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The factorial polynomials are to [finite] difference calculus what polynomials are to [infinitesimal] differential calculus.
Falling factorial and raising factorial
The falling factorial is defined as[1]
-
and the raising factorial is defined as[1]
-
where, in either case, for
we get the
empty product, i.e.
1.
Factorial polynomials
A factorial term (Boole, 1970: p. 6) or a factorial polynomial (Elaydi, 2005: p. 60) is defined as
-
and for negative
we have the definition
-
x ( − k ) := [x k ] − 1, k ∈ ℕ, |
where, in either case, for
we get the
empty product, i.e.
1.
The
factorial polynomials of degree
are defined as the sum of
factorial terms
-
P (x) := ak x (k ), n ∈ ℕ, |
where for
we get the constant polynomial
.
Generalized factorial polynomials
In a fashion similar to Laurent polynomials, we also have the generalized factorial polynomials
-
L (x) := ak x (k ), m ∈ ℤ, n ∈ ℤ, m ≤ n. |
Finite difference operator
With the above definitions, the [finite] difference operator
-
Δ f (x) := (E − I ) f (x) = f (x + 1) − f (x) |
where
is the
shift operator and
is the
identity operator, behaves with the [whether ordinary or generalized] factorial polynomials like the [infinitesimal] differential operator does with the [whether ordinary or Laurent] polynomials.
Difference operator
|
Differential operator
|
|
|
Δ x (k ) = k x (k − 1), k ∈ ℕ+ |
|
D x k = k x k − 1, k ∈ ℕ+ |
|
Δ x ( − k ) = ( − k ) x ( − k − 1), k ∈ ℕ+ |
|
D x − k = ( − k ) x − k − 1, k ∈ ℕ+ |
|
Δ{ ak x (k )} = ak Δ x (k ), m ∈ ℤ, n ∈ ℤ, m ≤ n |
|
D{ ak x k} = ak D x k, m ∈ ℤ, n ∈ ℤ, m ≤ n |
|
Converting from polynomial representation to factorial polynomial representation and vice versa
From polynomial representation to factorial polynomial representation, we have
-
(b0, b1, ..., bn )T = Fn + 1 × (a0, a1, ..., an )T, |
and from factorial polynomial representation to polynomial representation, we have
-
(a0, a1, ..., an )T = (Fn + 1) − 1 × (b0, b1, ..., bn )T, |
where
are the polynomial coefficients,
are the factorial polynomial coefficients (in both cases the degree being
), and where
is a
transformation (from polynomial to factorial polynomial) upper triangular matrix while
is the matrix inverse.
F1 = ,
F2 = ,
F3 = ,
F4 = 1 | | 0 | | 0 | | 0 |
0 | | 1 | | 1 | | 1 |
0 | | 0 | | 1 | | 3 |
0 | | 0 | | 0 | | 1 |
,
F5 = 1 | | 0 | | 0 | | 0 | | 0 |
0 | | 1 | | 1 | | 1 | | 1 |
0 | | 0 | | 1 | | 3 | | 7 |
0 | | 0 | | 0 | | 1 | | 6 |
0 | | 0 | | 0 | | 0 | | 1 |
,
F6 = 1 | | 0 | | 0 | | 0 | | 0 | | 0 |
0 | | 1 | | 1 | | 1 | | 1 | | 1 |
0 | | 0 | | 1 | | 3 | | 7 | | 15 |
0 | | 0 | | 0 | | 1 | | 6 | | 25 |
0 | | 0 | | 0 | | 0 | | 1 | | 10 |
0 | | 0 | | 0 | | 0 | | 0 | | 1 |
,
F7 = 1 | | 0 | | 0 | | 0 | | 0 | | 0 | | 0 |
0 | | 1 | | 1 | | 1 | | 1 | | 1 | | 1 |
0 | | 0 | | 1 | | 3 | | 7 | | 15 | | 31 |
0 | | 0 | | 0 | | 1 | | 6 | | 25 | | 90 |
0 | | 0 | | 0 | | 0 | | 1 | | 10 | | 65 |
0 | | 0 | | 0 | | 0 | | 0 | | 1 | | 15 |
0 | | 0 | | 0 | | 0 | | 0 | | 0 | | 1 |
, ..., S (0, 0) | | S (1, 0) | | S (2, 0) | | S (3, 0) | | S (4, 0) | | S (5, 0) | | S (6, 0) | | ⋯ |
0 | | S (1, 1) | | S (2, 1) | | S (3, 1) | | S (4, 1) | | S (5, 1) | | S (6, 1) | | ⋯ |
0 | | 0 | | S (2, 2) | | S (3, 2) | | S (4, 2) | | S (5, 2) | | S (6, 2) | | ⋯ |
0 | | 0 | | 0 | | S (3, 3) | | S (4, 3) | | S (5, 3) | | S (6, 3) | | ⋯ |
0 | | 0 | | 0 | | 0 | | S (4, 4) | | S (5, 4) | | S (6, 4) | | ⋯ |
0 | | 0 | | 0 | | 0 | | 0 | | S (5, 5) | | S (6, 5) | | ⋯ |
0 | | 0 | | 0 | | 0 | | 0 | | 0 | | S (6, 6) | | ⋯ |
⋮ | | ⋮ | | ⋮ | | ⋮ | | ⋮ | | ⋮ | | ⋮ | | ⋱ |
,
where
are Stirling numbers of the second kind.
