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A135225 Pascal's triangle A007318 augmented with a leftmost border column of 1's. 9
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

Row sums give A094373.

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

G. C. Greubel, Rows n = 0..100 of triangle, flattened

FORMULA

A103451 * A007318 * A000012(signed), where A000012(signed) = (1; -1,1; 1,-1,1; ...); as infinite lower triangular matrices.

Given A007318, binomial(n,k) is shifted to T(n+1,k+1) and a leftmost border of 1's is added.

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

  |...

  \

MAPLE

T:= proc(n, k) option remember;

      if k=0 then 1

    else binomial(n-1, k-1)

      fi; end:

seq(seq(T(n, k), k=0..n), n=0..10); # G. C. Greubel, Nov 19 2019

MATHEMATICA

T[n_, k_]:= T[n, k]= If[k==0, 1, Binomial[n-1, k-1]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 19 2019 *)

PROG

(PARI) T(n, k) = if(k==0, 1, binomial(n-1, k-1)); \\ G. C. Greubel, Nov 19 2019

(MAGMA)

T:= func< n, k | k eq 0 select 1 else Binomial(n-1, k-1) >;

[T(n, k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 19 2019

(Sage)

@CachedFunction

def T(n, k):

    if (k==0): return 1

    else: return binomial(n-1, k-1)

[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 19 2019

(GAP)

T:= function(n, k)

    if k=0 then return 1;

    else return Binomial(n-1, k-1);

    fi; end;

Flat(List([0..10], n-> List([0..n], k-> T(n, k) ))); # G. C. Greubel, Nov 19 2019

CROSSREFS

Cf. A007318, A027641, A027642, A062715, A094373, 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 January 23 03:15 EST 2022. Contains 350504 sequences. (Running on oeis4.)