A285595 -id:A285595 - cp's OEIS frontend

A285362 Sum T(n,k) of the entries in the k-th blocks of all set partitions of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

1, 4, 2, 15, 12, 3, 60, 58, 28, 4, 262, 273, 185, 55, 5, 1243, 1329, 1094, 495, 96, 6, 6358, 6839, 6293, 3757, 1148, 154, 7, 34835, 37423, 36619, 26421, 11122, 2380, 232, 8, 203307, 217606, 219931, 180482, 96454, 28975, 4518, 333, 9, 1257913, 1340597, 1376929, 1230737, 787959, 308127, 67898, 7995, 460, 10

Offset: 1

Alois P. Heinz, Apr 17 2017

			T(3,2) = 12 because the sum of the entries in the second blocks of all set partitions of [3] (123, 12|3, 13|2, 1|23, 1|2|3) is 0+3+2+5+2 = 12.
Triangle T(n,k) begins:
      1;
      4,     2;
     15,    12,     3;
     60,    58,    28,     4;
    262,   273,   185,    55,     5;
   1243,  1329,  1094,   495,    96,    6;
   6358,  6839,  6293,  3757,  1148,  154,   7;
  34835, 37423, 36619, 26421, 11122, 2380, 232, 8;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Wikipedia, Partition of a set

Columns k=1-10 give: A285363, A285364, A285365, A285366, A285367, A285368, A285369, A285370, A285371, A285372.
Row sums give A000110(n) * A000217(n) = A105488(n+3).
Main diagonal and first lower diagonal give: A000027, A006000 (for n>0).
T(2n+1,n+1) gives A285410.
Cf. A270236, A285439, A285595.

T:= proc(h) option remember; local b; b:=
      proc(n, m) option remember; `if`(n=0, [1, 0], add((p-> p
        +[0, (h-n+1)*p[1]*x^j])(b(n-1, max(m, j))), j=1..m+1))
      end: (p-> seq(coeff(p, x, i), i=1..n))(b(h, 0)[2])
    end:
seq(T(n), n=1..12);

T[h_] := T[h] = Module[{b}, b[n_, m_] := b[n, m] = If[n == 0, {1, 0}, Sum[# + {0, (h - n + 1)*#[[1]]*x^j}&[b[n - 1, Max[m, j]]], {j, 1, m + 1}]]; Table[Coefficient[#, x, i], {i, 1, n}]&[b[h, 0][[2]]]];
Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

A283424 Number T(n,k) of blocks of size >= k in all set partitions of [n], assuming that every set partition contains one block of size zero; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 2, 1, 5, 3, 1, 15, 10, 4, 1, 52, 37, 17, 5, 1, 203, 151, 76, 26, 6, 1, 877, 674, 362, 137, 37, 7, 1, 4140, 3263, 1842, 750, 225, 50, 8, 1, 21147, 17007, 9991, 4307, 1395, 345, 65, 9, 1, 115975, 94828, 57568, 25996, 8944, 2392, 502, 82, 10, 1

Offset: 0

Alois P. Heinz, May 14 2017

T(n,k) is defined for all n,k >= 0. The triangle contains only the terms with k<=n. T(n,k) = 0 for k>n.

			T(3,2) = 4 because the number of blocks of size >= 2 in all set partitions of [3] (123, 12|3, 13|2, 1|23, 1|2|3) is 1+1+1+1+0 = 4.
Triangle T(n,k) begins:
      1;
      2,     1;
      5,     3,    1;
     15,    10,    4,    1;
     52,    37,   17,    5,    1;
    203,   151,   76,   26,    6,   1;
    877,   674,  362,  137,   37,   7,  1;
   4140,  3263, 1842,  750,  225,  50,  8, 1;
  21147, 17007, 9991, 4307, 1395, 345, 65, 9, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Partition of a set

Columns k=0-10 give: A000110(n+1), A138378 or A005493(n-1), A124325, A288785, A288786, A288787, A288788, A288789, A288790, A288791, A288792.
Row sums give A124427(n+1).
T(2n,n) gives A286896.
Cf. A005493, A056857, A056860, A070071, A224271, A285595.