is obtained recursively from
:
F1 | = (1); |
Fn +1(i, j) | = Fn (i, j), n ≥ 1, 0 ≤ i ≤ n − 1, 0 ≤ j ≤ n − 1; |
Fn +1(n, j) | = 0, n ≥ 1, 0 ≤ j ≤ n − 1; |
Fn +1(0, n) | = 0, n ≥ 1; |
Fn +1(i, n) | = i Fn (i, n − 1) + Fn (i − 1, n − 1), n ≥ 1, 1 ≤ i ≤ n.
|
|
Starting with the central diagonal of the upper triangular matrix, the diagonals are given by:
- A000012 Stirling numbers of the second kind .
{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...}
- A000217 Stirling numbers of the second kind .
{0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, ...}
- A001296 Stirling numbers of the second kind .
{0, 1, 7, 25, 65, 140, 266, 462, 750, 1155, 1705, 2431, 3367, 4550, 6020, 7820, 9996, ...}
- A001297 Stirling numbers of the second kind .
{0, 1, 15, 90, 350, 1050, 2646, 5880, 11880, 22275, 39325, 66066, 106470, 165620, 249900, ...}
- A001298 Stirling numbers of the second kind .
{0, 1, 31, 301, 1701, 6951, 22827, 63987, 159027, 359502, 752752, 1479478, 2757118, ...}
- A112494 Stirling numbers of the second kind .
{0, 1, 63, 966, 7770, 42525, 179487, 627396, 1899612, 5135130, 12662650, 28936908, 62022324, ...}
- A144969 Stirling numbers of the second kind .
{0, 1, 127, 3025, 34105, 246730, 1323652, 5715424, 20912320, 67128490, 193754990, ...}
Stirling numbers
For
, we have
-
x (n) := x n = s (n, k) x k = S1(n, k) x k = (−1) n + k | S1(n, k) | x k, |
and
-
x ( − n ) := [x n̅ ] − 1 = [ (−1) n + k S1(n, k ) [x − k ] − 1] − 1 = [ (−1) n + k S1(n, k ) x k] − 1 = |
[ | S1(n, k ) | x k] − 1 = [ [ nk ] x k] − 1, |
where
or
are
Stirling numbers of the first kind, and
are
unsigned Stirling numbers of the first kind.
[2]
For
, we have
-
x n = { nk } x (k ) = S (n) k x (k ) = S (n, k ) x (k ) = S2(n, k ) x (k ), |
and
-
x − n := [x n ] − 1 = [ (−1) n + k S2(n, k ) [x ( − k ) ] − 1] − 1 = [ (−1) n + k S2(n, k ) x k̅] − 1, |
where
,
,
or
are
Stirling numbers of the second kind, and
are
signed Stirling numbers of the second kind.
[2]
Stirling numbers of the first kind
We have
-
involving the
triangle of Stirling numbers of the first kind, where the coefficients of row
are obtained by multiplying the polynomial of row
by
.
