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A135227 Triangle A000012 * A135225, read by rows. 2

%I #8 Mar 27 2022 19:03:26

%S 1,2,1,3,2,1,4,3,3,1,5,4,6,4,1,6,5,10,10,5,1,7,6,15,20,15,6,1,8,7,21,

%T 35,35,21,7,1,9,8,28,56,70,56,28,8,1,10,9,36,84,126,126,84,36,9,1,11,

%U 10,45,120,210,252,210,120,45,10,1,12,11,55,165,330,462,462,330,165,55,11,1

%N Triangle A000012 * A135225, read by rows.

%C Row sums = A006127: (1, 3, 6, 11, 20, 37, ...).

%H G. C. Greubel, <a href="/A135227/b135227.txt">Rows n = 0..100 of triangle, flattened</a>

%F A000012 * A135225 as infinite lower triangular matrices. Left border of 1's in Pascal's Triangle (A007318) is replaced with a column of (1,2,3,...).

%F T(n,k) = binomial(n,k), with T(n,0) = n+1. - _G. C. Greubel_, Nov 20 2019

%e First few rows of the triangle:

%e 1;

%e 2, 1;

%e 3, 2, 1;

%e 4, 3, 3, 1;

%e 5, 4, 6, 4, 1;

%e 6, 5, 10, 10, 5, 1;

%e 7, 6, 15, 20, 15, 6, 1;

%e ...

%p seq(seq( `if`(k=0, n+1, binomial(n,k)), k=0..n), n=0..12); # _G. C. Greubel_, Nov 20 2019

%t Table[If[k==0, n+1, Binomial[n, k]], {n, 0, 12}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Nov 20 2019 *)

%o (PARI) T(n,k) = if(k==0, n+1, binomial(n,k)); \\ _G. C. Greubel_, Nov 20 2019

%o (Magma) [k eq 0 select n+1 else Binomial(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 20 2019

%o (Sage)

%o def T(n, k):

%o if (k==0): return 1

%o else: return binomial(n, k)

%o [[T(n, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Nov 20 2019

%o (GAP)

%o T:= function(n,k)

%o if k=0 then return 1;

%o else return Binomial(n,k);

%o fi; end;

%o Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # _G. C. Greubel_, Nov 20 2019

%Y Cf. A006127, A007318, A135225.

%K nonn,tabl

%O 0,2

%A _Gary W. Adamson_, Nov 23 2007

%E More terms added by _G. C. Greubel_, Nov 20 2019

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Last modified March 28 12:26 EDT 2024. Contains 371254 sequences. (Running on oeis4.)