OFFSET
0,5
COMMENTS
This sequence and its companion sequences A143629 and A143630 may be viewed as generalizations of the Uppuluri-Carpenter numbers (complementary Bell numbers) A000587. Define E(n) = Sum_{k >= 0} (-1)^floor(k/3)*k^n/k! = 0^n/0! + 1^n/1! + 2^n/2! - 3^n/3! - 4^n/4! - 5^n/5! + + + - - - ... for n = 0,1,2,... . It is easy to see that E(n+3) = 3*E(n+2) - 2*E(n+1) - Sum_{i = 0..n} 3^i* binomial(n,i)*E(n-i) for n >= 0. Thus E(n) is an integral linear combination of E(0), E(1) and E(2). Some examples are given below.
This sequence lists the coefficients of E(0). See A143629 and A143630 for the sequence of coefficients of E(1) and E(2) respectively. The functions F(n) := Sum_{k >= 0} (-1)^floor((k+1)/3)*k^n/k! and G(n) = Sum_{k >= 0} (-1)^floor((k+2)/3)*k^n/k! both satisfy the above recurrence as well as the identities E(n+1) = Sum_{i = 0..n} binomial(n,i)*F(i), F(n+1) = Sum_{i = 0..n} binomial(n,i)*G(i) and G(n+1) = - Sum_{i = 0..n} binomial(n,i)*E(i). This leads to the precise result for E(n) as a linear combination of E(0), E(1) and E(2), namely, E(n) = A143628(n)*E(0) + A143629(n)*E(1) + A143630(n)*E(2). Compare with A121867 and A143815.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..578
Eric Weisstein's World of Mathematics, Bell Polynomial.
FORMULA
Define three sequences A(n), B(n) and C(n) by the relations: A(n+1) = - Sum_{i = 0..n} binomial(n,i)*C(i), B(n+1) = Sum_{i = 0..n} binomial(n,i)*A(i), C(n+1) = Sum_{i = 0..n} binomial(n,i)*B(i), with initial conditions A(0) = 1, B(0) = C(0) = 0. Then a(n) = A(n). The other sequences are B(n) = A143630 and C(n) = A143629. Compare with A143815. Also a(n) = A143629(n) + A000587(n).
From Seiichi Manyama, Oct 15 2022: (Start)
a(n) = Sum_{k = 0..floor(n/3)} (-1)^k * Stirling2(n,3*k).
a(n) = ( Bell_n(-1) + Bell_n(-w) + Bell_n(-w^2) )/3, where Bell_n(x) is n-th Bell polynomial and w = exp(2*Pi*i/3). (End)
EXAMPLE
E(n) as linear combination of E(i),
i = 0..2.
====================================
..E(n)..|.....E(0).....E(1)....E(2).
====================================
..E(3)..|......-1......-2........3..
..E(4)..|......-6......-7........7..
..E(5)..|.....-25.....-23.......14..
..E(6)..|.....-89.....-80.......16..
..E(7)..|....-280....-271......-77..
..E(8)..|....-700....-750.....-922..
..E(9)..|....-380....-647....-6660..
..E(10).|...13452...13039...-41264..
...
a(5) = -25 because E(5) = -25*E(0) - 23*E(1) + 14*E(2).
a(6) = -89 because E(6) = -89*E(0) - 80*E(1) + 16*E(2).
MAPLE
# Compare with A143815
#
M:=24: a:=array(0..100): b:=array(0..100): c:=array(0..100):
a[0]:=1: b[0]:=0: c[0]:=0:
for n from 1 to M do
a[n]:= -add(binomial(n-1, k)*c[k], k=0..n-1);
b[n]:= add(binomial(n-1, k)*a[k], k=0..n-1);
c[n]:= add(binomial(n-1, k)*b[k], k=0..n-1);
end do:
A143628:=[seq(a[n], n=0..M)];
MATHEMATICA
m = 23; a[0] = 1; b[0] = 0; c[0] = 0; For[n = 1, n <= m, n++, a[n] = -Sum[ Binomial[n-1, k]*c[k], {k, 0, n-1}]; b[n] = Sum[ Binomial[n-1, k]*a[k], {k, 0, n-1}]; c[n] = Sum[ Binomial[n-1, k]*b[k], {k, 0, n-1}]]; Table[a[n], {n, 0, m}] (* Jean-François Alcover, Mar 06 2013, after Maple *)
PROG
(PARI) Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
a(n) = my(w=(-1+sqrt(3)*I)/2); round(Bell_poly(n, -1)+Bell_poly(n, -w)+Bell_poly(n, -w^2))/3; \\ Seiichi Manyama, Oct 15 2022
CROSSREFS
KEYWORD
easy,sign,changed
AUTHOR
Peter Bala, Sep 05 2008
STATUS
approved