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

1,6

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

Links

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

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

Example

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

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 *)

Crossrefs

Cf. A008277, A024427.

Sequence in context: A097229 A097862 A097612 * A227984 A021991 A112132

Adjacent sequences: A136008 A136009 A136010 * A136012 A136013 A136014

Keyword

nonn,tabf

Author

Gary W. Adamson, Dec 09 2007

Status

approved