A209668 -id:A209668 - cp's OEIS frontend

A247551 Decimal expansion of Product_{k>=2} 1/(1-1/k!).

2, 5, 2, 9, 4, 7, 7, 4, 7, 2, 0, 7, 9, 1, 5, 2, 6, 4, 8, 1, 8, 0, 1, 1, 6, 1, 5, 4, 2, 5, 3, 9, 5, 4, 2, 4, 1, 1, 7, 8, 7, 0, 2, 3, 9, 4, 8, 4, 5, 9, 9, 7, 3, 3, 7, 5, 8, 4, 9, 3, 4, 9, 8, 2, 5, 0, 0, 2, 1, 1, 8, 7, 8, 0, 0, 8, 6, 6, 9, 9, 1, 2, 1, 9, 9, 8, 8, 3, 6, 4, 6, 2, 5, 2, 6, 1, 4, 9, 5, 5, 1, 6, 4, 3, 2

Offset: 1

Vaclav Kotesovec, Sep 19 2014

			2.5294774720791526481801161542539542411787023948459973375849349825...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 269.
A. Knopfmacher, A. M. Odlyzko, B. Pittel, L. B. Richmond, D. Stark, G. Szekeres, and N. C. Wormald, The Asymptotic Number of Set Partitions with Unequal Block Sizes, The Electronic Journal of Combinatorics, Volume 6 (1999), Research Paper #R2.

Cf. A005651, A035341, A072605, A140585, A183239, A209668, A261600, A282529, A292713, A320566, A327827, A335643.

evalf(1/product(1-1/k!,k=2..infinity),100); # Vaclav Kotesovec, Sep 19 2014

digits = 105;
RealDigits[NProduct[1/(1-1/k!), {k, 2, Infinity}, WorkingPrecision -> digits+10, NProductFactors -> digits], 10, digits][[1]] (* Jean-François Alcover, Nov 02 2020 *)

default(realprecision,150); 1/prodinf(k=2,1 - 1/k!) \\ Vaclav Kotesovec, Sep 21 2014

Product_{k>=2} 1/(1-1/k!).
Equals lim_{n -> oo} A005651(n) / n!.
Equals 1/A282529. - Amiram Eldar, Sep 15 2023

A212943 T(n,k)=Number of nXk 0..k-1 arrays with no column j greater than or equal to than column j-1 in all rows.

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 181, 37, 1, 1, 10311, 9019, 175, 1, 1, 1016501, 6470341, 331489, 781, 1, 1, 152747323, 10058484751, 2509306671, 10669771, 3367, 1, 1, 32383630189, 28744943858947, 52311221188001, 801439905901, 320396041, 14197, 1, 1

Offset: 1

R. H. Hardin May 31 2012

Table starts
.1...1........1............1..................1........................1
.1...7......181........10311............1016501................152747323
.1..37.....9019......6470341........10058484751...........28744943858947
.1.175...331489...2509306671.....52311221188001......2438624218076957695
.1.781.10669771.801439905901.212180664326328751.153322267564381742818531

			Some solutions for n=4 k=4
..1..3..2..2....0..0..0..3....1..3..0..1....1..3..2..1....1..3..3..3
..1..1..2..2....0..1..1..2....0..0..1..0....1..1..0..3....2..3..0..3
..0..1..3..1....0..2..2..1....0..1..1..3....2..3..3..0....0..2..0..2
..3..1..0..3....2..1..0..2....3..0..3..1....1..0..3..0....2..0..1..0

R. H. Hardin, Table of n, a(n) for n = 1..191

Column 2 is A005061

cf. A212930

Empirical for column k:
k=1: a(n) = a(n-1)
k=2: a(n) = 7*a(n-1) -12*a(n-2)
k=3: a(n) = 55*a(n-1) -936*a(n-2) +4860*a(n-3)
k=4: a(n) = 631*a(n-1) -144700*a(n-2) +15035200*a(n-3) -702208000*a(n-4) +11468800000*a(n-5)
The coefficient of a(n-1) is A209668(k) (through at least k=1..7)

A261718 Number A(n,k) of partitions of n where each part i is marked with a word of length i over a k-ary alphabet whose letters appear in alphabetical order; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 7, 3, 0, 1, 4, 15, 18, 5, 0, 1, 5, 26, 55, 50, 7, 0, 1, 6, 40, 124, 216, 118, 11, 0, 1, 7, 57, 235, 631, 729, 301, 15, 0, 1, 8, 77, 398, 1470, 2780, 2621, 684, 22, 0, 1, 9, 100, 623, 2955, 8001, 12954, 8535, 1621, 30, 0

