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

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, 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, 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

Offset: 1

Author

Gary W. Adamson, Jan 04 2007

Keywords

• nonn
• tabl

Comments

Also partial row sums of the n-th row of A126988 read left to right. - Reinhard Zumkeller, Jan 21 2014

Examples

			The triangle starts
   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, 1, 1, 1;
  13, 4, 4, 1, 1, 1, 1, 1, 1;
  18, 8, 3, 3, 3, 1, 1, 1, 1, 1; ...
		

Links

• Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Crossrefs

Cf. A126988, A127013, A000203, A038040.

Programs

Haskell

a127057 n k = a127057_tabl !! (n-1) !! (k-1)
a127057_row n = a127057_tabl !! (n-1)
a127057_tabl = map (scanr1 (+)) a126988_tabl
-- Reinhard Zumkeller, Jan 21 2014

Magma

A126988:= func< n,k | (n mod k) eq 0 select n/k else 0 >;
T:= func< n,k | (&+[A126988(n, j): j in [k..n]]) >;
[[T(n,k): k in [1..n]]: n in [1..12]]; // G. C. Greubel, Jun 03 2019

Mathematica

A126988[n_, m_]:= If[Mod[n, m]==0, n/m, 0];
T[n_, m_]:= Sum[A126988[n, j], {j,m,n}];
Table[T[n, m], {n,1,12}, {m,1,n}]//Flatten (* G. C. Greubel, Jun 03 2019 *)

PARI

A126988(n, k) = if(n%k==0, n/k, 0);
T(n,k) = sum(j=k,n, A126988(n,j));
for(n=1, 12, for(k=1,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Jun 03 2019

Sage

def A126988(n, k):
    if (n%k==0): return n/k
    else: return 0
def T(n,k): return sum(A126988(n,j) for j in (k..n))
[[T(n, k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jun 03 2019

Formula

T(n,k) = Sum_{i=1..n-k+1} A127013(n,i), n>=1, 1<=k<=n.

T(n,k) = Sum_{i=k..n} A126988(n,i).

Row sums: Sum_{k=1..n} T(n,k) = A038040(n).

T(n,1) = A000203(n).

T = A126988 * M as infinite lower triangular matrices, M = (1; 1, 1; 1, 1, 1; ...).

Extensions

Edited and extended by R. J. Mathar, Jul 23 2008