0,5

This sequence and its companion sequences A143629 and A143630 may be viewed as generalisations of the Uppuluri-Carpenter numbers (complementary Bell numbers) A000587. Define E(n) = sum {k = 0..inf} (-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..inf} (-1)^floor((k+1)/3)*k^n/k! and G(n) = sum {k = 0..inf} (-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.

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).

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).

# 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)];

A000587, A121867, A143629, A143630, A143631, A143815, A143816, A143817, A143818.

Sequence in context: A164271 A055585 A099625 this_sequence A056279 A055337 A001871

Adjacent sequences: A143625 A143626 A143627 this_sequence A143629 A143630 A143631

easy,sign

Peter Bala (pbala(AT)toucansurf.com), Sep 05 2008