Offset: 0

Alois P. Heinz, Aug 29 2015

			A(3,2) = 18: 3aaa, 3aab, 3abb, 3bbb, 2aa1a, 2aa1b, 2ab1a, 2ab1b, 2bb1a, 2bb1b, 1a1a1a, 1a1a1b, 1a1b1a, 1a1b1b, 1b1a1a, 1b1a1b, 1b1b1a, 1b1b1b.
Square array A(n,k) begins:
  1,  1,   1,    1,     1,      1,      1,       1, ...
  0,  1,   2,    3,     4,      5,      6,       7, ...
  0,  2,   7,   15,    26,     40,     57,      77, ...
  0,  3,  18,   55,   124,    235,    398,     623, ...
  0,  5,  50,  216,   631,   1470,   2955,    5355, ...
  0,  7, 118,  729,  2780,   8001,  19158,   40299, ...
  0, 11, 301, 2621, 12954,  45865, 130453,  317905, ...
  0, 15, 684, 8535, 55196, 241870, 820554, 2323483, ...

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

Columns k=0-10 give: A000007, A000041, A074141, A261737, A261738, A261739, A261740, A261741, A261742, A261743, A261744.
Rows n=0-2 give: A000012, A001477, A005449.
Main diagonal gives A209668.
Cf. A144064, A261719, A261780.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k)+`if`(i>n, 0, b(n-i, i, k)*binomial(i+k-1, k-1))))
    end:
A:= (n, k)-> b(n, n, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i > n, 0, b[n - i, i, k]*Binomial[i + k - 1, k - 1]]]]; A[n_, k_] := b[n, n, k]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz *)

A(n,k) = Sum_{i=0..k} C(k,i) * A261719(n,k-i).

A209673 a(n) = count of monomials, of degree k=n, in the Schur symmetric polynomials s(mu,k) summed over all partitions mu of n.

Original entry on oeis.org

1, 1, 4, 19, 116, 751, 5552, 43219, 366088, 3245311, 30569012, 299662672, 3079276708, 32773002718, 362512238272, 4136737592323, 48773665308176, 591313968267151, 7375591544495636, 94340754464144215, 1237506718985945656, 16608519982801477908, 228013066931927465872

Offset: 0

Wouter Meeussen, Mar 11 2012

nonn

Main diagonal of triangle A191714.

a(n) is also the number of semistandard Young tableaux of size and maximal entry n. - Christian Stump, Oct 09 2015

Alois P. Heinz, Table of n, a(n) for n = 0..500
Wikipedia, Symmetric Polynomials
FindStat - Combinatorial Statistic Finder, Semistandard Young tableaux

Cf. A191714, A209664, A209665, A209666, A209667, A209668, A209669, A209670, A209671, A209672, A209673.

Main diagonal of A210391. - Alois P. Heinz, Mar 22 2012

(* see A191714 *)
Tr /@ Table[(stanley[#, l] & /@ Partitions[l]), {l, 11}]
(* or *)
Table[SeriesCoefficient[1/((1-x)^(n*(n+1)/2) * (1+x)^(n*(n-1)/2)), {x, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Aug 06 2025 *)

a(12)-a(22) from Alois P. Heinz, Mar 11 2012

Typo in Mathematica program fixed by Vaclav Kotesovec, Mar 19 2015

A075197 Number of partitions of n balls of n colors.

Original entry on oeis.org

1, 1, 6, 38, 305, 2777, 28784, 330262, 4152852, 56601345, 829656124, 12992213830, 216182349617, 3804599096781, 70540645679070, 1373192662197632, 27982783451615363, 595355578447896291, 13193917702518844859, 303931339674133588444, 7263814501407389465610

Offset: 0

Christian G. Bower, Sep 07 2002

nonn

For each integer partition of n, consider each part of size k to be a box containing k balls of up to n color. Order among parts and especially among parts of the same size does not matter. - Olivier Gérard, Aug 26 2016

