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 A135225 Pascal's triangle A007318 augmented with a leftmost border column of 1's. 8
 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, 8, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 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) LINKS 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,...). 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 |... \ CROSSREFS Cf. A007318, A094373. A027641, A027642, A062715, A103438, A132440 Sequence in context: A124445 A124279 A297299 * A208891 A177767 A047030 Adjacent sequences:  A135222 A135223 A135224 * A135226 A135227 A135228 KEYWORD nonn,easy,tabl AUTHOR Gary W. Adamson, Nov 23 2007 EXTENSIONS Corrected by R. J. Mathar, Apr 16 2013 STATUS approved

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Last modified February 19 21:59 EST 2019. Contains 320328 sequences. (Running on oeis4.)