (1 + a(n-1-k)).
1, 2, 7, 28, 122, 565, 2735, 13682, 70188, 367248, 1952394, 10516141, 57265929, 314751625, 1743829163, 9728561418, 54604800126, 308137127382, 1747158309208, 9949001656704, 56872435967840, 326243091718978, 1877419829207578, 10835354636496321
Offset: 0
Keywords
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
- J. P. S. Kung and A. de Mier, Catalan lattice paths with rook, bishop and spider steps, Journal of Combinatorial Theory, Series A 120 (2013) 379-389. See P_{1,P}(t), p. 386. - From _N. J. A. Sloane_, Dec 27 2012
Programs
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Haskell
a215973 n = a215973_list !! n a215973_list = 1 : f [1] where f xs = y : f (y:xs) where y = sum $ zipWith (*) xs $ map (+ 1) $ reverse xs -
Mathematica
nmax = 30; CoefficientList[Series[(2*x - 1 + Sqrt[1-8*x+12*x^2-4*x^3]) / (2*x*(x-1)), {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 31 2017 *) -
Maxima
a(n):=sum(sum(binomial(j+1,i)*binomial(2*j-i,j-i)*binomial(n-j+i-1,n-j),i,0,j)/(j+1),j,0,n); /* Vladimir Kruchinin, May 04 2018 */
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PARI
a(n) = sum(j=0, n, sum(i=0,j, binomial(j+1,i)*binomial(2*j-i,j-i)*binomial(n-j+i-1,n-j)/(j+1))); \\ Altug Alkan, May 04 2018
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PARI
x='x+O('x^99); Vec((2*x-1+(1-8*x+12*x^2-4*x^3)^(1/2))/(2*x*(x-1))) \\ Altug Alkan, May 04 2018
Formula
G.f.: (2*x-1+sqrt( 1-8*x+12*x^2-4*x^3))/(2*x*(x-1)). - N. J. A. Sloane, Dec 27 2012
Conjecture: (n+1)*a(n) +3*(-3*n+1)*a(n-1) +4*(5*n-7)*a(n-2) +2*(-8*n+19)*a(n-3) +2*(2*n-7)*a(n-4)=0. - R. J. Mathar, Jun 14 2016
a(n) = Sum_{j=0..n} Sum_{i=0..j} C(j+1,i)*C(2*j-i,j-i)*C(n-j+i-1,n-j)/(j+1). - Vladimir Kruchinin, May 04 2018
G.f. A(x) satisfies: A(x) = 1 + x * A(x) / (1 - x) + x * A(x)^2. - Ilya Gutkovskiy, Nov 05 2021
Comments