OFFSET
0,2
COMMENTS
Inverse binomial transform of A188312.
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
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
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 *)
PROG
(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
(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 */
(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
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Aug 29 2012
STATUS
approved