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A143629
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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(1).
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12
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0, 1, 0, -2, -7, -23, -80, -271, -750, -647, 13039, 152011, 1232583, 8750796, 57405464, 349329354, 1899818951, 8008845556, 5981853002, -425732481925, -7285403175563, -89895756043392, -970910901819211, -9663021449412616
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| This sequence and its companion sequences A143628 and A143630 may be viewed as generalisations 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). This sequence lists the coefficients of E(1). Some examples are given below. The precise result for E(n) as a linear combination of E(0), E(1) and E(2) is E(n) = A143628(n)*E(0) + A143629(n)*E(1) + A143630(n)*E(2). Compare with A121867 and A143815.
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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) = B(n) - C(n). The other sequences are A(n) = A143628 and C(n) = A143630. The values of B(n) are recorded in A143631. Compare with A143818. Also a(n) = A143628(n) - A000587(n).
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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) = -23 because E(5) = -25*E(0) - 23*E(1) + 14*E(2).
a(6) = -80 because E(6) = -89*E(0) - 80*E(1) + 16*E(2).
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MAPLE
| # Compare with A143818
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:
A143629:=[seq(b[n]-c[n], n=0..M)];
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CROSSREFS
| A000587, A121867, A143628, A143630, A143631, A143815, A143816, A143817, A143818.
Sequence in context: A007717 A130567 A091514 * A176287 A119371 A151290
Adjacent sequences: A143626 A143627 A143628 * A143630 A143631 A143632
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KEYWORD
| easy,sign
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AUTHOR
| Peter Bala (pbala(AT)toucansurf.com), Sep 05 2008
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