A245790 -id:A245790 - cp's OEIS frontend

A245732 Number T(n,k) of endofunctions on [n] such that at least one preimage with cardinality >=k exists and a nonempty preimage of j implies that all i<=j have preimages with cardinality >=k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 1, 1, 4, 3, 1, 27, 13, 1, 1, 256, 75, 7, 1, 1, 3125, 541, 21, 1, 1, 1, 46656, 4683, 141, 21, 1, 1, 1, 823543, 47293, 743, 71, 1, 1, 1, 1, 16777216, 545835, 5699, 183, 71, 1, 1, 1, 1, 387420489, 7087261, 42241, 2101, 253, 1, 1, 1, 1, 1

Offset: 0

Alois P. Heinz, Jul 30 2014

T(0,0) = 1 by convention.

In general, column k > 1 is asymptotic to n! / ((1+r^(k-1)/(k-1)!) * r^(n+1)), where r is the root of the equation 2 - exp(r) + Sum_{j=1..k-1} r^j/j! = 0. - Vaclav Kotesovec, Aug 02 2014

			Triangle T(n,k) begins:
0 :         1;
1 :         1,      1;
2 :         4,      3,    1;
3 :        27,     13,    1,   1;
4 :       256,     75,    7,   1,  1;
5 :      3125,    541,   21,   1,  1, 1;
6 :     46656,   4683,  141,  21,  1, 1, 1;
7 :    823543,  47293,  743,  71,  1, 1, 1, 1;
8 :  16777216, 545835, 5699, 183, 71, 1, 1, 1, 1;

Alois P. Heinz, Rows n = 0..140, flattened

Column k=0 gives A000312.
Columns k=1-10 give (for n>0): A000670, A032032, A102233, A232475, A245790, A245791, A245792, A245793, A245794, A245795.
T(2n,n) gives A244174(n) or 1+A007318(2n,n) = 1+A000984(n) for n>0.
Cf. A245733.

b:= proc(n, k) option remember; `if`(n=0, 1,
      add(b(n-j, k)*binomial(n, j), j=k..n))
    end:
T:= (n, k)-> `if`(k=0, n^n, `if`(n=0, 0, b(n, k))):
seq(seq(T(n, k), k=0..n), n=0..12);

b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[b[n-j, k]*Binomial[n, j], {j, k, n}]]; T[n_, k_] := If[k == 0, n^n, If[n == 0, 0, b[n, k]]]; T[0, 0] = 1; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 05 2015, after Alois P. Heinz *)

E.g.f. (for column k > 0): 1/(2 -exp(x) +Sum_{j=1..k-1} x^j/j!) -1. - Vaclav Kotesovec, Aug 02 2014

A343542 Number of ways to partition n labeled elements into sets of different sizes of at least 5.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 463, 793, 3004, 5006, 14444, 23817, 62323, 14805403, 35175993, 177791475, 745977222, 2333540804, 7589340982, 29027728612, 81515120641, 23232813583331, 69799133324911, 436678552247551, 2215090972333651, 13529994077951557, 48863594588239153

Offset: 0

Ilya Gutkovskiy, Apr 28 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..700

Cf. A007837, A025150, A032311, A057814, A245790, A341283, A343319.

b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i>n, 0, b(n, i+1)+binomial(n, i)*b(n-i, i+1)))
    end:
a:= n-> b(n, 5):
seq(a(n), n=0..31);  # Alois P. Heinz, Apr 28 2021

nmax = 31; CoefficientList[Series[Product[(1 + x^k/k!), {k, 5, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -(n - 1)! Sum[DivisorSum[k, # (-#!)^(-k/#) &, # > 4 &] a[n - k]/(n - k)!, {k, 1, n}]; Table[a[n], {n, 0, 31}]

E.g.f.: Product_{k>=5} (1 + x^k/k!).

A365915 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(2k+5) / (2k+5)! ).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 252, 1, 1584, 1, 7436, 756757, 31616, 14702689, 129404, 189559657, 11733266992, 2062481617, 516242875084, 20611819933, 14135172627712, 623557476714481, 312148517693820, 52096977907924561, 6121122865591920

Offset: 0

Seiichi Manyama, Sep 23 2023

Seiichi Manyama, Table of n, a(n) for n = 0..529

Cf. A245790, A365916, A365917.
Cf. A006154, A332258.
Cf. A365896.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+x+x^3/6-sinh(x))))

a(0) = 1; a(n) = Sum_{k=0..floor((n-5)/2)} binomial(n,2*k+5) * a(n-2*k-5).

