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%I #22 Nov 09 2022 19:21:37
%S 1,0,0,-1,-6,-25,-89,-280,-700,-380,13452,149831,1214852,8700263,
%T 57515640,351296151,1909757620,8017484274,5703377941,-428273438434,
%U -7295220035921,-89868583754993,-970185398785810,-9657428906237364
%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(0).
%C 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.
%C 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.
%H Seiichi Manyama, <a href="/A143628/b143628.txt">Table of n, a(n) for n = 0..578</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%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) = A(n). The other sequences are B(n) = A143630 and C(n) = A143629. Compare with A143815. Also a(n) = A143629(n) + A000587(n).
%F From _Seiichi Manyama_, Oct 15 2022: (Start)
%F a(n) = Sum_{k = 0..floor(n/3)} (-1)^k * Stirling2(n,3*k).
%F 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)
%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) = -25 because E(5) = -25*E(0) - 23*E(1) + 14*E(2).
%e a(6) = -89 because E(6) = -89*E(0) - 80*E(1) + 16*E(2).
%p # Compare with A143815
%p #
%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 A143628:=[seq(a[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}]]; Table[a[n], {n, 0, m}] (* _Jean-François Alcover_, Mar 06 2013, after Maple *)
%o (PARI) Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
%o 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
%Y A000587, A121867, A143629, A143630, A143631, A143815, A143816, A143817, A143818.
%K easy,sign
%O 0,5
%A _Peter Bala_, Sep 05 2008
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