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%I #12 Sep 08 2022 08:45:29
%S 1,3,1,4,1,1,7,3,1,1,6,1,1,1,1,12,6,3,1,1,1,8,1,1,1,1,1,1,15,7,3,3,1,
%T 1,1,1,13,4,4,1,1,1,1,1,1,18,8,3,3,3,1,1,1,1,1,12,1,1,1,1,1,1,1,1,1,1,
%U 28,16,10,6,3,3,1,1,1,1,1,1,14,1,1,1,1,1,1,1,1,1,1,1,1,24,10,3,3,3,3,3,1,1
%N Triangle T(n,k), partial row sums of the n-th row of A127013 read right to left.
%C Also partial row sums of the n-th row of A126988 read left to right. - _Reinhard Zumkeller_, Jan 21 2014
%H Reinhard Zumkeller, <a href="/A127057/b127057.txt">Rows n = 1..125 of triangle, flattened</a>
%F T(n,k) = Sum_{i=1..n-k+1} A127013(n,i), n>=1, 1<=k<=n.
%F T(n,k) = Sum_{i=k..n} A126988(n,i).
%F Row sums: Sum_{k=1..n} T(n,k) = A038040(n).
%F T(n,1) = A000203(n).
%F T = A126988 * M as infinite lower triangular matrices, M = (1; 1, 1; 1, 1, 1; ...).
%e The triangle starts
%e 1;
%e 3, 1;
%e 4, 1, 1;
%e 7, 3, 1, 1;
%e 6, 1, 1, 1, 1;
%e 12, 6, 3, 1, 1, 1;
%e 8, 1, 1, 1, 1, 1, 1;
%e 15, 7, 3, 3, 1, 1, 1, 1;
%e 13, 4, 4, 1, 1, 1, 1, 1, 1;
%e 18, 8, 3, 3, 3, 1, 1, 1, 1, 1; ...
%t A126988[n_, m_]:= If[Mod[n, m]==0, n/m, 0];
%t T[n_, m_]:= Sum[A126988[n, j], {j,m,n}];
%t Table[T[n, m], {n,1,12}, {m,1,n}]//Flatten (* _G. C. Greubel_, Jun 03 2019 *)
%o (Haskell)
%o a127057 n k = a127057_tabl !! (n-1) !! (k-1)
%o a127057_row n = a127057_tabl !! (n-1)
%o a127057_tabl = map (scanr1 (+)) a126988_tabl
%o -- _Reinhard Zumkeller_, Jan 21 2014
%o (PARI)
%o A126988(n, k) = if(n%k==0, n/k, 0);
%o T(n,k) = sum(j=k,n, A126988(n,j));
%o for(n=1, 12, for(k=1,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Jun 03 2019
%o (Magma)
%o A126988:= func< n,k | (n mod k) eq 0 select n/k else 0 >;
%o T:= func< n,k | (&+[A126988(n, j): j in [k..n]]) >;
%o [[T(n,k): k in [1..n]]: n in [1..12]]; // _G. C. Greubel_, Jun 03 2019
%o (Sage)
%o def A126988(n, k):
%o if (n%k==0): return n/k
%o else: return 0
%o def T(n,k): return sum(A126988(n,j) for j in (k..n))
%o [[T(n, k) for k in (1..n)] for n in (1..12)] # _G. C. Greubel_, Jun 03 2019
%Y Cf. A126988, A127013, A000203, A038040.
%K nonn,tabl
%O 1,2
%A _Gary W. Adamson_, Jan 04 2007
%E Edited and extended by _R. J. Mathar_, Jul 23 2008
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