OFFSET
1,2
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.
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
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Fouad IBN MAJDOUB HASSANI, Apr 14 2003
EXTENSIONS
More terms from Vladeta Jovovic, Sep 18 2003
STATUS
approved