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A143628 Define E(n) = sum {k = 0..inf} (-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). 15
1, 0, 0, -1, -6, -25, -89, -280, -700, -380, 13452, 149831, 1214852, 8700263, 57515640, 351296151, 1909757620, 8017484274, 5703377941, -428273438434, -7295220035921, -89868583754993, -970185398785810, -9657428906237364 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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..inf} (-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(0). See A143629 and A143630 for the sequence of coefficients of E(1) and E(2) respectively. The functions F(n) := sum {k = 0..inf} (-1)^floor((k+1)/3)*k^n/k! and G(n) = sum {k = 0..inf} (-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.

LINKS

Table of n, a(n) for n=0..23.

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) = A(n). The other sequences are B(n) = A143630 and C(n) = A143629. Compare with A143815. Also a(n) = A143629(n) + A000587(n).

EXAMPLE

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) = -25 because E(5) = -25*E(0) - 23*E(1) + 14*E(2).

a(6) = -89 because E(6) = -89*E(0) - 80*E(1) + 16*E(2).

MAPLE

# Compare with A143815

#

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:

A143628:=[seq(a[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}]]; Table[a[n], {n, 0, m}] (* Jean-Fran├žois Alcover, Mar 06 2013, after Maple *)

CROSSREFS

A000587, A121867, A143629, A143630, A143631, A143815, A143816, A143817, A143818.

Sequence in context: A055585 A099625 A209243 * A056279 A055337 A001871

Adjacent sequences:  A143625 A143626 A143627 * A143629 A143630 A143631

KEYWORD

easy,sign

AUTHOR

Peter Bala, Sep 05 2008

STATUS

approved

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Last modified May 19 22:32 EDT 2019. Contains 323411 sequences. (Running on oeis4.)