A309973 -id:A309973 - cp's OEIS frontend

A005651 Sum of multinomial coefficients (n_1+n_2+...)!/(n_1!n_2!...) where (n_1, n_2, ...) runs over all integer partitions of n.

1, 1, 3, 10, 47, 246, 1602, 11481, 95503, 871030, 8879558, 98329551, 1191578522, 15543026747, 218668538441, 3285749117475, 52700813279423, 896697825211142, 16160442591627990, 307183340680888755, 6147451460222703502, 129125045333789172825, 2841626597871149750951

Offset: 0

Simon Plouffe

This is the total number of hierarchies of n labeled elements arranged on 1 to n levels. A distribution of elements onto levels is "hierarchical" if a level l+1 contains <= elements than level l. Thus for n=4 the arrangement {1,2}:{3}{4} is not allowed. See also A140585. Examples: Let the colon ":" separate two consecutive levels l and l+1. Then n=2 --> 3: {1}{2}, {1}:{2}, {2}:{1}, n=3 --> 10: {1}{2}{3}, {1}{2}:{3}, {3}{1}:{2}, {2}{3}:{1}, {1}:{2}:{3}, {3}:{1}:{2}, {2}:{3}:{1}, {1}:{3}:{2}, {2}:{1}:{3}, {3}:{2}:{1}. - Thomas Wieder, May 17 2008
n identical objects are painted by dipping them into a long row of cans of paint of distinct colors. Begining with the first can and not skipping any cans k, 1<=k<=n, objects are dipped (painted) and not more objects are dipped into any subsequent can than were dipped into the previous can. The painted objects are then linearly ordered. - Geoffrey Critzer, Jun 08 2009
a(n) is the number of partitions 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 and all n letters occur exactly once in the partition. a(3) = 10: 3abc, 2ab1c, 2ac1b, 2bc1a, 1a1b1c, 1a1c1b, 1b1a1c, 1b1c1a, 1c1a1b, 1c1b1a. - Alois P. Heinz, Aug 30 2015
Also the number of ordered set partitions of {1,...,n} with weakly decreasing block sizes. - Gus Wiseman, Sep 03 2018
The parity of a(n) is that of A000110(A000120(n)), so a(n) is even if and only if A000120(n) == 2 (mod 3). - Álvar Ibeas, Aug 11 2020

			For n=3, say the first three cans in the row contain red, white, and blue paint respectively. The objects can be painted r,r,r or r,r,w or r,w,b and then linearly ordered in 1 + 3 + 6 = 10 ways. - _Geoffrey Critzer_, Jun 08 2009
From _Gus Wiseman_, Sep 03 2018: (Start)
The a(3) = 10 ordered set partitions with weakly decreasing block sizes:
  {{1},{2},{3}}
  {{1},{3},{2}}
  {{2},{1},{3}}
  {{2},{3},{1}}
  {{3},{1},{2}}
  {{3},{2},{1}}
  {{2,3},{1}}
  {{1,2},{3}}
  {{1,3},{2}}
  {{1,2,3}}
(End)

Abramowitz and Stegun, Handbook, p. 831, column labeled "M_1".
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 126.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..450 (first 101 terms from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. E. Hoffman, Updown categories: Generating functions and universal covers, arXiv preprint arXiv:1207.1705 [math.CO], 2012.
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, 6 (1999), R2.
S. Schreiber & N. J. A. Sloane, Correspondence, 1980.

Main diagonal of: A226873, A261719, A309973.
Row sums of: A226874, A262071, A327803.
Column k=1 of A309951.
Column k=0 of A327801.
Cf. A000041, A000110, A000258, A000670, A007837, A008277, A008480, A036038, A140585, A178682, A212855, A247551, A300335, A318762.

A005651b := proc(k) add( d/(d!)^(k/d),d=numtheory[divisors](k)) ; end proc:
A005651 := proc(n) option remember; local k ; if n <= 1 then 1; else (n-1)!*add(A005651b(k)*procname(n-k)/(n-k)!, k=1..n) ; end if; end proc:
seq(A005651(k), k=0..10) ; # R. J. Mathar, Jan 03 2011
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0 or i=1, n!,
      b(n, i-1) +binomial(n, i)*b(n-i, min(n-i, i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 29 2015, Dec 12 2016

Table[Total[n!/Map[Function[n, Apply[Times, n! ]], IntegerPartitions[n]]], {n, 0, 20}] (* Geoffrey Critzer, Jun 08 2009 *)
Table[Total[Apply[Multinomial, IntegerPartitions[n], {1}]], {n, 0, 20}] (* Jean-François Alcover and Olivier Gérard, Sep 11 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[t==1, 1/n!, Sum[b[n-j, j, t-1]/j!, {j, i, n/t}]]; a[n_] := If[n==0, 1, n!*b[n, 0, n]]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Nov 20 2015, after Alois P. Heinz *)

a(m,n):=if n=m then 1 else sum(binomial(n,k)*a(k,n-k),k,m,(n/2))+1;
makelist(a(1,n),n,0,17); /* Vladimir Kruchinin, Sep 06 2014 */

a(n)=my(N=n!,s);forpart(x=n,s+=N/prod(i=1,#x,x[i]!));s \\ Charles R Greathouse IV, May 01 2015

