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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. 6
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 (list; graph; refs; listen; history; text; internal format)
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: A153931 A214049 A273034 * A130377 A260956 A322113

Adjacent sequences:  A208434 A208435 A208436 * A208438 A208439 A208440

KEYWORD

nonn,tabf

AUTHOR

Geoffrey Critzer, Feb 26 2012

STATUS

approved

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Last modified January 23 16:48 EST 2020. Contains 331173 sequences. (Running on oeis4.)