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

1,2

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

Links

Alois P. Heinz, Rows n = 1..220, flattened

Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 180.

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

Example

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

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

Crossrefs

Cf. A022915, A116608, A218868.

Sequence in context: A214049 A273034 A374735 * A130377 A260956 A360299

Adjacent sequences: A208434 A208435 A208436 * A208438 A208439 A208440

Keyword

nonn,tabf

Author

Geoffrey Critzer, Feb 26 2012

Status

approved