			Illustration of first terms, ordered by number of parts, size of parts and smallest color of parts, etc.
a(1) = 1:
  {{1}}
a(2) = 6 = 3+3:
  {{1,1}},{{1,2}},{{2,2}},
  {{1},{1}},{{1},{2}},{{2},{2}}
a(3) = 38 = 10+18+10:
  {{1,1,1}},{{1,1,2}},{{1,1,3}},{{1,2,2}},{{1,2,3}},{{1,3,3}},
  {{2,2,2}},{{2,2,3}},{{2,3,3}},{{3,3,3}},
  {{1},{1,1}},{{1},{1,2}},{{1},{1,3}},{{1},{2,2}},{{1},{2,3}},{{1},{3,3}},
  {{2},{1,1}},{{2},{1,2}},{{2},{1,3}},{{2},{2,2}},{{2},{2,3}},{{2},{3,3}},
  {{3},{1,1}},{{3},{1,2}},{{3},{1,3}},{{3},{2,2}},{{3},{2,3}},{{3},{3,3}},
  {{1},{1},{1}},{{1},{1},{2}},{{1},{1},{3}},{{1},{2},{2}},{{1},{2},{3}},{{1},{3},{3}},
  {{2},{2},{2}},{{2},{2},{3}},{{2},{3},{3}},{{3},{3},{3}}

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

Main diagonal of A075196.
Cf. A001700 (n balls of one color in n unlabeled boxes).
Cf. A209668 (boxes are ordered by size but not by content among a given size: order among boxes of the same size matters.),
Cf. A261783 (compositions of balls of n colors: boxes are labeled)
Cf. A252654 (lists instead of boxes : order of balls matter)
Cf. A000262 (lists instead of boxes and all n colors are used)
Cf. A255906 (the c colors used form the interval [1,c])
Cf. A255951 (the n-1 colors used form the interval [1,n-1])
Cf. A255942 (0/1 binary coloring)
Cf. A066186 (only 1 color among n = n * p(n))
Cf. A000110 (the n possible colors are used : set partitions of [n])
Cf. A005651 (the n possible colors are used and order of parts of the same size matters)
Cf. A000670 (the n possible colors are used and order of all parts matters)

with(numtheory):
A:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*
      binomial(d+k-1, k-1), d=divisors(j))*A(n-j, k), j=1..n)/n)
    end:
a:= n-> A(n, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 26 2012

A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[Sum[d*Binomial[d+k-1, k-1], {d, Divisors[j]}]*A[n-j, k], {j, 1, n}]/n]; a[n_] := A[n, n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

a(n) = [x^n] Product_{k>=1} 1 / (1 - x^k)^binomial(k+n-1,n-1). - Ilya Gutkovskiy, May 09 2021

A209666 T(n,k) = count of degree k monomials in the complete homogeneous symmetric polynomials h(mu,k) summed over all partitions mu of n.

Original entry on oeis.org

1, 2, 7, 3, 18, 55, 5, 50, 216, 631, 7, 118, 729, 2780, 8001, 11, 301, 2621, 12954, 45865, 130453, 15, 684, 8535, 55196, 241870, 820554, 2323483, 22, 1621, 28689, 241634, 1307055, 5280204, 17353028, 48916087, 30, 3620, 91749, 1012196, 6783210, 32711022, 124991685, 401709720, 1129559068

Offset: 1

Wouter Meeussen, Mar 11 2012

			Table starts as:
  1;
  2,   7;
  3,  18, 55;
  5,  50, 216,  631;
  7, 118, 729, 2780, 8001;

Alois P. Heinz, Rows n = 1..141, flattened
Wikipedia, Symmetric Polynomials

Main diagonal is A209668; row sums are A209667.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1, k)*binomial(i+k-1, k-1)^j, j=0..n/i)))
    end:
T:= (n, k)-> b(n$2, k):
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Mar 04 2016

h[n_, v_] := Tr@ Apply[Times, Table[Subscript[x, j], {j, v}]^# & /@ Compositions[n, v], {1}]; h[par_?PartitionQ, v_] := Times @@ (h[#, v] & /@ par); Table[Tr[(h[#, k] & /@ Partitions[l]) /. Subscript[x, _] -> 1], {l, 10}, {k, l}]

A209667 a(n) = count of monomials, of degrees k=0 to n, in the complete homogeneous symmetric polynomials h(mu,k) summed over all partitions mu of n.

