A088142 -id:A088142 - cp's OEIS frontend

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

Geoffrey Critzer, Feb 26 2012

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

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

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

Cf. A022915, A116608, A218868.

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

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

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

A131661 Number of compositions of n such that the cardinality of the set of parts is 2.

Original entry on oeis.org

0, 0, 2, 5, 14, 22, 44, 68, 107, 172, 261, 396, 606, 950, 1414, 2238, 3418, 5411, 8368, 13297, 20840, 33268, 52549, 84120, 133775, 214611, 343025, 551064, 883600, 1421767, 2284870, 3680296, 5924725, 9551161, 15393855, 24834827, 40061700

Offset: 1

Vladeta Jovovic, Sep 13 2007

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000005, A002133, A005772, A088142.
Column k=2 of A235998.
Cf. A242900 (with distinct multiplicities).

with(numtheory):
a:= n-> add(add(add(binomial(j+(n-i*j)/d, j), d=select(x->xAlois P. Heinz, Feb 01 2014

Rest@ CoefficientList[ Series[ Sum[ x^(i + j)*(x^i + x^j - 2)/((x^i - 1)*(x^j - 1)*(x^i + x^j - 1)), {i, 2, 37}, {j, i - 1}], {x, 0, 37}], x] (* Robert G. Wilson v, Sep 16 2007 *)

G.f.: Sum(Sum(x^(i+j)*(x^i+x^j-2)/((x^i-1)*(x^j-1)*(x^i+x^j-1)), j=1..i-1), i=2..infinity).

a(n) ~ 1/sqrt(5) * ((1+sqrt(5))/2)^(n+1). - Vaclav Kotesovec, May 01 2014

More terms from Robert G. Wilson v, Sep 16 2007

A133118 Number of partitions of n-set with 3 block sizes.

Original entry on oeis.org

60, 315, 2268, 14742, 72180, 464640, 2676366, 16400098, 94209206, 673282610, 4095231104, 29371828846, 197547348216, 1513916607683, 10904464442572, 87070803499372, 673555061736062, 5718121102062336, 47028289679340734, 418812093667530755, 3680961843042545490, 34161428275433710485

Offset: 6

Vladeta Jovovic, Sep 18 2007

Alois P. Heinz, Table of n, a(n) for n = 6..300

Cf. A038041, A088142, A122404, A005225, A005772.

Column k=3 of A208437.

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[multinomial[n, Prepend[Table[i, {j}], n - i*j]]/j!*b[n - i*j, i - 1]*If[j == 0, 1, x], {j, 0, n/i}]]];
a[n_] := Coefficient[b[n, n], x, 3];
Array[a, 22, 6] (* Jean-François Alcover, May 24 2019, after Alois P. Heinz in A208437 *)

We obtain e.g.f. for number of partitions of n-set with m block sizes if we substitute x(i) with -Sum_{k>0} (1-exp(x^k/k!))^i in cycle index Z(S(m); x(1),x(2),...,x(n)) of symmetric group S(m) of degree m.

More terms from Max Alekseyev, Jun 17 2011

A133119 Number of permutations of [n] with 3 cycle lengths.

Original entry on oeis.org

120, 1050, 12712, 141876, 1418400, 17061660, 212254548, 2735287698, 37354035628, 581350330470, 8895742806480, 151305163230480, 2659183039338192, 50112909523522476, 976443721325014300, 20413628375979803370, 434137453618439716068

Offset: 6

Vladeta Jovovic, Sep 18 2007

Alois P. Heinz, Table of n, a(n) for n = 6..200

Cf. A005225, A005772, A038041, A088142.

Column k=3 of A218868.

We obtain e.g.f. for number of permutations of [n] with m cycle lengths if we substitute x(i) with -Sum_{k>0} ((1-exp(x^k/k))^i in cycle index Z(S(m); x(1),x(2),..,x(m)) of symmetric group S(m) of degree m.

More terms from Max Alekseyev, Feb 08 2010

cp's OEIS Frontend

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.

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A131661 Number of compositions of n such that the cardinality of the set of parts is 2.

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A133118 Number of partitions of n-set with 3 block sizes.

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A133119 Number of permutations of [n] with 3 cycle lengths.

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