{ my(n=25); Vec(serlaplace(prod(k=1, n, 1/(1-x^k/k!) + O(x*x^n)))) } \\ Andrew Howroyd, Dec 20 2017

E.g.f.: 1 / Product (1 - x^k/k!).
a(n) = Sum_{k=1..n} (n-1)!/(n-k)!*b(k)*a(n-k), where b(k) = Sum_{d divides k} d*d!^(-k/d). - Vladeta Jovovic, Oct 14 2002
a(n) ~ c * n!, where c = Product_{k>=2} 1/(1-1/k!) = A247551 = 2.52947747207915264... . - Vaclav Kotesovec, May 09 2014
a(n) = S(n,1), where S(n,m) = sum(k=m..n/2 , binomial(n,k)*S(n-k,k))+1, S(n,n)=1, S(n,m)=0 for nVladimir Kruchinin, Sep 06 2014
E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} x^(j*k)/(k*(j!)^k)). - Ilya Gutkovskiy, Jun 18 2018

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003

A327116 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that all parts have different color patterns and a pattern for part i has i colors in (weakly) increasing order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 2, 6, 5, 0, 2, 15, 27, 15, 0, 3, 32, 102, 124, 52, 0, 4, 65, 319, 656, 600, 203, 0, 5, 124, 897, 2780, 4210, 3084, 877, 0, 6, 230, 2346, 10305, 23040, 27567, 16849, 4140, 0, 8, 414, 5818, 34864, 108135, 188284, 186095, 97640, 21147

Offset: 0

Alois P. Heinz, Sep 13 2019

			T(3,2) = 6; 3aab, 3abb, 2aa1b, 2ab1a, 2ab1b, 2bb1a.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,   2;
  0, 2,   6,    5;
  0, 2,  15,   27,    15;
  0, 3,  32,  102,   124,     52;
  0, 4,  65,  319,   656,    600,    203;
  0, 5, 124,  897,  2780,   4210,   3084,    877;
  0, 6, 230, 2346, 10305,  23040,  27567,  16849,  4140;
  0, 8, 414, 5818, 34864, 108135, 188284, 186095, 97640, 21147;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Partition (number theory)

Columns k=0-2 give: A000007, A000009 (for n>0), A327598.
Main diagonal gives A000110.
Row sums give A317776.
T(2n,n) gives A327556.
Cf. A255903, A326914, A326962, A327117, A327557, A309973.

C:= binomial:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      b(n-i*j, min(n-i*j, i-1), k)*C(C(k+i-1, i), j), j=0..n/i)))
    end:
T:= (n, k)-> add(b(n$2, i)*(-1)^(k-i)*C(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..12);

c = Binomial;
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i j, Min[n - i j, i - 1], k] c[c[k + i - 1, i], j], {j, 0, n/i}]]];
T[n_, k_] := Sum[b[n, n, i] (-1)^(k - i) c[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 27 2020, after Alois P. Heinz *)

Sum_{k=1..n} k * T(n,k) = A327557(n).

A327679 Number of colored integer partitions of n using all colors of an initial interval of the color palette such that parts i have distinct color patterns in arbitrary order and each pattern for a part i has i colors in (weakly) increasing order.

Original entry on oeis.org

1, 1, 4, 18, 112, 732, 6156, 53720, 559584, 6138216, 76636080, 1006039320, 14693223032, 224774090592, 3756082129296, 65650522695344, 1236568354232176, 24299076684879264, 509677108276779168, 11124779898457678240, 257204596479739401760, 6174928911548312072704

Offset: 0

Alois P. Heinz, Sep 21 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..200

Row sums of A309973.

b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(b(n-i*j, min(n-i*j, i-1), k)*
       binomial(binomial(k+i-1, i), j)*j!, j=0..n/i)))
    end:
a:= n-> add(add(b(n$2, i)*(-1)^(k-i)*binomial(k, i), i=0..k), k=0..n):
seq(a(n), n=0..22);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i j, Min[n - i j, i-1], k]Binomial[Binomial[k+i-1, i], j] j!, {j, 0, n/i}]]];
a[n_] := Sum[Sum[b[n, n, i](-1)^(k-i)Binomial[k, i], {i, 0, k}], {k, 0, n}];
a /@ Range[0, 22] (* Jean-François Alcover, Dec 18 2020, after Alois P. Heinz *)

A327680 Total number of colors used in all colored integer partitions of n using all colors of an initial interval of the color palette such that parts i have distinct color patterns in arbitrary order and each pattern for a part i has i colors in (weakly) increasing order.