E.g.f.: 1 / ( 1 + x + x^3/6 - sinh(x) ).

A365917 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(4k+5) / (4k+5)! ).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 252, 0, 0, 1, 4004, 756756, 0, 1, 65756, 69837768, 11732745024, 1, 1047508, 5772957036, 3957845988096, 623360743125121, 16781260, 475191562560, 1078063276530240, 587517500395425601, 88832646060056769732, 38604505286340

Offset: 0

Seiichi Manyama, Sep 23 2023

Cf. A245790, A365915, A365916.
Cf. A352429, A365911.
Cf. A365898.

my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1+x-(sinh(x)+sin(x))/2)))

a(0) = 1; a(n) = Sum_{k=0..floor((n-5)/4)} binomial(n,4*k+5) * a(n-4*k-5).

E.g.f.: 1 / ( 1 + x - (sinh(x) + sin(x))/2 ).

A245857 Number of preferential arrangements of n labeled elements such that the minimal number of elements per rank equals 4.

Original entry on oeis.org

1, 0, 0, 0, 70, 252, 420, 660, 35640, 271700, 1389388, 5137860, 79463020, 905649500, 7336909980, 48400150764, 573924746400, 7735300382250, 85942063340210, 795156908528290, 9670781421636258, 143772253669334950, 1993964186469438950, 24015169625528033550

Offset: 4

Alois P. Heinz, Aug 04 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..400

Column k=4 of A245733.

Cf. A232475, A245790, A245732.

b:= proc(n, k) option remember; `if`(n=0, 1,
      add(b(n-j, k)*binomial(n, j), j=k..n))
    end:
a:= n-> b(n, 4) -b(n, 5):
seq(a(n), n=4..30);

E.g.f.: 1/(1-Sum_{j>=4} x^j/j!) - 1/(1-Sum_{j>=5} x^j/j!).

a(n) = A232475(n) - A245790(n) = A245732(n,4) - A245732(n,5).

A245858 Number of preferential arrangements of n labeled elements such that the minimal number of elements per rank equals 5.

Original entry on oeis.org

1, 0, 0, 0, 0, 252, 924, 1584, 2574, 4004, 762762, 6062784, 31868200, 121314312, 399096216, 12936646128, 167685283332, 1429020461484, 9754485257594, 55756633204272, 905519956068420, 14816352889289380, 179362257853420980, 1745771827872126600

Offset: 5

Alois P. Heinz, Aug 04 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..400

Column k=5 of A245733.

Cf. A245790, A245791, A245732.

b:= proc(n, k) option remember; `if`(n=0, 1,
      add(b(n-j, k)*binomial(n, j), j=k..n))
    end:
a:= n-> b(n, 5) -b(n, 6):
seq(a(n), n=5..30);

E.g.f.: 1/(1-Sum_{j>=5} x^j/j!) - 1/(1-Sum_{j>=6} x^j/j!).

a(n) = A245790(n) - A245791(n) = A245732(n,5) - A245732(n,6).

A365916 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(3k+5) / (3k+5)! ).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 252, 1, 0, 2574, 1, 756756, 21606, 1, 33081048, 174420, 11732745025, 1052328186, 1397640, 1484192245537, 30223445274, 623360754309330, 126750660276241, 835509726090, 182333017453575330, 9309138073555321

Offset: 0

Seiichi Manyama, Sep 23 2023

Cf. A245790, A365915, A365917.

Cf. A365897.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=0, N\3, x^(3*k+5)/(3*k+5)!))))

a(0) = 1; a(n) = Sum_{k=0..floor((n-5)/3)} binomial(n,3*k+5) * a(n-3*k-5).

cp's OEIS Frontend

A245732 Number T(n,k) of endofunctions on [n] such that at least one preimage with cardinality >=k exists and a nonempty preimage of j implies that all i<=j have preimages with cardinality >=k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A343542 Number of ways to partition n labeled elements into sets of different sizes of at least 5.

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A365915 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(2*k+5) / (2*k+5)! ).

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A365917 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(4*k+5) / (4*k+5)! ).

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A245857 Number of preferential arrangements of n labeled elements such that the minimal number of elements per rank equals 4.

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Maple

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A245858 Number of preferential arrangements of n labeled elements such that the minimal number of elements per rank equals 5.

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Maple

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A365916 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(3*k+5) / (3*k+5)! ).

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PARI

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A365915 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(2k+5) / (2k+5)! ).

A365917 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(4k+5) / (4k+5)! ).

A365916 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(3k+5) / (3k+5)! ).