Triangle of Stirling numbers of the first kind
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0
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1
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1
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1
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0
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1
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1
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2
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0
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−1
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1
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0
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3
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0
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2
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−3
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1
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0
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4
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0
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−6
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11
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−6
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1
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0
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5
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0
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24
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−50
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35
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−10
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1
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0
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6
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0
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−120
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274
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−225
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85
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−15
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1
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0
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7
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0
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720
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−1764
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1624
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−735
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175
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−21
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1
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0
|
8
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0
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−5040
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13068
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−13132
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6769
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−1960
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322
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−28
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1
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0
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9
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0
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40320
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−109584
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118124
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−67284
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22449
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−4536
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546
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−36
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1
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0
|
10
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0
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−362880
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1026576
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−1172700
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723680
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−269325
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63273
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−9450
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870
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−45
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1
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0
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1
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2
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3
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4
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5
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6
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7
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8
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9
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10
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A048994 Triangle read by rows of Stirling numbers of the first kind,
S1(n, k), n ≥ 0, 0 ≤ k ≤ n |
.
-
{1, 0, 1, 0, −1, 1, 0, 2, −3, 1, 0, − 6, 11, − 6, 1, 0, 24, −50, 35, −10, 1, 0, −120, 274, −225, 85, −15, 1, 0, 720, −1764, 1624, −735, 175, −21, 1, 0, −5040, 13068, −13132, 6769, −1960, 322, −28, 1, ...}
A008275 Triangle read by rows of Stirling numbers of the first kind,
S1(n, k), n ≥ 1, 1 ≤ k ≤ n |
.
-
{1, −1, 1, 2, −3, 1, − 6, 11, − 6, 1, 24, −50, 35, −10, 1, −120, 274, −225, 85, −15, 1, 720, −1764, 1624, −735, 175, −21, 1, −5040, 13068, −13132, 6769, −1960, 322, −28, 1, ...}
Row sums of Stirling numbers of the first kind
The row sums of Stirling numbers of the first kind are
-
S1(n, k) = 0 n (2) = 0 n (n − 1), n ≥ 0. |
Unsigned Stirling numbers of the first kind
We have
-
involving the
triangle of unsigned Stirling numbers of the first kind, also called the
factorial triangle,
[3] where the coefficients of row
are obtained by multiplying the polynomial of row
by
.
A132393 Triangle of unsigned Stirling numbers of the first kind (see
A048994), read by rows,
| S (n, k) | , n ≥ 0, 0 ≤ k ≤ n |
.
-
{1, 0, 1, 0, 1, 1, 0, 2, 3, 1, 0, 6, 11, 6, 1, 0, 24, 50, 35, 10, 1, 0, 120, 274, 225, 85, 15, 1, 0, 720, 1764, 1624, 735, 175, 21, 1, 0, 5040, 13068, 13132, 6769, 1960, 322, 28, 1, ...}
Row sums of triangle of unsigned Stirling numbers of the first kind
The row sums of unsigned Stirling numbers of the first kind are
-
| S1(n, k) | = | S1(n + 1, 1) | = n!, n ≥ 0. |
Falling diagonals of triangle of unsigned Stirling numbers of the first kind
The coefficients of the
th falling diagonal, where
refers to the rightmost one, are given by a polynomial of degree
.
A000012 Stirling numbers of the first kind:
(The simplest sequence of positive numbers: the all
1’s sequence.)
[
] G.f.:
.
-
{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...}
A000217 Stirling numbers of the first kind:
(
Triangular numbers:
a (n) = ( n + 12 ) = n (n + 1) / 2 = 0 + 1 + 2 + ... + n |
.)
[
a1(n) = (n + n 2 ) / 2 = n (1 + n) / 2 |
] G.f.:
.
-
{0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, ...}
A000914 Stirling numbers of the first kind:
.
[
a2(n) = (10 n + 21 n 2 + 14 n 3 + 3 n 4 ) / 24 = n (1 + n) (2 + n) (5 + 3 n) / 24 |
] G.f.:
.
-
{0, 2, 11, 35, 85, 175, 322, 546, 870, 1320, 1925, 2717, 3731, 5005, 6580, 8500, 10812, 13566, 16815, 20615, 25025, 30107, 35926, 42550, 50050, 58500, 67977, 78561, 90335, ...}
A001303 Stirling numbers of first kind,
negated.
[
a3(n) = (36 n + 96 n 2 + 97 n 3 + 47 n 4 + 11 n 5 + n 6 ) / 48 = n (1 + n) (2 + n) 2 (3 + n) 2 / 48 |
] G.f.:
(6 x + 8 x 2 + x 3) / (1 − x) 7 |
.
-
{6, 50, 225, 735, 1960, 4536, 9450, 18150, 32670, 55770, 91091, 143325, 218400, 323680, 468180, 662796, 920550, 1256850, 1689765, 2240315, 2932776, 3795000, 4858750, 6160050, ...}
A000915 Stirling numbers of first kind
. [
] G.f.:
(24 x + 58 x 2 + 22 x 3 + x 4) / (1 − x) 9 |
.
