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A135223 Triangle A000012 * A127648 * A103451, read by rows.

%I A135223 #13 Mar 27 2022 18:25:29
%S A135223 1,3,2,6,2,3,10,2,3,4,15,2,3,4,5,21,2,3,4,5,6,28,2,3,4,5,6,7,36,2,3,4,
%T A135223 5,6,7,8,45,2,3,4,5,6,7,8,9,55,2,3,4,5,6,7,8,9,10,66,2,3,4,5,6,7,8,9,
%U A135223 10,11,78,2,3,4,5,6,7,8,9,10,11,12,91,2,3,4,5,6,7,8,9,10,11,12,13
%N A135223 Triangle A000012 * A127648 * A103451, read by rows.
%C A135223 Row sums = A028387.
%H A135223 G. C. Greubel, <a href="/A135223/b135223.txt">Rows n = 1..100 of triangle, flattened</a>
%F A135223 T(n,k) = A000012(n,k) * A127648(n,k) * A103451(n,k) as infinite lower triangular matrices. Replace left border of 1's in A002260 with (1, 3, 6, 10, 15, ...).
%F A135223 T(n, k) = k with T(n,1) = binomial(n+1, 2). - _G. C. Greubel_, Nov 20 2019
%e A135223 First few rows of the triangle are:
%e A135223    1;
%e A135223    3, 2;
%e A135223    6, 2, 3;
%e A135223   10, 2, 3, 4;
%e A135223   15, 2, 3, 4, 5;
%e A135223 ...
%p A135223 seq(seq( `if`(k=1, binomial(n+1,2), k), k=1..n), n=1..15); # _G. C. Greubel_, Nov 20 2019
%t A135223 T[n_, k_]:= T[n, k]= If[k==1, Binomial[n+1, 2], k]; Table[T[n, k], {n, 15}, {k,n}]//Flatten (* _G. C. Greubel_, Nov 20 2019 *)
%o A135223 (PARI) T(n,k) = if(k==1, binomial(n+1,2), k); \\ _G. C. Greubel_, Nov 20 2019
%o A135223 (Magma) [k eq 1 select Binomial(n+1,2) else k: k in [1..n], n in [1..15]]; // _G. C. Greubel_, Nov 20 2019
%o A135223 (Sage)
%o A135223 @CachedFunction
%o A135223 def T(n,k):
%o A135223     if (k==1): return binomial(n+1, 2)
%o A135223     else: return k
%o A135223 [[T(n,k) for k in (1..n)] for n in (1..15)] # _G. C. Greubel_, Nov 20 2019
%o A135223 (GAP)
%o A135223 T:= function(n,k)
%o A135223     if k=1 then return Binomial(n+1,2);
%o A135223     else return k;
%o A135223     fi; end;
%o A135223 Flat(List([1..15], n-> List([1..n], k-> T(n,k) ))); # _G. C. Greubel_, Nov 20 2019
%Y A135223 Cf. A002260, A103451, A127648.
%K A135223 nonn,tabl
%O A135223 1,2
%A A135223 _Gary W. Adamson_, Nov 23 2007
%E A135223 More terms added by _G. C. Greubel_, Nov 20 2019