A143631 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).

0, 1, 1, 1, 0, -9, -64, -348, -1672, -7307, -28225, -81817, 14191, 3143571, 38184875, 353727284, 2916494333, 22260343389, 157677357255, 1007259846130, 5241783274713, 12146415146776, -210638381350012, -4813155361775252

Offset: 0

Author

Peter Bala, Sep 05 2008

Keywords

• easy
• sign

Comments

The other sequences are A(n) = A143628(n) and C(n) = A143630(n). Compare with A121867 and A121868. See also A143816.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..578
• Eric Weisstein's World of Mathematics, Bell Polynomial.

Crossrefs

A121867, A121868, A143628, A143629, A143630, A143816.

Programs

Maple

# Compare with A143816
#
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:
A143631:=[seq(b[n], n=0..M)];

Mathematica

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 *)

PARI

Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
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

Formula

a(n) = A143629(n) + A143630(n).

From Seiichi Manyama, Oct 15 2022: (Start)

a(n) = Sum_{k = 0..floor((n-1)/3)} (-1)^k * Stirling2(n,3*k+1).

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)