-
{24, 274, 1624, 6769, 22449, 63273, 157773, 357423, 749463, 1474473, 2749747, 4899622, 8394022, 13896582, 22323822, 34916946, 53327946, 79721796, 116896626, 16842387, ...}
Triangle of polynomials for the numerator of the o.g.f.s for unsigned Stirling numbers of first kind along the falling diagonals:
-
A112007 Coefficient triangle for polynomials used for o.g.f.s for unsigned Stirling1 diagonals, for
.
-
{1, 2, 1, 6, 8, 1, 24, 58, 22, 1, 120, 444, 328, 52, 1, 720, 3708, 4400, 1452, 114, 1, 5040, 33984, 58140, 32120, 5610, 240, 1, 40320, 341136, 785304, 644020, 195800, 19950, ...}
Stirling numbers of the second kind
We have
-
involving the triangle of Stirling numbers of the second kind.
Triangle of Stirling numbers of the second kind
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0
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1
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1
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1
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0
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1
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1
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2
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0
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1
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1
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2
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3
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0
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1
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3
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1
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5
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4
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0
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1
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7
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6
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1
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15
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5
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0
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1
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15
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25
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10
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1
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52
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6
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0
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1
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31
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90
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65
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15
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1
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203
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7
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0
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1
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63
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301
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350
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140
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21
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1
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877
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8
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0
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1
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127
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966
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1701
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1050
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266
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28
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1
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4140
|
9
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0
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1
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255
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3025
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7770
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6951
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2646
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462
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36
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1
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21147
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10
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0
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1
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511
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9330
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34105
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42525
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22827
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5880
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750
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45
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1
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115975
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1
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2
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3
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4
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5
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6
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7
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8
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9
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10
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A048993 Triangle of Stirling numbers of second kind,
S 2(n, k ), n ≥ 0, 0 ≤ k ≤ n |
.
-
{1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 7, 6, 1, 0, 1, 15, 25, 10, 1, 0, 1, 31, 90, 65, 15, 1, 0, 1, 63, 301, 350, 140, 21, 1, 0, 1, 127, 966, 1701, 1050, 266, 28, 1, 0, 1, 255, 3025, 7770, 6951, 2646, 462, 36, 1, ...}
A106800 Triangle of Stirling numbers of second kind,
S 2(n, n − k ), n ≥ 0, 0 ≤ k ≤ n |
.
-
{1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 7, 1, 0, 1, 10, 25, 15, 1, 0, 1, 15, 65, 90, 31, 1, 0, 1, 21, 140, 350, 301, 63, 1, 0, 1, 28, 266, 1050, 1701, 966, 127, 1, 0, 1, 36, 462, 2646, 6951, 7770, 3025, 255, 1, ...}
A008277 Triangle read by rows of Stirling numbers of the second kind,
S 2(n, k ), n ≥ 1, 1 ≤ k ≤ n |
.
-
{1, 1, 1, 1, 3, 1, 1, 7, 6, 1, 1, 15, 25, 10, 1, 1, 31, 90, 65, 15, 1, 1, 63, 301, 350, 140, 21, 1, 1, 127, 966, 1701, 1050, 266, 28, 1, 1, 255, 3025, 7770, 6951, 2646, 462, 36, 1, ...}
Row sums of triangle of Stirling numbers of the second kind
The row sums of Stirling numbers of the second kind are
-
where
are the
Bell (or exponential) numbers.
A000110 Bell or exponential numbers: number of ways to partition a set of
labeled elements.
-
{1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597, 27644437, 190899322, 1382958545, 10480142147, 82864869804, 682076806159, 5832742205057, 51724158235372, ...}
Bell polynomials
By replacing the
by
in
-
x n = S2(n, k ) x (k), n ≥ 0, |
one obtains the Bell polynomials
-
Bn(x) := S2(n, k ) x k, n ≥ 0, |
for which
yields the
Bell numbers and
yields the
complementary Bell numbers.
Signed Stirling numbers of the second kind
We have
-
involving the triangle of signed Stirling numbers of the second kind.
A?????? Triangle of signed Stirling numbers of second kind,
( − 1) n + k S2(n, k ), n ≥ 0, 0 ≤ k ≤ n |
.