T:= proc(n, k) option remember; `if`(k>n, 0,
      binomial(n, k)*combinat[bell](n-k)+T(n, k+1))
    end:
seq(seq(T(n, k), k=0..n), n=0..14);

T[n_, k_] := Sum[Binomial[n, j]*BellB[j], {j, 0, n - k}];
Table[T[n, k], {n, 0, 14}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 30 2018 *)

T(n,k) = Sum_{j=0..n-k} binomial(n,j) * Bell(j).
T(n,k) = Bell(n+1) - Sum_{j=0..k-1} binomial(n,j) * Bell(n-j).
T(n,k) = Sum_{j=k..n} A056857(n+1,j) = Sum_{j=k..n} A056860(n+1,n+1-j).
Sum_{k=0..n} T(n,k) = n*Bell(n)+Bell(n+1) = A124427(n+1).
Sum_{k=1..n} T(n,k) = n*Bell(n) = A070071(n).
T(n,0)-T(n,1) = Bell(n).
Sum_{k=0..n} (-1)^k * T(n,k) = A224271(n+1). - Alois P. Heinz, May 17 2023

A278677 a(n) = Sum_{k=0..n} A011971(n, k)*(k + 1). The Aitken-Bell triangle considered as a linear transform applied to the positive numbers.

Original entry on oeis.org

1, 5, 23, 109, 544, 2876, 16113, 95495, 597155, 3929243, 27132324, 196122796, 1480531285, 11647194573, 95297546695, 809490850313, 7126717111964, 64930685865768, 611337506786061, 5940420217001199, 59502456129204083, 613689271227219015, 6510381400140132872

Offset: 0

Sergey Kirgizov, Nov 26 2016

nonn

Original name: Popularity of left children in treeshelves avoiding pattern T231 (with offset 2).
Treeshelves are ordered binary (0-1-2) increasing trees where every child is connected to its parent by a left or a right link. Classical Françon's bijection maps bijectively treeshelves into permutations. Pattern T231 illustrated below corresponds to a treeshelf constructed from permutation 231. Popularity is the sum of a certain statistic (number of left children, in this case) over all objects of size n.
a(n) is also the sum of the last entries in all blocks of all set partitions of [n-1]. a(4) = 23 because the sum of the last entries in all blocks of all set partitions of [3] (123, 12|3, 13|2, 1|23, 1|2|3) is 3+5+5+4+6  = 23. - Alois P. Heinz, Apr 24 2017
a(n-2) is the number of lines that rhyme (with at least one earlier line) across all rhyme schemes counted by A000110. - Martin Fuller, Apr 20 2025

			Treeshelves of size 3:
      1  1          1    1       1        1
     /    \        /      \     / \      / \
    2      2      /        \   2   \    /   2
   /        \    2          2       3  3
  3          3    \        /
                   3      3
Pattern T231:
     1
    /
   /
  2
   \
    3
Treeshelves of size 3 that avoid pattern T231:
      1  1      1       1        1
     /    \      \     / \      / \
    2      2      \   2   \    /   2
   /        \      2       3  3
  3          3    /
                 3
Popularity of left children here is 5.

Alois P. Heinz, Table of n, a(n) for n = 0..572
Jean-Luc Baril, Sergey Kirgizov, and Vincent Vajnovszki, Patterns in treeshelves, arXiv:1611.07793 [cs.DM], 2016.
J. Françon, Arbres binaires de recherche : propriétés combinatoires et applications, Revue française d'automatique informatique recherche opérationnelle, Informatique théorique, 10 no. 3 (1976), pp. 35-50.

Cf. A000110, A000111, A000142, A001286, A008292, A011971, A131178, A278678, A278679, A285595, A286897, A367955.

b:= proc(n, m) option remember; `if`(n=0, [1, 0],
     (p-> p+[0, p[1]*n])(b(n-1, m+1))+m*b(n-1, m))
    end:
a:= n-> b(n+1, 0)[2]:
seq(a(n), n=0..22);  # Alois P. Heinz, Dec 15 2023
# Using the generating function:
gf := ((exp(z + exp(z)-1)*(z-1)) + exp(exp(z)-1))/z^2: ser := series(gf, z, 25):
seq((n+2)!*coeff(ser, z, n), n=0..22);  # Peter Luschny, Feb 01 2025

a[n_] := (n+3) BellB[n+2] - BellB[n+3];
Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Dec 01 2018 *)

from sympy import bell
HOW_MANY = 30
print([(n + 3) * bell(n+2) - bell(n + 3) for n in range(HOW_MANY)])

E.g.f.: ((exp(z + exp(z)-1)*(z-1)) + exp(exp(z)-1))/z^2.
a(n) = (n + 3)*b(n + 2) - b(n + 3) where b(n) is the n-th Bell number (see A000110).
Asymptotics: a(n) ~ n*b(n).
a(n) = Sum_{k=1..n+1} A285595(n+1,k)/k. - Alois P. Heinz, Apr 24 2017
a(n) = Sum_{k=0..n} Stirling2(n+2, k+1) * (n+1-k). - Ilya Gutkovskiy, Apr 06 2021
a(n) ~ n*Bell(n)*(1 - 1/LambertW(n)). - Vaclav Kotesovec, Jul 28 2021
a(n) = Sum_{k=n+1..(n+1)*(n+2)/2} k * A367955(n+1,k). - Alois P. Heinz, Dec 11 2023

