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).

0, 0, 1, 3, 7, 14, 16, -77, -922, -6660, -41264, -233828, -1218392, -5607225, -19220589, 4397930, 1016675382, 14251497833, 151695504253, 1432992328055, 12527186450276, 102042171190168, 760272520469199, 4849866087637364

Offset: 0

Author

Peter Bala, Sep 05 2008

Keywords

• easy
• sign

Comments

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.

Examples

			E(n) as linear combination of E(i),
i = 0..2.
====================================
..E(n)..|.....E(0)....E(1).....E(2).
====================================
..E(3)..|......-1......-2........3..
..E(4)..|......-6......-7........7..
..E(5)..|.....-25.....-23.......14..
..E(6)..|.....-89.....-80.......16..
..E(7)..|....-280....-271......-77..
..E(8)..|....-700....-750.....-922..
..E(9)..|....-380....-647....-6660..
..E(10).|...13452...13039...-41264..
...
a(5) = 14 because E(5) = -25*E(0) - 23*E(1) + 14*E(2).
a(6) = 16 because E(6) = -89*E(0) - 80*E(1) + 16*E(2).
		

Links

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

Crossrefs

A000587, A121867, A143628, A143629, A143631, A143815, A143816, A143817, A143818.

Programs

Maple

# Compare with A143817
#
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:
A143630:=[seq(c[n], n=0..M)];

Mathematica

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 *)

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*Bell_poly(n, -w)+w^2*Bell_poly(n, -w^2))/3; \\ Seiichi Manyama, Oct 15 2022

Formula

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.

From Seiichi Manyama, Oct 15 2022: (Start)

a(n) = Sum_{k = 0..floor((n-2)/3)} (-1)^k * Stirling2(n,3*k+2).

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)