A125232 Triangle T(n,k) read by rows: the (n-k)-th term of the k-fold iterated partial sum of the pentagonal numbers.

1, 5, 1, 12, 6, 1, 22, 18, 7, 1, 35, 40, 25, 8, 1, 51, 75, 65, 33, 9, 1, 70, 126, 140, 98, 42, 10, 1, 92, 196, 266, 238, 140, 52, 11, 1, 117, 288, 462, 504, 378, 192, 63, 12, 1, 145, 405, 750, 966, 882, 570, 255, 75, 13, 1, 176, 550, 1155, 1716, 1848, 1452, 825, 330, 88, 14, 1

Offset

1,2

References

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

Links

Robert Israel, Table of n, a(n) for n = 1..10011(rows 0 to 140, flattened)

Formula

T(n,0)=A000326(n). T(n,k)=T(n-1,k) + T(n-1,k-1), k>0. - R. J. Mathar, Jun 09 2008

G.f. as triangle: (1+2*x)/((1-x)^2*(1-x-x*y)). - Robert Israel, Nov 07 2016

Example

First few rows of the triangle are:
   1;
   5,   1;
  12,   6,   1;
  22,  18,   7,   1;
  35,  40,  25,   8,   1;
  51,  75,  65,  33,   9,   1;
  70, 126, 140,  98,  42,  10,   1;
  ...
Example: (5,3) = 65 = 25 + 40 = (4,3) + (4,2).

Maple

A125232 := proc(n, k) option remember ; if k = 0 then A000326(n) ; elif k = n-1 then 1 ; else procname(n-1, k)+procname(n-1, k-1) ; fi : end: # R. J. Mathar, Jun 09 2008

Mathematica

nmax = 11; col[1] = Table[n(3n-1)/2, {n, 1, nmax}]; col[k_] := col[k] = Prepend[Accumulate[col[k-1]], 0]; Table[col[k][[n]], {n, 1, nmax}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 25 2019 *)

Crossrefs

Columns: A000326 (pentagonal numbers), A002411, A001296, A051836, A051923.

Cf. A095264 (row sums).

Sequence in context: A146993 A343223 A104572 * A116923 A327797 A062264

Adjacent sequences: A125229 A125230 A125231 * A125233 A125234 A125235

Keyword

nonn,tabl

Author

Gary W. Adamson, Nov 24 2006

Extensions

Edited and extended by R. J. Mathar, Jun 09 2008

Status

approved