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

Author

Gary W. Adamson, Nov 24 2006

Keywords

• nonn
• tabl

Examples

			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).
		

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)

Crossrefs

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

Cf. A095264 (row sums).

Programs

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 *)

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

Extensions

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