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A135225
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Pascal's triangle A007318 augmented with a leftmost border column of 1's.
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5
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1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 4, 6, 4, 1, 1, 1, 5, 10, 10, 5, 1, 1, 1, 6, 15, 20, 15, 6, 1, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 1, 8, 28, 56, 70, 56, 28, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,9
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COMMENTS
| From Peter Bala, Sep 08 2011: (Start)
This augmented Pascal array, call it P, has interesting connections with the Bernoulli polynomials B(n,x). The infinitesimal generator S of P is the array such that exp(S) = P. The array S is obtained by augmenting the infinitesimal generator A132440 of the Pascal triangle with an initial column [0,0,1/2,1/6,0,-1/30,...] on the left. The entries in this column, after the first two zeros, are the Bernoulli values B(n,1), n>=1.
The array P is also connected with the problem of summing powers of consecutive integers. In the array P^n, the entry in position p+1 of the first column is equal to sum {k = 1..n} k^p - see the Example section below.
For similar results for the square of Pascal's triangle see A062715.
Note: If we augment Pascal's triangle with the column [1,1,x,x^2,x^3,...] on the left, the resulting lower unit triangular array has the Bernoulli polynomials B(n,x) in the first column of its infinitesimal generator. The present case is when x = 1.
(End)
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FORMULA
| A103451 * A007318 * A000012(signed), where A000012(signed) = (1; -1,1; 1,-1,1;...); as infinite lower triangular matrices. Given A007318, bin(n,k) is shifted to T(n+1,k+1) and a leftmost border of 1's is added. Row sums = A094373: (1, 2, 3, 5, 9, 17, 33,...).
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EXAMPLE
| First few rows of the triangle are:
1;
1, 1;
1, 1, 1;
1, 1, 2, 1;
1, 1, 3, 3, 1;
1, 1, 4, 6, 4, 1;
...
The infinitesimal generator for P begins
/0
|0.......0
|1/2.....1...0
|1/6.....0...2....0
|0.......0...0....3....0
|-1/30...0...0....0....4....0
|0.......0...0....0....0....5....0
|1/42....0...0....0....0....0....6....0
|...
\
The array P^n begins
/1
|1+1+...+1........1
|1+2+...+n........n.........1
|1+2^2+...+n^2....n^2.....2*n........1
|1+2^3+...+n^3....n^3.....3*n^2....3*n.......1
|...
\
More generally, the array P^t, defined as exp(t*S) for complex t, begins
/1
|B(1,1+t)-B(1,1)..........1
|1/2*(B(2,1+t)-B(2,1))....t.........1
|1/3*(B(3,1+t)-B(3,1))....t^2.....2*t........1
|1/4*(B(4,1+t)-B(4,1))....t^3.....3*t^2....3*t.......1
|...
\
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CROSSREFS
| Cf. A007318, A094373. A027641, A027642, A062715, A103438, A132440
Sequence in context: A193516 A124445 A124279 * A177767 A047030 A047120
Adjacent sequences: A135222 A135223 A135224 * A135226 A135227 A135228
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KEYWORD
| nonn,tabl
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AUTHOR
| Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 23 2007
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