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%I #13 Oct 20 2017 17:09:39
%S 0,1,0,-2,-7,-23,-80,-271,-750,-647,13039,152011,1232583,8750796,
%T 57405464,349329354,1899818951,8008845556,5981853002,-425732481925,
%U -7285403175563,-89895756043392,-970910901819211,-9663021449412616
%N Define E(n) = Sum_{k>=0} (-1)^floor(k/3)*k^n/k! for n = 0,1,2,... . Then E(n) is an integral linear combination of E(0), E(1) and E(2). This sequence lists the coefficients of E(1).
%C 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.
%F 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 E(n) as linear combination of E(i),
%e i = 0..2.
%e ====================================
%e ..E(n)..|.....E(0).....E(1)....E(2).
%e ====================================
%e ..E(3)..|......-1......-2........3..
%e ..E(4)..|......-6......-7........7..
%e ..E(5)..|.....-25.....-23.......14..
%e ..E(6)..|.....-89.....-80.......16..
%e ..E(7)..|....-280....-271......-77..
%e ..E(8)..|....-700....-750.....-922..
%e ..E(9)..|....-380....-647....-6660..
%e ..E(10).|...13452...13039...-41264..
%e ...
%e a(5) = -23 because E(5) = -25*E(0) - 23*E(1) + 14*E(2).
%e a(6) = -80 because E(6) = -89*E(0) - 80*E(1) + 16*E(2).
%p # Compare with A143818
%p M:=24: a:=array(0..100): b:=array(0..100): c:=array(0..100):
%p a[0]:=1: b[0]:=0: c[0]:=0:
%p for n from 1 to M do
%p a[n]:= -add(binomial(n-1,k)*c[k], k=0..n-1);
%p b[n]:= add(binomial(n-1,k)*a[k], k=0..n-1);
%p c[n]:= add(binomial(n-1,k)*b[k], k=0..n-1);
%p end do:
%p A143629:=[seq(b[n]-c[n], n=0..M)];
%t 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 *)
%Y A000587, A121867, A143628, A143630, A143631, A143815, A143816, A143817, A143818.
%K easy,sign
%O 0,4
%A _Peter Bala_, Sep 05 2008
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