A245732 -id:A245732 - cp's OEIS frontend

A245795 Number of preferential arrangements of n labeled elements when at least k=10 elements per rank are required.

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 184757, 705433, 1998725, 4992289, 11618957, 25852921, 55791791, 117832681, 245039011, 503891821, 5552024604991, 46933238932021, 261680950107511, 1205121760579981, 4959685199012641, 18947093053200193

Offset: 0

Alois P. Heinz, Aug 01 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. column k=10 of A245732.

a:= proc(n) option remember; `if`(n=0, 1,
       add(a(n-j)*binomial(n, j), j=10..n))
    end:
seq(a(n), n=0..40);

CoefficientList[Series[1/(2 + x - E^x + x^2/2! + x^3/3! + x^4/4! + x^5/5! + x^6/6! + x^7/7! + x^8/8! + x^9/9!),{x,0,40}],x]*Range[0,40]! (* Vaclav Kotesovec, Aug 02 2014 *)

E.g.f.: 1/(2 + x - exp(x) + x^2/2! + x^3/3! + x^4/4! + x^5/5! + x^6/6! + x^7/7! + x^8/8! + x^9/9!). - Vaclav Kotesovec, Aug 02 2014

a(n) ~ n! / ((1+r^9/9!) * r^(n+1)), where r = 4.320434975980068857383128... is the root of the equation 2 + r - exp(r) + r^2/2! + r^3/3! + r^4/4! + r^5/5! + r^6/6! + r^7/7! + r^8/8! + r^9/9! = 0. - Vaclav Kotesovec, Aug 02 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).

A245862 Number of preferential arrangements of n labeled elements such that the minimal number of elements per rank equals 9.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 48620, 184756, 335920, 587860, 994840, 1634380, 2615008, 4085950, 6249100, 227882805150, 1914150638400, 10597377540750, 42894094729200, 150967391072550, 488846715676800, 1495608303532200, 4389524294884872, 12479799500904120

Offset: 9

Alois P. Heinz, Aug 04 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 9..400

Column k=9 of A245733.

Cf. A245794, A245795, 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, 9) -b(n, 10):
seq(a(n), n=9..40);

E.g.f.: 1/(1-Sum_{j>=9} x^j/j!) - 1/(1-Sum_{j>=10} x^j/j!).

a(n) = A245794(n) - A245795(n) = A245732(n,9) - A245732(n,10).

cp's OEIS Frontend

A245795 Number of preferential arrangements of n labeled elements when at least k=10 elements per rank are required.

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

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Maple

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

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Maple

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

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