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

0,2

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

Cf. A000108, A188312.

Sequence in context: A269228 A005435 A291091 * A143927 A253787 A060379

Adjacent sequences: A215970 A215971 A215972 * A215974 A215975 A215976

Keyword

nonn

Author

Reinhard Zumkeller, Aug 29 2012

Status

approved