A125234 Triangle T(n,k) read by rows: the k-th column contains the k-fold iterated partial sum of A000566.

1, 7, 1, 18, 8, 1, 34, 26, 9, 1, 55, 60, 35, 10, 1, 81, 115, 95, 45, 11, 1, 112, 196, 210, 140, 56, 12, 1, 148, 308, 406, 350, 196, 68, 13, 1, 189, 456, 714, 756, 546, 264, 81, 14, 1, 235, 645, 1170, 1470, 1302, 810, 345, 95, 15, 1, 286, 880, 1815, 2640, 2772, 2112, 1155, 440, 110, 16, 1

Offset

1,2

Comments

The leftmost column contains the heptagonal numbers A000566.

The adjacent columns to the right are A002413, A002418, A027800, A051946, A050484.

Row sums = 1, 8, 27, 70, 161, 348, 727, ... = 6*(2^n-1)-5*n.

References

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, 1966, p. 189.

Links

Table of n, a(n) for n=1..66.

Formula

T(n,0) = A000566(n). T(n,k) = T(n-1,k) + T(n-1,k-1), k>0.

Example

First few rows of the triangle are:
  1;
  7, 1;
  18, 8, 1;
  34, 26, 9, 1;
  55, 60, 35, 10, 1;
  81, 115, 95, 45, 11, 1;
  112, 196, 210, 140, 56, 12, 1;
Example: T(6,2) = 95 = 35 + 60 = T(5,2) + T(5,1).

Maple

A000566 := proc(n) n*(5*n-3)/2 ; end: A125234 := proc(n, k) if k = 0 then A000566(n); elif k>= n then 0 ; else procname(n-1, k-1)+procname(n-1, k) ; fi; end: seq(seq(A125234(n, k), k=0..n-1), n=1..16) ; # R. J. Mathar, Sep 09 2009

Mathematica

A000566[n_] := n(5n-3)/2;
T[n_, k_] := Which[k == 0, A000566[n], k >= n, 0, True, T[n-1, k-1] + T[n-1, k] ];
Table[Table[T[n, k], {k, 0, n-1}], {n, 1, 11}] // Flatten (* Jean-François Alcover, Oct 26 2023, after R. J. Mathar *)

Crossrefs

Cf. A000566, A002413, A002418, A027800, A051946, A050484.

Analogous triangles for the hexagonal and pentagonal numbers are A125233 and A125232.

Sequence in context: A013614 A179530 A098081 * A028325 A245484 A215503

Adjacent sequences: A125231 A125232 A125233 * A125235 A125236 A125237

Keyword

nonn,tabl

Author

Gary W. Adamson, Nov 24 2006

Extensions

Edited and extended by R. J. Mathar, Sep 09 2009

Status

approved