A336195 -id:A336195 - cp's OEIS frontend

A212856 Number of 3 X n arrays with rows being permutations of 0..n-1 and no column j greater than column j-1 in all rows.

1, 1, 7, 163, 8983, 966751, 179781181, 53090086057, 23402291822743, 14687940716402023, 12645496977257273257, 14490686095184389113277, 21557960797148733086439949, 40776761007750226749220637461, 96332276574683758035941025907591

Offset: 0

R. H. Hardin, May 28 2012

nonn

			Some solutions for n=3:
  2 1 0   2 0 1   1 2 0   0 2 1   2 0 1   2 1 0   2 1 0
  0 2 1   2 0 1   0 2 1   2 1 0   2 1 0   2 1 0   2 0 1
  0 2 1   2 1 0   2 0 1   2 0 1   0 1 2   1 2 0   2 0 1

Alois P. Heinz, Table of n, a(n) for n = 0..183 (terms n=1..19 from R. H. Hardin)
Morton Abramson and David Promislow, Enumeration of arrays by column rises, J. Combinatorial Theory Ser. A 24(2) (1978), 247-250.

Row 3 of A212855.

Cf. A000275, A212857, A212858, A212859, A212860, A336195.

A212856 := proc(n) sum(z^k/k!^3, k = 0..infinity);
series(%^x, z=0, n+1): n!^3*coeff(%,z,n); add(abs(coeff(%,x,k)), k=0..n) end:
seq(A212856(n), n=0..14); # Peter Luschny, May 27 2017
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, -add(
      binomial(n, j)^3*(-1)^j*a(n-j), j=1..n))
    end:
seq(a(n), n=0..15);  # Alois P. Heinz, Apr 26 2020

f[0] = 1; f[n_] := f[n] = Sum[(-1)^(n+k+1)*f[k]*Binomial[n, k]^2/(n-k)!, {k, 0, n-1}]; a[n_] := f[n]*n!; Array[a, 14] (* Jean-François Alcover, Feb 27 2018, after Daniel Suteu *)

a(n) = f(n) * n!, where f(0) = 1, f(n) = Sum_{k=0..n-1} (-1)^(n+k+1) * f(k) * binomial(n, k)^2 / (n-k)!. - Daniel Suteu, Feb 23 2018
a(n) = (n!)^3 * [x^n] 1 / (1 + Sum_{k>=1} (-x)^k / (k!)^3). - Seiichi Manyama, Jul 18 2020
a(n) ~ c * n!^3 / r^n, where r = 1.16151549806386358435938834554462085598002... is the root of the equation HypergeometricPFQ[{}, {1, 1}, -r] = 0 and c = 1.182760720067731330743886867947078139186402925891650811631774628... - Vaclav Kotesovec, Sep 16 2020

a(0)=1 prepended by Alois P. Heinz, Apr 26 2020

A336196 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n,k)^4 * a(k).

Original entry on oeis.org

1, 1, 17, 1459, 395793, 262131251, 359993423843, 915919888063853, 3975467425523532305, 27639424688447366285203, 292886774320942590679779267, 4544030770812055230064359134573, 99847457331663057820508375752459491, 3021907600842518917755426740899056448141

Offset: 0

Ilya Gutkovskiy, Jul 11 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..143

Column k=4 of A326322.

Cf. A000670, A102221, A336195, A336197.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k]^4 a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 13}]
nmax = 13; CoefficientList[Series[1/(1 - Sum[x^k/(k!)^4, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!^4

a(n) = (n!)^4 * [x^n] 1 / (1 - Sum_{k>=1} x^k / (k!)^4).

A336197 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n,k)^5 * a(k).

Original entry on oeis.org

1, 1, 33, 8263, 8718945, 28076306251, 224968772934303, 3896175006605313013, 131557135159637950535265, 8004845815916146011992853811, 824857614282973828473497207276283, 136888961901974254918775560412316183913, 35099479542762449254288789631427310686677535

Offset: 0

Ilya Gutkovskiy, Jul 11 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..120

Column k=5 of A326322.

Cf. A000670, A102221, A336195, A336196.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k]^5 a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 12}]
nmax = 12; CoefficientList[Series[1/(1 - Sum[x^k/(k!)^5, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!^5

a(n) = (n!)^5 * [x^n] 1 / (1 - Sum_{k>=1} x^k / (k!)^5).

A336259 a(0) = 1; a(n) = (n!)^3 * Sum_{k=0..n-1} a(k) / (k! * (n-k))^3.

Original entry on oeis.org

1, 1, 9, 278, 20464, 2948824, 735078968, 291153023664, 172201253334528, 145044581320046592, 167609226267379703808, 257816558769660828601344, 514890814087717253133447168, 1307445058678686737908660752384, 4146656933568759002389401276616704

Offset: 0

Ilya Gutkovskiy, Jul 15 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..180

Cf. A007840, A193436, A336195, A336258, A336260, A336261.

