cp's OEIS Frontend

A127057 Triangle T(n,k), partial row sums of the n-th row of A127013 read right to left.

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