A245733 -id:A245733 - 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

A133286 a(n) is the difference by which n^n overestimates the value of (1/2) Sum_{k>=0} k^n/2^k.

Original entry on oeis.org

0, 0, 1, 14, 181, 2584, 41973, 776250, 16231381, 380333228, 9897752437, 283689038038, 8888008880661, 302348248243872, 11101365482587573, 437663607189881522, 18441428419027570261, 827109891119307628276, 39343022540633280730101, 1978326854072994260712846

Offset: 0

Ramesh L. Srigiriraju (rsrigir(AT)vt.edu), Oct 16 2007

nonn

			a(3) = 3^3 - (1/2) Sum_{k>=0} k^3/2^k = 27 - 1/2 * 26 = 27 - 13 = 14.

Alois P. Heinz, Table of n, a(n) for n = 0..380

Cf. A000312, A000670.

Cf. column k=0 of A245733.

a:= n-> n^n -sum(k^n/2^k, k=0..infinity)/2:
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 29 2014
# second Maple program:
b:= proc(n) b(n):= `if`(n=0, 1, add(b(n-j)/j!, j=1..n)) end:
a:= n-> n^n- b(n)*n!:
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 29 2014

a[n_] := If[n==0, 0, n^n - HurwitzLerchPhi[1/2, -n, 0]/2];
a /@ Range[0, 25] (* Jean-François Alcover, Nov 10 2020 *)

a(n) = n^n - (1/2) Sum_{k>=0} k^n/2^k.
a(n) = A000312(n) - A000670(n). - Alois P. Heinz, Jul 29 2014
E.g.f.: 1/(1+LambertW(-x)) - 1/(2-exp(x)). - Alois P. Heinz, Aug 03 2014

More terms and a(14)-a(17) corrected by Alois P. Heinz, Jul 29 2014

A245854 Number of preferential arrangements of n labeled elements such that the minimal number of elements per rank equals 1.

Original entry on oeis.org

1, 2, 12, 68, 520, 4542, 46550, 540136, 7045020, 101865410, 1619046418, 28053492348, 526430246264, 10636085523910, 230214619661790, 5314695463338704, 130356558777712468, 3385311352838750538, 92797887464933030762, 2677623216872061223780, 81123642038690958720048

Offset: 1

Alois P. Heinz, Aug 04 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..400

Column k=1 of A245733.

Cf. A000670, A032032, 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, 1) -b(n, 2):
seq(a(n), n=1..25);

With[{nn=30},CoefficientList[Series[1/(2-Exp[x])-1/(2-Exp[x]+x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 29 2024 *)

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

a(n) = A000670(n) - A032032(n) = A245732(n,1) - A245732(n,2).

A245855 Number of preferential arrangements of n labeled elements such that the minimal number of elements per rank equals 2.

Original entry on oeis.org

1, 0, 6, 20, 120, 672, 5516, 40140, 368640, 3521870, 37445298, 422339502, 5215454426, 68144100780, 954428684280, 14160968076584, 222769496190060, 3692874342747114, 64493471050666430, 1181830474135532130, 22692074431844298558, 455404848204906308984

Offset: 2

Alois P. Heinz, Aug 04 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..400

Column k=2 of A245733.

Cf. A032032, A102233, 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, 2) -b(n, 3):
seq(a(n), n=2..25);

E.g.f.: 1/(2-exp(x)+x) -1/(2-exp(x)+x+x^2/2).

a(n) = A032032(n) - A102233(n) = A245732(n,2) - A245732(n,3).

A245856 Number of preferential arrangements of n labeled elements such that the minimal number of elements per rank equals 3.

Original entry on oeis.org

1, 0, 0, 20, 70, 112, 1848, 12840, 62700, 591800, 5484908, 40589276, 421291780, 4704380800, 46345716880, 533446290384, 6931113219780, 85313661653400, 1121432682942740, 16310909250477380, 237534778732260548, 3578871132644512672, 57980168196079811800

Offset: 3

Alois P. Heinz, Aug 04 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..400

Column k=3 of A245733.

Cf. A102233, A232475, 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, 3) -b(n, 4):
seq(a(n), n=3..30);

With[{nn=30},CoefficientList[Series[1/(2-Exp[x]+x+x^2/2)-1/(2-Exp[x]+ x+ x^2/2+ x^3/6),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 14 2016 *)

E.g.f.: 1/(2-exp(x)+x+x^2/2)-1/(2-exp(x)+x+x^2/2+x^3/6).

a(n) = A102233(n) - A232475(n) = A245732(n,3) - A245732(n,4).

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).

A245859 Number of preferential arrangements of n labeled elements such that the minimal number of elements per rank equals 6.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 924, 3432, 6006, 10010, 16016, 24752, 17190264, 139729800, 748339320, 2910015528, 9794896188, 30251595066, 2396910064472, 33228482071400, 291616291666700, 2036218597884900, 11895959650285620, 61536913327513260, 1662981928016982300

Offset: 6

Alois P. Heinz, Aug 04 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..400

Column k=6 of A245733.

a(n) = A245791, A245792, 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, 6) -b(n, 7):
seq(a(n), n=6..35);

E.g.f.: 1/(1-Sum_{j>=6} x^j/j!) - 1/(1-Sum_{j>=7} x^j/j!).

a(n) = A245791(n) - A245792(n) = A245732(n,6) - A245732(n,7).

A245860 Number of preferential arrangements of n labeled elements such that the minimal number of elements per rank equals 7.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 3432, 12870, 22880, 38896, 63648, 100776, 155040, 399305520, 3292693008, 17879790324, 70676513424, 242216077400, 762341522800, 2264840592300, 478970960616720, 6869326015894680, 61426122596911800, 435982960069722000, 2589856033041531072

Offset: 7

Alois P. Heinz, Aug 04 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..400

Column k=7 of A245733.

Cf. A245792, A245793, 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, 7) -b(n, 8):
seq(a(n), n=7..35);

E.g.f.: 1/(1-Sum_{j>=7} x^j/j!) - 1/(1-Sum_{j>=8} x^j/j!).

a(n) = A245792(n) - A245793(n) = A245732(n,7) - A245732(n,8).

A245861 Number of preferential arrangements of n labeled elements such that the minimal number of elements per rank equals 8.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 12870, 48620, 87516, 151164, 251940, 406980, 639540, 980628, 9466982712, 78881427900, 432962644400, 1733914096200, 6029537213700, 19273224716460, 58178097911700, 168431757261300, 100033451495909100, 1461521434059544572

Offset: 8

Alois P. Heinz, Aug 04 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..400

Column k=8 of A245733.

Cf. A245793, A245794, 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, 8) -b(n, 9):
seq(a(n), n=8..35);

E.g.f.: 1/(1-Sum_{j>=8} x^j/j!) - 1/(1-Sum_{j>=9} x^j/j!).

a(n) = A245793(n) - A245794(n) = A245732(n,8) - A245732(n,9).

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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A133286 a(n) is the difference by which n^n overestimates the value of (1/2) Sum_{k>=0} k^n/2^k.

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A245854 Number of preferential arrangements of n labeled elements such that the minimal number of elements per rank equals 1.

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A245855 Number of preferential arrangements of n labeled elements such that the minimal number of elements per rank equals 2.

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A245856 Number of preferential arrangements of n labeled elements such that the minimal number of elements per rank equals 3.

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

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A245859 Number of preferential arrangements of n labeled elements such that the minimal number of elements per rank equals 6.

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