New name and offset 0 by Peter Luschny, Feb 01 2025

A285793 Sum T(n,k) of the k-th entries in all cycles of all permutations of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

Original entry on oeis.org

1, 4, 2, 18, 13, 5, 96, 83, 43, 18, 600, 582, 342, 192, 84, 4320, 4554, 2874, 1824, 1068, 480, 35280, 39672, 26232, 17832, 11784, 7080, 3240, 322560, 382248, 261288, 185688, 131256, 88920, 54360, 25200, 3265920, 4044240, 2834640, 2078640, 1534320, 1110960, 765360, 473760, 221760

Offset: 1

Alois P. Heinz, Apr 26 2017

Each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.

			T(3,2) = 13 because the sum of the second entries in all cycles of all permutations of [3] ((123), (132), (12)(3), (13)(2), (1)(23), (1)(2)(3)) is 2+3+2+3+3+0 = 13.
Triangle T(n,k) begins:
:      1;
:      4,      2;
:     18,     13,      5;
:     96,     83,     43,     18;
:    600,    582,    342,    192,     84;
:   4320,   4554,   2874,   1824,   1068,   480;
:  35280,  39672,  26232,  17832,  11784,  7080,  3240;
: 322560, 382248, 261288, 185688, 131256, 88920, 54360, 25200;

Wikipedia, Permutation

Columns k=1-2 give: A001563, A285795.
Main diagonal and first lower diagonal give: A038720(n-1) (for n>1), A286175.
Row sums give A000142 * A000217 = A180119.
Cf. A285439, A285595, A286231.

T(n,1) = n * n!.

T(n,n) = floor((n-1)!*(n+2)/2).

A124325 Number of blocks of size >1 in all partitions of an n-set.

Original entry on oeis.org

0, 0, 1, 4, 17, 76, 362, 1842, 9991, 57568, 351125, 2259302, 15288000, 108478124, 805037105, 6233693772, 50257390937, 421049519856, 3659097742426, 32931956713294, 306490813820239, 2945638599347760, 29198154161188501

Offset: 0

Emeric Deutsch, Oct 28 2006

nonn

Sum of the first entries in all blocks of all set partitions of [n-1]. a(4) = 17 because the sum of the first entries in all blocks of all set partitions of [3] (123, 12|3, 13|2, 1|23, 1|2|3) is 1+4+3+3+6 = 17. - Alois P. Heinz, Apr 24 2017

			a(3) = 4 because in the partitions 123, 12|3, 13|2, 1|23, 1|2|3 we have four blocks of size >1.

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

Cf. A000110, A123324, A124327, A285595.

Column k=2 of A283424.

with(combinat): c:=n->bell(n+1)-bell(n)-n*bell(n-1): seq(c(n),n=0..23);

nn=22;Range[0,nn]!CoefficientList[Series[(Exp[x]-1-x)Exp[Exp[x]-1],{x,0,nn}],x]  (* Geoffrey Critzer, Mar 28 2013 *)

N = 66;  x = 'x + O('x^N);
egf = (exp(x)-1-x)*exp(exp(x)-1) + 'c0;
gf = serlaplace(egf);
v = Vec(gf);  v[1]-='c0;  v
/* Joerg Arndt, Mar 29 2013 */

a(n) = B(n+1)-B(n)-n*B(n-1), where B(q) are the Bell numbers (A000110).
E.g.f.: (exp(z)-1-z)*exp(exp(z)-1).
a(n) = Sum_{k=0..floor(n/2)} k*A124324(n,k).
a(n) = A285595(n-1,1). - Alois P. Heinz, Apr 24 2017
a(n) = Sum_{k=1..n*(n-1)/2} k * A124327(n-1,k) for n>1. - Alois P. Heinz, Dec 05 2023

A175757 Triangular array read by rows: T(n,k) is the number of blocks of size k in all set partitions of {1,2,...,n}.

Original entry on oeis.org

1, 2, 1, 6, 3, 1, 20, 12, 4, 1, 75, 50, 20, 5, 1, 312, 225, 100, 30, 6, 1, 1421, 1092, 525, 175, 42, 7, 1, 7016, 5684, 2912, 1050, 280, 56, 8, 1, 37260, 31572, 17052, 6552, 1890, 420, 72, 9, 1, 211470, 186300, 105240, 42630, 13104, 3150, 600, 90, 10, 1

Offset: 1

Geoffrey Critzer, Dec 04 2010

The row sums of this triangle equal A005493. Equals A056857 without its leftmost column.

