A136011 Irregular triangle read by rows, Stirling numbers of the second kind: columns shifted to allow (1, 1, 2, 2, 3, 3, ...) terms per row.

1, 1, 1, 1, 1, 3, 1, 7, 1, 1, 15, 6, 1, 31, 25, 1, 1, 63, 90, 10, 1, 127, 301, 65, 1, 1, 255, 966, 350, 15, 1, 511, 3025, 1701, 140, 1, 1, 1023, 9330, 7770, 1050, 21, 1, 2047, 28501, 34105, 6951, 266, 1, 1, 4095, 86526, 145750, 42525, 2646, 28

Offset: 1

Author

Gary W. Adamson, Dec 09 2007

Keywords

• nonn
• tabf

Comments

Row sums = A024427: (1, 1, 2, 4, 9, 22, 58, 164, 495, 1587, ...).

T(n,k) is the number of ways to partition {1,2,...,n+1} into exactly k blocks such that each block contains at least 2 elements and the smallest 2 elements in each block are consecutive integers. - Geoffrey Critzer, Dec 02 2013

Examples

			First few rows of the triangle:
  1;
  1;
  1,   1;
  1,   3;
  1,   7,   1;
  1,  15,   6;
  1,  31,  25,   1;
  1,  63,  90,  10;
  1, 127, 301,  65,  1;
  1, 255, 966, 350, 15;
  ...
T(5,3) = 1 because we have {1,2},{3,4},{5,6}.
T(6,3) = 6 because we have {1,2,7},{3,4},{5,6}; {1,2},{3,4,7},{5,6}; {1,2},{3,4},{5,6,7}; {1,2},{3,4,5},{6,7}; {1,2,3},{4,5},{6,7}; {1,2,5},{3,4},{6,7}. - _Geoffrey Critzer_, Dec 02 2013
		

Links

• Alois P. Heinz, Rows n = 1..200, flattened

Crossrefs

Cf. A008277, A024427.

Programs

Maple

T:= (n, k)-> Stirling2(n+1-k, k):
seq(seq(T(n, k), k=1..(n+1)/2), n=1..20); # Alois P. Heinz, Dec 04 2013

Mathematica

nn=15;Range[0,nn]!;Map[Select[#,#>0&]&,Drop[Transpose[Table[CoefficientList[Series[Product[x^2/(1-i x),{i,1,k}],{x,0,nn}],x],{k,1,nn/2}]],2]]//Grid (* Geoffrey Critzer, Dec 02 2013 *)

Formula

Given A008277, the Stirling number of the second kind triangle, left column = (1, 1, 1, ...); all other columns start at 3rd term of previous column.

O.g.f. for column k: Product_{i=1..k} x^2/(1 - i*x). - Geoffrey Critzer, Dec 02 2013

T(n,k) = Stirling2(n+1-k,k). - Alois P. Heinz, Dec 04 2013