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A082397 Number of directed aggregates of height <= 2 with n cells.

%I A082397 #33 Apr 12 2023 11:18:05
%S A082397 1,2,5,11,26,62,153,385,988,2573,6786,18084,48621,131718,359193,
%T A082397 985185,2715972,7521567,20915256,58373586,163462815,459136809,
%U A082397 1293223230,3651864606,10336625731,29321683082,83344398533,237344961291
%N A082397 Number of directed aggregates of height <= 2 with n cells.
%C A082397 Conjecture: partial sums of A342912. - _Sean A. Irvine_, Jul 16 2022
%D A082397 Fouad Ibn-Majdoub-Hassani. Combinatoire de polyominos et des tableaux décalés oscillants. Thèse de Doctorat. 1991. Laboratoire de Recherche en Informatique, Université Paris-Sud XI, France.
%H A082397 Rigoberto Flórez and José L. Ramírez, <a href="https://doi.org/10.26493/1855-3974.3061.5bf">Enumerating symmetric pyramids in Motzkin paths</a>, Ars Math. Contemporanea (2023) Vol. 23, #P4.06.
%F A082397 a(n) = Sum_{k=1..n}(-1)^(k+1)*binomial(n+1, k+1)*binomial(k, floor((k-1)/2)). E.g.f.: -exp(x)*int(-BesselI(1, 2*x)+BesselI(2, 2*x), x)-exp(x)*(-BesselI(1, 2*x)+BesselI(2, 2*x)). - _Vladeta Jovovic_, Sep 18 2003
%F A082397 Conjecture D-finite with recurrence +(n+2)*a(n) +(-3*n-2)*a(n-1) -n*a(n-2) +3*n*a(n-3)=0. - _R. J. Mathar_, Jun 27 2022
%p A082397 A082397 := proc(n)
%p A082397     add( (-1)^(k+1)*binomial(n+1,k+1)*binomial(k,floor((k-1)/2)),k=1..n) ;
%p A082397 end proc:
%p A082397 seq(A082397(n),n=1..30) ; # _R. J. Mathar_, Jun 27 2022
%t A082397 Table[Sum[(-1)^(i+1)*Binomial[k+1, i+1] Binomial[i, Floor[(i-1)/2]], {i,1,k}], {k,1,20}] (* _Rigoberto Florez_, Dec 10 2022 *)
%o A082397 (Python)
%o A082397 import math
%o A082397 def Sum(k):
%o A082397     S= sum((-1)**(i+1)*math.comb(k,i+1)*math.comb(i,math.floor((i-1)/2)) for i in range(1,k))
%o A082397     return S
%o A082397 for i in range (2,20): print(Sum(i))
%o A082397 # _Rigoberto Florez_, Dec 10 2022
%K A082397 nonn
%O A082397 1,2
%A A082397 _Fouad IBN MAJDOUB HASSANI_, Apr 14 2003
%E A082397 More terms from _Vladeta Jovovic_, Sep 18 2003