0,4

This sequence and its companion sequences A143628 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). This sequence lists the coefficients of E(1). Some examples are given below. The precise result for E(n) as a linear combination of E(0), E(1) and E(2) is 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) = B(n) - C(n). The other sequences are A(n) = A143628(n) and C(n) = A143630(n). The values of B(n) are recorded in A143631. Compare with A143818. Also a(n) = A143628(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) = -23 because E(5) = -25*E(0) - 23*E(1) + 14*E(2).

a(6) = -80 because E(6) = -89*E(0) - 80*E(1) + 16*E(2).

# Compare with A143818

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:

A143629:=[seq(b[n]-c[n], n=0..M)];

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}] ]; A143629 = Table[b[n] - c[n], {n, 0, m}] (* Jean-François Alcover, Mar 06 2013, after Maple *)

A000587, A121867, A143628, A143630, A143631, A143815, A143816, A143817, A143818.

Sequence in context: A007717 A130567 A091514 * A176287 A119371 A151290

Adjacent sequences: A143626 A143627 A143628 * A143630 A143631 A143632

easy,sign

Peter Bala, Sep 05 2008

approved