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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)).

%I A215973 #37 Nov 05 2021 20:26:46
%S A215973 1,2,7,28,122,565,2735,13682,70188,367248,1952394,10516141,57265929,
%T A215973 314751625,1743829163,9728561418,54604800126,308137127382,
%U A215973 1747158309208,9949001656704,56872435967840,326243091718978,1877419829207578,10835354636496321
%N A215973 a(0) = 1, for n > 0: a(n) = Sum_{k=0..n-1} a(k) * (1 + a(n-1-k)).
%C A215973 Inverse binomial transform of A188312.
%H A215973 Reinhard Zumkeller, <a href="/A215973/b215973.txt">Table of n, a(n) for n = 0..1000</a>
%H A215973 J. P. S. Kung and A. de Mier, <a href="http://dx.doi.org/10.1016/j.jcta.2012.08.010">Catalan lattice paths with rook, bishop and spider steps</a>, 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
%F A215973 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
%F A215973 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
%F A215973 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
%F A215973 G.f. A(x) satisfies: A(x) = 1 + x * A(x) / (1 - x) + x * A(x)^2. - _Ilya Gutkovskiy_, Nov 05 2021
%t A215973 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 *)
%o A215973 (Haskell)
%o A215973 a215973 n = a215973_list !! n
%o A215973 a215973_list = 1 : f [1] where
%o A215973    f xs = y : f (y:xs) where
%o A215973      y = sum $ zipWith (*) xs $ map (+ 1) $ reverse xs
%o A215973 (Maxima)
%o A215973 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 */
%o A215973 (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
%o A215973 (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
%Y A215973 Cf. A000108, A188312.
%K A215973 nonn
%O A215973 0,2
%A A215973 _Reinhard Zumkeller_, Aug 29 2012