%I #39 Jul 20 2023 13:13:08
%S 1,1,2,3,6,12,27,63,154,386,989,2574,6787,18085,48622,131719,359194,
%T 985186,2715973,7521568,20915257,58373587,163462816,459136810,
%U 1293223231,3651864607,10336625732,29321683083,83344398534,237344961292
%N Number of shifted Young tableaux with height <= 3.
%H Vincenzo Librandi, <a href="/A082395/b082395.txt">Table of n, a(n) for n = 1..1000</a>
%H FouadIbn-Majdoub-Hassani, <a href="https://www.theses.fr/1996PA112406">Combinatoire de polyominos et des tableaux décalés oscillants</a>, Thèse de Doctorat, 1996, Laboratoire de Recherche en Informatique, Université Paris-Sud XI, France.
%F a(n) = Sum_{k=1..n}(-1)^(k+1)*binomial(n, k)*binomial(k-1, floor(k/2)). - _Vladeta Jovovic_, Sep 18 2003
%F Recurrence: 3*(n-2)*a(n-3)+(2-n)*a(n-2)+(4-3*n)*a(n-1)+n*a(n)=0. - _Vaclav Kotesovec_, Oct 02 2012
%F Asymptotic: a(n) ~ 3^(n+3/2)/(16*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 02 2012
%F From _Paul D. Hanna_, Jul 08 2023: (Start)
%F G.f. A(x) = (1 + x - sqrt(1 - 2*x - 3*x^2))/(2*(1-x^2)).
%F G.f. A(x) = A(x)^2 + (1 - A(x)^2)*x + (A(x) - A(x)^2)*x^2 + A(x)^2*x^3.
%F G.f. A(x) satisfies 0 = Sum_{n>=0} (-1)^n * x^(n*(n-3)/2) * A(x)^n / Product_{k=0..n+1} (1 - x^k*A(x)). (End)
%e G.f.: A(x) = x + x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 12*x^6 + 27*x^7 + 63*x^8 + 154*x^9 + 386*x^10 + 989*x^11 + 2574*x^12 + ...
%t Table[Sum[(-1)^(k+1)*Binomial[n, k]*Binomial[k-1, Floor[k/2]],{k,1,n}],{n,1,20}]
%t RecurrenceTable[{3*(n-2)*a[n-3]+(2-n)*a[n-2]+(4-3n)*a[n-1]+n*a[n]==0,a[1]==1,a[2]==1,a[3]==2},a,{n,20}] (* _Vaclav Kotesovec_, Oct 02 2012 *)
%o (Sage)
%o def A082395():
%o a, b, s, n = 1, 0, 1, 1
%o yield a
%o while True:
%o s += b
%o yield s
%o n += 1
%o a, b = b, (2*b+3*a)*(n-1)/(n+1)
%o A082395_list = A082395()
%o [next(A082395_list) for i in range(30)] # _Peter Luschny_, Sep 24 2014
%Y Partial sums of A005043.
%Y Cf. A363555.
%K nonn
%O 1,3
%A _Fouad IBN MAJDOUB HASSANI_, Apr 14 2003
%E More terms from _Vladeta Jovovic_, Sep 18 2003