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A215973
a(0) = 1, for n > 0: a(n) = Sum_{k=0..n-1} a(k) * (1 + a(n-1-k)).
5
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
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
Sequence in context: A269228 A005435 A291091 * A143927 A253787 A060379
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Aug 29 2012
STATUS
approved