T(n,k) = binomial(n,k)*B(n-k) where B is the Bell number.

			The set {1,2,3} has 5 partitions, {{1, 2, 3}}, {{2, 3}, {1}}, {{1, 3}, {2}}, {{1, 2}, {3}}, and {{2}, {3}, {1}}, and there are a total of 3 blocks of size 2, so T(3,2)=3.
Triangle begins:
    1;
    2,   1;
    6,   3,   1;
   20,  12,   4,  1;
   75,  50,  20,  5, 1;
  312, 225, 100, 30, 6, 1;
  ...

Alois P. Heinz, Rows n = 1..141, flattened

Cf. A000110, A056860, A052889, A070071, A105479, A105480, A105481, A105482, A285595.

b:= proc(n) option remember; `if`(n=0, [1, 0],
      add((p-> p+[0, p[1]*x^j])(b(n-j)*
      binomial(n-1, j-1)), j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n)[2]):
seq(T(n), n=1..12);  # Alois P. Heinz, Apr 24 2017

Table[Table[Length[Select[Level[SetPartitions[m],{2}],Length[#]==n&]],{n,1,m}],{m,1,10}]//Grid

E.g.f. for column k is x^k/k!*exp(exp(x)-1).

Sum_{k=1..n} k * T(n,k) = A070071(n). - Alois P. Heinz, Mar 03 2020

A286897 Sum T(n,k) of the k-th last entries in all blocks of all set partitions of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

Original entry on oeis.org

1, 5, 1, 23, 6, 1, 109, 33, 7, 1, 544, 182, 45, 8, 1, 2876, 1034, 284, 59, 9, 1, 16113, 6122, 1815, 420, 75, 10, 1, 95495, 37927, 11931, 2987, 595, 93, 11, 1, 597155, 246030, 81205, 21620, 4665, 814, 113, 12, 1, 3929243, 1669941, 573724, 160607, 36900, 6979, 1082, 135, 13, 1

Offset: 1

Alois P. Heinz, May 15 2017

			T(3,2) = 6 because the sum of the second last entries in all blocks of all set partitions of [3] (123, 12|3, 13|2, 1|23, 1|2|3) is 2+1+1+2 = 6.
Triangle T(n,k) begins:
      1;
      5,     1;
     23,     6,     1;
    109,    33,     7,    1;
    544,   182,    45,    8,   1;
   2876,  1034,   284,   59,   9,  1;
  16113,  6122,  1815,  420,  75, 10,  1;
  95495, 37927, 11931, 2987, 595, 93, 11, 1;
  ...

Alois P. Heinz, Row n = 1..50, flattened
Wikipedia, Partition of a set

Column k=1 gives A278677(n-1).
Row sums give A000110(n) * A000217(n) = A105488(n+3).
Cf. A285595.

b:= proc(n, l) option remember; `if`(n=0, [1, 0],
      (p-> p+[0, n*p[1]*x^1])(b(n-1, [l[], 1]))+
       add((p-> p+[0, n*p[1]*x^(l[j]+1)])(b(n-1,
       sort(subsop(j=l[j]+1, l), `>`))), j=1..nops(l)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n, [])[2]):
seq(T(n), n=1..14);

b[0, ] = {1, 0}; b[n, l_] := b[n, l] = Function[p, p + {0, n*p[[1]]*x^1} ][b[n - 1, Append[l, 1]]] + Sum[Function[p, p + {0, n*p[[1]]*x^(l[[j]] + 1)}][b[n - 1, Reverse @ Sort[ReplacePart[l, j -> l[[j]] + 1]]]], {j, 1, Length[l]}];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, n}]][b[n, {}][[2]]];
Table[T[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, May 26 2018, from Maple *)

cp's OEIS Frontend

A285362 Sum T(n,k) of the entries in the k-th blocks of all set partitions of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

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A283424 Number T(n,k) of blocks of size >= k in all set partitions of [n], assuming that every set partition contains one block of size zero; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A278677 a(n) = Sum_{k=0..n} A011971(n, k)*(k + 1). The Aitken-Bell triangle considered as a linear transform applied to the positive numbers.

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A285793 Sum T(n,k) of the k-th entries in all cycles of all permutations of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

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A124325 Number of blocks of size >1 in all partitions of an n-set.

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A175757 Triangular array read by rows: T(n,k) is the number of blocks of size k in all set partitions of {1,2,...,n}.

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A286897 Sum T(n,k) of the k-th last entries in all blocks of all set partitions of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

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