Original entry on oeis.org

1, 1, 9, 76, 902, 11635, 192205, 3450337, 73128340, 1696862300, 44414258862, 1264163699189, 39640715859359, 1340191402045395, 49097854149726795, 1924982506686743639, 80831323253459088871, 3607487926962810556542, 170964537623741430399076

Offset: 0

Wouter Meeussen, Mar 11 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..250
Wikipedia, Symmetric Polynomials

Cf. A209664, A209665, A209666, A209667, A209668, A209669, A209670, A209671, A209672, A209673.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1, k)*binomial(i+k-1, k-1)^j, j=0..n/i)))
    end:
a:= n-> add(b(n$2, k), k=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 04 2016

h[n_, v_] := Tr@ Apply[Times, Table[Subscript[x, j], {j, v}]^# & /@ Compositions[n, v], {1}]; h[par_?PartitionQ, v_] := Times @@ (h[#, v] & /@ par); Tr/@ Table[Tr[(h[#, k] & /@ Partitions[l]) /. Subscript[x, _] -> 1], {l, 10}, {k, l}]

Row sums of table A209666.

a(0), a(11)-a(18) from Alois P. Heinz, Mar 04 2016

A261783 Number of compositions of n where each part i is marked with a word of length i over an n-ary alphabet whose letters appear in alphabetical order.

Original entry on oeis.org

1, 1, 7, 73, 1031, 18501, 403495, 10366833, 306717703, 10271072557, 384058268507, 15861842372465, 717135437119271, 35228475333207937, 1868440035684996207, 106412817671933423073, 6477200889282232394759, 419626639092214594301373, 28829330550533269570699411

Offset: 0

Alois P. Heinz, Aug 31 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..365

Main diagonal of A261780.

Cf. A209668.

A:= proc(n, k) option remember; `if`(n=0, 1,
      add(A(n-j, k)*binomial(j+k-1, k-1), j=1..n))
    end:
a:= n-> A(n$2):
seq(a(n), n=0..20);

A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[A[n - j, k]*Binomial[j + k - 1, k - 1], {j, 1, n}]]; a[n_] := A[n, n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 24 2017, translated from Maple *)

a(n) = A261780(n,n).
a(n) = [x^n] (1-x)^n/(2*(1-x)^n-1).
a(n) ~ n^n / (sqrt(2) * (log(2))^(n+1)). - Vaclav Kotesovec, Sep 21 2019
a(n) = Sum_{k>=1} (1/2)^k * binomial(k*n-1,n). - Seiichi Manyama, Aug 06 2024

A344098 a(n) = [x^n] Product_{k>=1} (1 + x^k)^binomial(k+n-1,n-1).

Original entry on oeis.org

1, 1, 4, 29, 221, 2027, 21022, 242209, 3060262, 41936745, 618154670, 9735013136, 162892047930, 2882449728121, 53727527279464, 1051276401060921, 21529017626095851, 460231878244308738, 10246160509840187387, 237067632496414877363, 5689786581042000827057, 141415234722601777758232

Offset: 0

Ilya Gutkovskiy, May 09 2021

nonn

Cf. A075197, A209668, A219555, A305206, A338645, A343200, A344097.

Table[SeriesCoefficient[Product[(1 + x^k)^Binomial[k + n - 1, n - 1], {k, 1, n}], {x, 0, n}], {n, 0, 21}]
A[n_, k_] := A[n, k] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(j/d + 1) d Binomial[d + k - 1, k - 1], {d, Divisors[j]}] A[n - j, k], {j, 1, n}]]; a[n_] := A[n, n]; Table[a[n], {n, 0, 21}]

cp's OEIS Frontend

A247551 Decimal expansion of Product_{k>=2} 1/(1-1/k!).

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A212943 T(n,k)=Number of nXk 0..k-1 arrays with no column j greater than or equal to than column j-1 in all rows.

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A261718 Number A(n,k) of partitions of n where each part i is marked with a word of length i over a k-ary alphabet whose letters appear in alphabetical order; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A209673 a(n) = count of monomials, of degree k=n, in the Schur symmetric polynomials s(mu,k) summed over all partitions mu of n.

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Mathematica

Extensions

A075197 Number of partitions of n balls of n colors.

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Maple

Mathematica

Formula

A209666 T(n,k) = count of degree k monomials in the complete homogeneous symmetric polynomials h(mu,k) summed over all partitions mu of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A209667 a(n) = count of monomials, of degrees k=0 to n, in the complete homogeneous symmetric polynomials h(mu,k) summed over all partitions mu of n.

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Keywords

Links

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Programs

Maple

Mathematica

Formula

Extensions

A261783 Number of compositions of n where each part i is marked with a word of length i over an n-ary alphabet whose letters appear in alphabetical order.

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