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

1, 2, 5, 11, 26, 62, 153, 385, 988, 2573, 6786, 18084, 48621, 131718, 359193, 985185, 2715972, 7521567, 20915256, 58373586, 163462815, 459136809, 1293223230, 3651864606, 10336625731, 29321683082, 83344398533, 237344961291

Offset: 1

Author

Fouad IBN MAJDOUB HASSANI, Apr 14 2003

Keywords

• nonn

Comments

Conjecture: partial sums of A342912. - Sean A. Irvine, Jul 16 2022

References

• 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.

Links

• Rigoberto Flórez and José L. Ramírez, Enumerating symmetric pyramids in Motzkin paths, Ars Math. Contemporanea (2023) Vol. 23, #P4.06.

Programs

Maple

A082397 := proc(n)
    add( (-1)^(k+1)*binomial(n+1,k+1)*binomial(k,floor((k-1)/2)),k=1..n) ;
end proc:
seq(A082397(n),n=1..30) ; # R. J. Mathar, Jun 27 2022

Mathematica

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 *)

Python

import math
def Sum(k):
    S= sum((-1)**(i+1)*math.comb(k,i+1)*math.comb(i,math.floor((i-1)/2)) for i in range(1,k))
    return S
for i in range (2,20): print(Sum(i))
# Rigoberto Florez, Dec 10 2022

Formula

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

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

Extensions

More terms from Vladeta Jovovic, Sep 18 2003