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

%I A135225 #26 Mar 27 2022 19:02:48
%S A135225 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,
%T A135225 20,15,6,1,1,1,7,21,35,35,21,7,1,1,1,8,28,56,70,56,28,8,1
%N A135225 Pascal's triangle A007318 augmented with a leftmost border column of 1's.
%C A135225 Row sums give A094373.
%C A135225 From _Peter Bala_, Sep 08 2011: (Start)
%C A135225 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.
%C A135225 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.
%C A135225 For similar results for the square of Pascal's triangle see A062715.
%C A135225 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.
%C A135225 (End)
%H A135225 G. C. Greubel, <a href="/A135225/b135225.txt">Rows n = 0..100 of triangle, flattened</a>
%F A135225 A103451 * A007318 * A000012(signed), where A000012(signed) = (1; -1,1; 1,-1,1; ...); as infinite lower triangular matrices.
%F A135225 Given A007318, binomial(n,k) is shifted to T(n+1,k+1) and a leftmost border of 1's is added.
%e A135225 First few rows of the triangle:
%e A135225   1;
%e A135225   1, 1;
%e A135225   1, 1, 1;
%e A135225   1, 1, 2, 1;
%e A135225   1, 1, 3, 3, 1;
%e A135225   1, 1, 4, 6, 4, 1;
%e A135225 ...
%e A135225 The infinitesimal generator for P begins:
%e A135225   /0
%e A135225   |0.......0
%e A135225   |1/2.....1...0
%e A135225   |1/6.....0...2....0
%e A135225   |0.......0...0....3....0
%e A135225   |-1/30...0...0....0....4....0
%e A135225   |0.......0...0....0....0....5....0
%e A135225   |1/42....0...0....0....0....0....6....0
%e A135225   |...
%e A135225   \
%e A135225 The array P^n begins:
%e A135225   /1
%e A135225   |1+1+...+1........1
%e A135225   |1+2+...+n........n.........1
%e A135225   |1+2^2+...+n^2....n^2.....2*n........1
%e A135225   |1+2^3+...+n^3....n^3.....3*n^2....3*n.......1
%e A135225   |...
%e A135225   \
%e A135225 More generally, the array P^t, defined as exp(t*S) for complex t, begins:
%e A135225   /1
%e A135225   |B(1,1+t)-B(1,1)..........1
%e A135225   |1/2*(B(2,1+t)-B(2,1))....t.........1
%e A135225   |1/3*(B(3,1+t)-B(3,1))....t^2.....2*t........1
%e A135225   |1/4*(B(4,1+t)-B(4,1))....t^3.....3*t^2....3*t.......1
%e A135225   |...
%e A135225   \
%p A135225 T:= proc(n, k) option remember;
%p A135225       if k=0 then 1
%p A135225     else binomial(n-1, k-1)
%p A135225       fi; end:
%p A135225 seq(seq(T(n, k), k=0..n), n=0..10); # _G. C. Greubel_, Nov 19 2019
%t A135225 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 *)
%o A135225 (PARI) T(n,k) = if(k==0, 1, binomial(n-1, k-1)); \\ _G. C. Greubel_, Nov 19 2019
%o A135225 (Magma)
%o A135225 T:= func< n, k | k eq 0 select 1 else Binomial(n-1, k-1) >;
%o A135225 [T(n,k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Nov 19 2019
%o A135225 (Sage)
%o A135225 @CachedFunction
%o A135225 def T(n,k):
%o A135225     if (k==0): return 1
%o A135225     else: return binomial(n-1, k-1)
%o A135225 [[T(n,k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Nov 19 2019
%o A135225 (GAP)
%o A135225 T:= function(n,k)
%o A135225     if k=0 then return 1;
%o A135225     else return Binomial(n-1,k-1);
%o A135225     fi; end;
%o A135225 Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # _G. C. Greubel_, Nov 19 2019
%Y A135225 Cf. A007318, A027641, A027642, A062715, A094373, A103438, A132440.
%K A135225 nonn,easy,tabl
%O A135225 0,9
%A A135225 _Gary W. Adamson_, Nov 23 2007
%E A135225 Corrected by _R. J. Mathar_, Apr 16 2013