Original entry on oeis.org

0, 1, 7, 44, 358, 2904, 29112, 296448, 3520568, 43482208, 602603120, 8712724080, 138736978208, 2302036052128, 41417364992160, 776413790063328, 15597709327298944, 325945020056535968, 7238587734613470208, 166897326948551436384, 4061690336695535982048

Offset: 0

Alois P. Heinz, Sep 21 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A309973.

b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(b(n-i*j, min(n-i*j, i-1), k)*
       binomial(binomial(k+i-1, i), j)*j!, j=0..n/i)))
    end:
a:= n-> add(add(k*b(n$2, i)*(-1)^(k-i)*
        binomial(k, i), i=0..k), k=0..n):
seq(a(n), n=0..22);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, Min[n - i*j, i-1], k] Binomial[Binomial[k+i-1, i], j] j!, {j, 0, n/i}]]];
a[n_] := Sum[Sum[k b[n, n, i](-1)^(k-i)Binomial[k, i], {i, 0, k}], {k, 0, n}];
a /@ Range[0, 22] (* Jean-François Alcover, Dec 18 2020, after_Alois P. Heinz_ *)

a(n) = Sum_{k=1..n} k * A309973(n,k).

A327681 Number of colored integer partitions of 2n using all colors of an n-set such that parts i have distinct color patterns in arbitrary order and each pattern for a part i has i colors in (weakly) increasing order.

Original entry on oeis.org

1, 1, 21, 619, 32621, 2619031, 298688151, 45747815408, 9130881915237, 2302153903685914, 716914926484850891, 270654298469985496639, 121905995767297357401683, 64616493201145984241278851, 39838866068219563302546530228, 28277347692301453998991014108124

Offset: 0

Alois P. Heinz, Sep 21 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..200 (terms 0..100 from Alois P. Heinz)

Cf. A309973.

b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(b(n-i*j, min(n-i*j, i-1), k)*
       binomial(binomial(k+i-1, i), j)*j!, j=0..n/i)))
    end:
a:= n-> add(b(2*n$2, i)*(-1)^(n-i)*binomial(n, i), i=0..n):
seq(a(n), n=0..17);

b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i < 1, 0, Sum[b[n - i*j, Min[n - i*j, i - 1], k] Binomial[Binomial[k + i - 1, i], j]*j!, {j, 0, n/i}]]];
a[n_] := Sum[b[2n, 2n, i] (-1)^(n-i) Binomial[n, i], {i, 0, n}];
a /@ Range[0, 17] (* Jean-François Alcover, Dec 18 2020, after Alois P. Heinz *)

a(n) = A309973(2n,n).

A327890 Number of colored integer partitions of n using all colors of a 2-set such that parts i have distinct color patterns in arbitrary order and each pattern for a part i has i colors in (weakly) increasing order.

Original entry on oeis.org

0, 0, 3, 6, 21, 42, 90, 176, 348, 640, 1203, 2152, 3848, 6692, 11701, 19968, 33966, 56952, 95300, 157326, 258736, 421240, 683804, 1099830, 1762867, 2805154, 4446826, 7005486, 10999634, 17172894, 26716627, 41362952, 63837722, 98079482, 150216194, 229155682

Offset: 0

Alois P. Heinz, Sep 29 2019

nonn

			a(3) = 6: 3aab, 3abb, 2aa1b, 2ab1a, 2ab1b, 2bb1a.

Vaclav Kotesovec, Table of n, a(n) for n = 0..4000 (terms 0..1000 from Alois P. Heinz)

Column k=2 of A309973.

b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(b(n-i*j, min(n-i*j, i-1), k)*
       binomial(binomial(k+i-1, i), j)*j!, j=0..n/i)))
    end:
a:= n-> (k-> add(b(n$2, i)*(-1)^(k-i)*binomial(k, i), i=0..k))(2):
seq(a(n), n=0..44);

b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, Sum[b[n - i*j, Min[n - i*j, i-1], k] Binomial[Binomial[k+i-1, i], j] j!, {j, 0, n/i}]]];
a[n_] := With[{k = 2}, Sum[b[n, n, i](-1)^(k-i)Binomial[k, i], {i, 0, k}]];
a /@ Range[0, 44] (* Jean-François Alcover, Dec 18 2020, after Alois P. Heinz *)

cp's OEIS Frontend

A005651 Sum of multinomial coefficients (n_1+n_2+...)!/(n_1!*n_2!*...) where (n_1, n_2, ...) runs over all integer partitions of n.

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A327116 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that all parts have different color patterns and a pattern for part i has i colors in (weakly) increasing order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A327679 Number of colored integer partitions of n using all colors of an initial interval of the color palette such that parts i have distinct color patterns in arbitrary order and each pattern for a part i has i colors in (weakly) increasing order.

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A327680 Total number of colors used in all colored integer partitions of n using all colors of an initial interval of the color palette such that parts i have distinct color patterns in arbitrary order and each pattern for a part i has i colors in (weakly) increasing order.

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A327681 Number of colored integer partitions of 2n using all colors of an n-set such that parts i have distinct color patterns in arbitrary order and each pattern for a part i has i colors in (weakly) increasing order.

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A327890 Number of colored integer partitions of n using all colors of a 2-set such that parts i have distinct color patterns in arbitrary order and each pattern for a part i has i colors in (weakly) increasing order.

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A005651 Sum of multinomial coefficients (n_1+n_2+...)!/(n_1!n_2!...) where (n_1, n_2, ...) runs over all integer partitions of n.