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-i)/i^3, i=1..n))
    end:
a:= n-> n!^3*b(n):
seq(a(n), n=0..14);  # Alois P. Heinz, Jan 04 2024

a[0] = 1; a[n_] := a[n] = (n!)^3 Sum[a[k]/(k! (n - k))^3, {k, 0, n - 1}]; Table[a[n], {n, 0, 14}]
nmax = 14; CoefficientList[Series[1/(1 - PolyLog[3, x]), {x, 0, nmax}], x] Range[0, nmax]!^3

a(n) = (n!)^3 * [x^n] 1 / (1 - polylog(3,x)).

a(n) ~ (n!)^3 / (polylog(2,r) * r^n), where r = 0.86512013798076629268795131756... is the root of the equation polylog(3,r) = 1. - Vaclav Kotesovec, Jul 15 2020

A342197 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k)^3 * a(k-1).

Original entry on oeis.org

1, 1, 9, 63, 919, 18919, 505639, 18602319, 877402487, 51212704151, 3688010412503, 321523601578079, 33283248550719793, 4050897039400696253, 574469890816237292037, 93943844587040615104177, 17565329004174205621822169, 3730161837629377369026433019

Offset: 0

Ilya Gutkovskiy, Mar 04 2021

nonn

Cf. A040027, A061684, A336195, A342196, A342198, A342199.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k]^3 a[k - 1], {k, 1, n}]; Table[a[n], {n, 0, 17}]

A352468 a(0) = 1; a(n) = Sum_{k=1..n} binomial(2n,2k)^3 * a(n-k).

Original entry on oeis.org

1, 1, 217, 735751, 16225658905, 1485378967457251, 429009059656530602767, 324779065084721999818137709, 563805297587600177760431368896025, 2028620600892240327820781003315525267467, 13978450121866685445815888094629703793828769467

Offset: 0

Ilya Gutkovskiy, Mar 17 2022

nonn

Cf. A094088, A336195, A352467, A352471.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[2 n, 2 k]^3 a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 10}]
nmax = 20; Take[CoefficientList[Series[1/(1 - Sum[x^(2 k)/(2 k)!^3, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!^3, {1,-1, 2}]

Sum_{n>=0} a(n) * x^(2*n) / (2*n)!^3 = 1 / (1 - Sum_{n>=1} x^(2*n) / (2*n)!^3).

A352471 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n,2k+1)^3 a(n-2*k-1).

Original entry on oeis.org

1, 1, 8, 217, 13952, 1752001, 380168432, 130996038265, 67377689108480, 49343690620021249, 49570079811804165008, 66280482720537078211945, 115058150837606807142692096, 253942526419333142443328522689, 700015299612132412448976873339008

Offset: 0

Ilya Gutkovskiy, Mar 17 2022

nonn

Cf. A006154, A336195, A352468, A352470.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, 2 k + 1]^3 a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 14}]
nmax = 14; CoefficientList[Series[1/(1 - Sum[x^(2 k + 1)/(2 k + 1)!^3, {k, 0, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!^3

Sum_{n>=0} a(n) * x^n / n!^3 = 1 / (1 - Sum_{n>=0} x^(2*n+1) / (2*n+1)!^3).

A346185 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n-1,k)^3 * a(k).

Original entry on oeis.org

1, 1, 2, 11, 93, 1294, 26045, 714391, 26109426, 1224739755, 71807248783, 5173027197636, 450173748220033, 46617339568635115, 5677430539873463470, 804907754967314483801, 131598260940217897338131, 24609634809861999705338820, 5226508081059269450476666513

Offset: 0

Ilya Gutkovskiy, Jul 09 2021

nonn

Cf. A000110, A061684, A101514, A181543, A336195.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k]^3 a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]

A346187 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n,k)^3 * a(k) * a(n-k-1).

Original entry on oeis.org

1, 1, 9, 279, 20655, 2997405, 753171615, 300907367775, 179603100388215, 152724638158940925, 178223093773584811875, 276909379421415142992975, 558708999223935430219474125, 1433526175506213101913925750425, 4594607165137022427482460137390625, 18114938314838093950739712451059177375

Offset: 0

Ilya Gutkovskiy, Jul 09 2021

nonn

Cf. A001059, A001147, A181543, A336195, A346186.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k]^3 a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 15}]

cp's OEIS Frontend

A212856 Number of 3 X n arrays with rows being permutations of 0..n-1 and no column j greater than column j-1 in all rows.

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A336196 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n,k)^4 * a(k).

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A336197 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n,k)^5 * a(k).

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A336259 a(0) = 1; a(n) = (n!)^3 * Sum_{k=0..n-1} a(k) / (k! * (n-k))^3.

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A342197 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k)^3 * a(k-1).

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Mathematica

A352468 a(0) = 1; a(n) = Sum_{k=1..n} binomial(2*n,2*k)^3 * a(n-k).

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A352471 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n,2*k+1)^3 * a(n-2*k-1).

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Mathematica

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A346185 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n-1,k)^3 * a(k).

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Mathematica

A346187 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n,k)^3 * a(k) * a(n-k-1).

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Programs

Mathematica

A352468 a(0) = 1; a(n) = Sum_{k=1..n} binomial(2n,2k)^3 * a(n-k).

A352471 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n,2k+1)^3 a(n-2*k-1).