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A143630 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(2).

%I A143630 #21 Feb 16 2025 08:33:08
%S A143630 0,0,1,3,7,14,16,-77,-922,-6660,-41264,-233828,-1218392,-5607225,
%T A143630 -19220589,4397930,1016675382,14251497833,151695504253,1432992328055,
%U A143630 12527186450276,102042171190168,760272520469199,4849866087637364
%N A143630 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(2).
%C A143630 This sequence and its companion sequences A143628 and A143629 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(2). The precise result is E(n) = A143628(n)*E(0) + A143629(n)*E(1) + A143630(n)*E(2). Compare with A121867 and A143815.
%H A143630 Seiichi Manyama, <a href="/A143630/b143630.txt">Table of n, a(n) for n = 0..578</a>
%H A143630 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%F A143630 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) = C(n). The other sequences are A(n) = A143628 and B(n) = A143631. Compare with A143817.
%F A143630 From _Seiichi Manyama_, Oct 15 2022: (Start)
%F A143630 a(n) = Sum_{k = 0..floor((n-2)/3)} (-1)^k * Stirling2(n,3*k+2).
%F A143630 a(n) = ( Bell_n(-1) + w * Bell_n(-w) + w^2 * Bell_n(-w^2) )/3, where Bell_n(x) is n-th Bell polynomial and w = exp(2*Pi*i/3). (End)
%e A143630 E(n) as linear combination of E(i),
%e A143630 i = 0..2.
%e A143630 ====================================
%e A143630 ..E(n)..|.....E(0)....E(1).....E(2).
%e A143630 ====================================
%e A143630 ..E(3)..|......-1......-2........3..
%e A143630 ..E(4)..|......-6......-7........7..
%e A143630 ..E(5)..|.....-25.....-23.......14..
%e A143630 ..E(6)..|.....-89.....-80.......16..
%e A143630 ..E(7)..|....-280....-271......-77..
%e A143630 ..E(8)..|....-700....-750.....-922..
%e A143630 ..E(9)..|....-380....-647....-6660..
%e A143630 ..E(10).|...13452...13039...-41264..
%e A143630 ...
%e A143630 a(5) = 14 because E(5) = -25*E(0) - 23*E(1) + 14*E(2).
%e A143630 a(6) = 16 because E(6) = -89*E(0) - 80*E(1) + 16*E(2).
%p A143630 # Compare with A143817
%p A143630 #
%p A143630 M:=24: a:=array(0..100): b:=array(0..100): c:=array(0..100):
%p A143630 a[0]:=1: b[0]:=0: c[0]:=0:
%p A143630 for n from 1 to M do
%p A143630 a[n]:= -add(binomial(n-1,k)*c[k], k=0..n-1);
%p A143630 b[n]:= add(binomial(n-1,k)*a[k], k=0..n-1);
%p A143630 c[n]:= add(binomial(n-1,k)*b[k], k=0..n-1);
%p A143630 end do:
%p A143630 A143630:=[seq(c[n], n=0..M)];
%t A143630 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}]]; A143630 = Table[c[n], {n, 0, m}] (* _Jean-François Alcover_, Mar 06 2013, after Maple *)
%o A143630 (PARI) Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
%o A143630 a(n) = my(w=(-1+sqrt(3)*I)/2); round(Bell_poly(n, -1)+w*Bell_poly(n, -w)+w^2*Bell_poly(n, -w^2))/3; \\ _Seiichi Manyama_, Oct 15 2022
%Y A143630 A000587, A121867, A143628, A143629, A143631, A143815, A143816, A143817, A143818.
%K A143630 easy,sign
%O A143630 0,4
%A A143630 _Peter Bala_, Sep 05 2008