A208437 Triangular array read by rows: T(n,k) is the number of set partitions of {1,2,...,n} that have exactly k distinct block sizes.

1, 2, 2, 3, 5, 10, 2, 50, 27, 116, 60, 2, 560, 315, 142, 1730, 2268, 282, 6123, 14742, 1073, 30122, 72180, 12600, 2, 116908, 464640, 97020, 32034, 507277, 2676366, 997920, 2, 2492737, 16400098, 8751600, 136853, 15328119, 94209206, 81225144, 1527528, 56182092, 673282610, 614128515, 37837800

Offset: 1

Author

Geoffrey Critzer, Feb 26 2012

Keywords

• nonn
• tabf

Comments

Column 1 = A038041.

Column 2 = A088142.

Column 3 = A133118.

Row sums = A000110 (Bell numbers).

Row n has floor([sqrt(1+8n)-1]/2) terms (number of terms increases by one at each triangular number). - Franklin T. Adams-Watters, Feb 26 2012

Examples

			:    1;
:    2;
:    2,      3;
:    5,     10;
:    2,     50;
:   27,    116,     60;
:    2,    560,    315;
:  142,   1730,   2268;
:  282,   6123,  14742;
: 1073,  30122,  72180,   12600;
		

Links

• Alois P. Heinz, Rows n = 1..220, flattened
• Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 180.

Crossrefs

Cf. A022915, A116608, A218868.

Programs

Maple

with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)/j!*
      b(n-i*j, i-1)*`if`(j=0, 1, x), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2)):
seq(T(n), n=1..16);  # Alois P. Heinz, Aug 21 2014

Mathematica

nn = 15; p = Product[1 + y (Exp[x^i/i!] - 1), {i, 1, nn}];f[list_] := Select[list, # > 0 &];
Map[f, Drop[ Range[0, nn]! CoefficientList[Series[p, {x, 0, nn}], {x, y}], 1]] // Flatten

Formula

E.g.f.: Product_{i>=1} 1 + y *(exp(x^i/i!)-1).

T(n*(n+1)/2,n) = A022915(n). - Alois P. Heinz, Apr 08 2016