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%I #19 Dec 29 2023 10:26:04
%S 0,1,1,1,0,-9,-64,-348,-1672,-7307,-28225,-81817,14191,3143571,
%T 38184875,353727284,2916494333,22260343389,157677357255,1007259846130,
%U 5241783274713,12146415146776,-210638381350012,-4813155361775252
%N Let A(0) = 1, B(0) = 0 and C(0) = 0. Let A(n+1) = - Sum_{k = 0..n} binomial(n,k)*C(k), B(n+1) = Sum_{k = 0..n} binomial(n,k)*A(k) and C(n+1) = Sum_{k = 0..n} binomial(n,k)*B(k). This entry gives the sequence B(n).
%C The other sequences are A(n) = A143628(n) and C(n) = A143630(n). Compare with A121867 and A121868. See also A143816.
%H Seiichi Manyama, <a href="/A143631/b143631.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 a(n) = A143629(n) + A143630(n).
%F From _Seiichi Manyama_, Oct 15 2022: (Start)
%F a(n) = Sum_{k = 0..floor((n-1)/3)} (-1)^k * Stirling2(n,3*k+1).
%F a(n) = -( Bell_n(-1) + w^2 * Bell_n(-w) + w * Bell_n(-w^2) )/3, where Bell_n(x) is n-th Bell polynomial and w = exp(2*Pi*i/3). (End)
%p # Compare with A143816
%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 A143631:=[seq(b[n], n=0..M)];
%t m = 23; a[0] = 1; b[0] = 0; c[0] = 0; For[n = 1, n <= m, n++, 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}]; a[n] = Sum[Binomial[n - 1, k]*c[k], {k, 0, n - 1}]]; A143631 = Table[ -b[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)+w^2*Bell_poly(n, -w)+w*Bell_poly(n, -w^2))/3; \\ _Seiichi Manyama_, Oct 15 2022
%Y A121867, A121868, A143628, A143629, A143630, A143816.
%K easy,sign
%O 0,6
%A _Peter Bala_, Sep 05 2008
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