-
{1, 0, 1, 0, −1, 1, 0, 1, −3, 1, 0, −1, 7, − 6, 1, 0, 1, −15, 25, −10, 1, 0, −1, 31, − 90, 65, −15, 1, 0, 1, − 63, 301, −350, 140, −21, 1, 0, −1, 127, − 966, 1701, −1050, 266, −28, 1, ...}
A?????? Triangle read by rows of signed Stirling numbers of the second kind,
( − 1) n + k S2(n, k ), n ≥ 1, 1 ≤ k ≤ n |
.
-
{?, ...}
Row sums of triangle of signed Stirling numbers of the second kind
The row sums of signed Stirling numbers of the second kind
-
(−1) n + k S2(n, k ) = (−1) n (−1) k S2(n, k ) = (−1) n B̃n , n ≥ 0, |
where
are the
Rao Uppuluri–Carpenter numbers (or complementary Bell numbers), yield the sequence
-
{1, (−) −1, 0, (−) 1, 1, (−) −2, − 9, (−) − 9, 50, (−) 267, 413, (−) −2180, −17731, (−) −50533, 110176, (−) 1966797, 9938669, (−) 8638718, −278475061, (−) −2540956509, − 9816860358, (−) 27172288399, ...}
A000587 Rao Uppuluri–Carpenter numbers (or complementary Bell numbers),
: e.g.f. =
.
-
{1, −1, 0, 1, 1, −2, − 9, − 9, 50, 267, 413, −2180, −17731, −50533, 110176, 1966797, 9938669, 8638718, −278475061, −2540956509, − 9816860358, 27172288399, 725503033401, 5592543175252, ...}
Lah numbers
The Lah numbers (also called “Stirling numbers of the third kind”), discovered by Ivo Lah[4] in 1954, are defined as
-
L (n, k ) := | s (n, m) | S (m, k ) , |
where
and
are unsigned Stirling numbers of the first kind and Stirling numbers of the second kind, respectively.
The Lah numbers express the “rising factorials” in terms of the “falling factorials” and vice versa, i.e.
-
-
x n = (−1) n − k L (n, k ) x k. |
Triangle of Lah numbers
|
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1
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−1
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−1
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2
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2
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1
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3
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3
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−6
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−6
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−1
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−13
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4
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24
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36
|
12
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1
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|
73
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5
|
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−120
|
−240
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−120
|
−20
|
−1
|
|
−501
|
6
|
|
720
|
1800
|
1200
|
300
|
30
|
1
|
|
4051
|
7
|
|
−5040
|
−15120
|
−12600
|
−4200
|
−630
|
−42
|
−1
|
|
−37633
|
8
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|
40320
|
141120
|
141120
|
58800
|
11760
|
1176
|
56
|
1
|
|
394353
|
9
|
|
−362880
|
−1451520
|
−1693440
|
−846720
|
−211680
|
−28224
|
−2016
|
−72
|
−1
|
|
−4596553
|
10
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|
3628800
|
16329600
|
21772800
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12700800
|
3810240
|
635040
|
60480
|
3240
|
90
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1
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58941091
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11
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|
−39916800
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−199584000
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−299376000
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−199584000
|
−69854400
|
−13970880
|
−1663200
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−118800
|
−4950
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−110
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−1
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−824073141
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2
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3
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4
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5
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6
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7
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8
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9
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10
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11
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A008297 Triangle of Lah numbers (concatenated rows).
-
{−1, 2, 1, − 6, − 6, −1, 24, 36, 12, 1, −120, −240, −120, −20, −1, 720, 1800, 1200, 300, 30, 1, −5040, −15120, −12600, − 4200, − 630, − 42, −1, 40320, 141120, 141120, 58800, 11760, ...}
Row sums of triangle of Lah numbers
A293125 Expansion of e.g.f.:
: for
, gives row sums of Lah triangle.
-
{−1, 3, −13, 73, −501, 4051, −37633, 394353, − 4596553, 58941091, −824073141, 12470162233, −202976401213, 3535017524403, − 65573803186921, 1290434218669921, ...}
Factorial polynomials inspired from ordinary binomial expansions
Factorial polynomials inspired from ordinary binomial expansion of (x + 1)n
By replacing the
by
in the
binomial expansion
(x + 1) n = () x k, n ≥ 0, |
we get
Pn (x) := () x (k ), n ≥ 0, |
which is given by the recurrence relation
P0 (x) | = 1; |
P1(x) | = 1 + x; |
Pn (x) | = (x − (n − 2)) Pn − 1(x) + (n − 1) Pn − 2(x), n ≥ 2.
|
|
Examples:
-
A?????? Triangle read by rows of ordinary polynomial coefficients resulting from replacing
by
in the binomial expansion of
, for
.
-
{1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 0, 5, −2, 1, 1, 9, −15, 15, −5, 1, 1, −35, 94, −85, 40, − 9, 1, ...}
Related to (although the signs differ):
A269953 Triangle read by rows,
T (n, k ) = ( − j − 1 − n − 1 ) S1( j, k ) |
where
are the Stirling cycle numbers
A132393, for
and
.
-
{1, −1, 1, 1, −1, 1, −1, 2, 0, 1, 1, 0, 5, 2, 1, −1, 9, 15, 15, 5, 1, 1, 35, 94, 85, 40, 9, 1, −1, 230, 595, 609, 315, 91, 14, 1, 1, 1624, 4458, 4844, 2779, 924, 182, 20, 1, −1, ...}
Triangle row sums
It appears that the row sums yield:
-
Triangle columns
It appears that we might have for column 1:
T (n, 1) = ( − 1) [n belongs to some set] A002741 (n), n ≥ 0, |
where
[⋅] is the
Iverson bracket.
-
{0, 1, 1, 2, 0, 9, −35, ...}
A002741 Logarithmic numbers: expansion of
− log (1 − x) exp ( − x), n ≥ 0 |
.
-
{0, 1, −1, 2, 0, 9, 35, 230, 1624, 13209, 120287, 1214674, 13469896, 162744945, 2128047987, 29943053062, 451123462672, 7245940789073, ...}
Factorial polynomials inspired from ordinary binomial expansion of (x − 1)n
By replacing the
by
in the
binomial expansion
(x − 1) n = () (−1) n − k x k, n ≥ 0, |
we get
Pn (x) := () (−1) n − k x (k ), n ≥ 0, |
which is given by the recurrence relation
P0 (x) | = 1; |
P1(x) | = − 1 + x; |
Pn (x) | = (x − n) Pn − 1(x) − (n − 1) Pn − 2(x), n ≥ 2.
|
|
Examples:
-
A?????? Triangle read by rows of ordinary polynomial coefficients resulting from replacing
by
in the binomial expansion of
, for
.
-
{1, −1, 1, 1, −3, 1, −1, 8, − 6, 1, 1, −24, 29, −10, 1, −1, 89, −145, 75, −15, 1, 1, − 415, 814, −545, 160, −21, 1, ...}
This absolute values are given by:
A094816 Triangle read by rows:
equals coefficients of Charlier polynomials:
A046716 transposed.
-
{1, 1, 1, 1, 3, 1, 1, 8, 6, 1, 1, 24, 29, 10, 1, 1, 89, 145, 75, 15, 1, 1, 415, 814, 545, 160, 21, 1, 1, 2372, 5243, 4179, 1575, 301, ...}
Triangle row sums
It appears that the row sums yield:
Pn (1) = ( − 1) n +1 (n − 1), n ≥ 0. |
-
{1, 0, −1, 2, −3, 4, −5, ...}
Triangle columns
It appears that we have for column 1:
T (n, 1) = ( − 1) n +1 A002104 (n), n ≥ 0. |
-
{0, 1, −3, 8, −24, 89, − 415, ...}
A002104 Logarithmic numbers: expansion of
− log (1 − x) exp (x), n ≥ 0 |
.
-
{0, 1, 3, 8, 24, 89, 415, 2372, 16072, 125673, 1112083, 10976184, 119481296, 1421542641, 18348340127, 255323504932, 3809950977008, ...}
Notes
- ↑ 1.0 1.1 Using the and notation proposed in (Graham, Knuth, and Patashnik, 1994: p. 48).
- ↑ 2.0 2.1 Named after the Scottish mathematician James Stirling.
- ↑ Steven Schwartzman, “The Factorial Triangle and Polynomial Sequences,” The College Mathematics Journal, Vol. 15, No. 5 (Nov., 1984), pp. 424–426.
- ↑ O’Connor, John J.; Robertson, Edmund F., “Ivo Lah”, MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Lah.html .
References
- Boole, G. Finite Differences, 5th ed. New York, NY: Chelsea, 1970.
- Elaydi, S.N. An Introduction to Difference Equations, 3rd ed. Springer, 2005.
- Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994.
External links