A270702 -id:A270702 - cp's OEIS frontend

A270701 Total sum T(n,k) of the sizes of all blocks with maximal element k in all set partitions of {1,2,...,n}; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

1, 1, 3, 2, 4, 9, 5, 9, 16, 30, 15, 25, 41, 67, 112, 52, 82, 127, 195, 299, 463, 203, 307, 456, 670, 979, 1429, 2095, 877, 1283, 1845, 2623, 3702, 5204, 7307, 10279, 4140, 5894, 8257, 11437, 15717, 21485, 29278, 39848, 54267, 21147, 29427, 40338, 54692, 73561, 98367, 131007, 174029, 230884, 306298

Offset: 1

Alois P. Heinz, Mar 21 2016

			Row n=3 is [2, 4, 9] = [0+0+0+1+1, 0+2+1+0+1, 3+1+2+2+1] because the set partitions of {1,2,3} are: 123, 12|3, 13|2, 1|23, 1|2|3.
Triangle T(n,k) begins:
:     1;
:     1,    3;
:     2,    4,    9;
:     5,    9,   16,    30;
:    15,   25,   41,    67,   112;
:    52,   82,  127,   195,   299,   463;
:   203,  307,  456,   670,   979,  1429,  2095;
:   877, 1283, 1845,  2623,  3702,  5204,  7307, 10279;
:  4140, 5894, 8257, 11437, 15717, 21485, 29278, 39848, 54267;

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

Columns k=1-10 give: A000110(n-1), A270756, A270757, A270758, A270759, A270760, A270761, A270762, A270763, A270764.
Main and lower diagonals give: A124427, A270765, A270766, A270767, A270768, A270769, A270770, A270771, A270772, A270773.
Row sums give A070071.
Reflected triangle gives A270702.
T(2n-1,n) gives A270703.

b:= proc(n, m, t) option remember; `if`(n=0, [1, 0], add(
     `if`(t=1 and j<>m+1, 0, (p->p+`if`(j=-t or t=1 and j=m+1,
      [0, p[1]], 0))(b(n-1, max(m, j), `if`(t=1 and j=m+1, -j,
     `if`(t<0, t, `if`(t>0, t-1, 0)))))), j=1..m+1))
    end:
T:= (n, k)-> b(n, 0, max(0, 1+n-k))[2]:
seq(seq(T(n, k), k=1..n), n=1..12);

b[n_, m_, t_] := b[n, m, t] = If[n == 0, {1, 0}, Sum[If[t == 1 && j != m+1, 0, Function[p, p + If[j == -t || t == 1 && j == m+1, {0, p[[1]]}, 0]][b[ n-1, Max[m, j], If[t == 1 && j == m+1, -j, If[t < 0, t, If[t > 0, t-1, 0] ]]]]], {j, 1, m+1}]];
T[n_, k_] := b[n, 0, Max[0, 1+n-k]][[2]];
Table[Table[T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Apr 24 2016, translated from Maple *)

T(n,k) = A270702(n,n-k+1).

A070071 a(n) = n*B(n), where B(n) are the Bell numbers, A000110.

Original entry on oeis.org

0, 1, 4, 15, 60, 260, 1218, 6139, 33120, 190323, 1159750, 7464270, 50563164, 359377681, 2672590508, 20744378175, 167682274352, 1408702786668, 12277382510862, 110822101896083, 1034483164707440, 9972266139291771, 99147746245841106, 1015496134666939958

Offset: 0

Karol A. Penson, Apr 19 2002

nonn

a(n) is the total number of successions among all partitions of {1,2,...,n+1}; a succession is a pair (i,i+1) of consecutive integers lying in a block. For example, a(3)=15 because {1,2,3,4} has 6 partitions with 1 succession - 1/2/34, 1/23/4, 12/3/4, 14/23, 134/2, 124/3, 3 partitions with 2 successions - 1/234, 123/4, 12/34 and 1 partition with 3 successions - 1234. Thus a(3) = 6*1 + 3*2 + 1*3 = 15. - Augustine O. Munagi, Jul 01 2008
a(n) is the number of occurrences of integers in a list of all partitions of the set {1,...,n}. For example, the list 123, 1/23, 2/13, 3/12, 1/2/3 of all partitions of the set {1,2,3} requires 15 occurrences of integers each belonging to that set. [From Michael Hardy (hardy(AT)math.umn.edu), Nov 08 2008]
The bijection between the two foregoing characterizations is as follows: Fix x in {1,2,...,n} and associate x with the succession (x,x+1) which appears in some partitions of {1,2,...,n+1}. Replace x,x+1 by x and partition the n-set {1,2,...,x,x+2,...,n+1}, giving B(n) partitions. Thus the succession (x,x+1) occurs among partitions of {1,2,...,n+1} exactly B(n) times. - Augustine O. Munagi, Jun 02 2010

Alois P. Heinz, Table of n, a(n) for n = 0..574 (terms n=0..200 from Vincenzo Librandi)
Augustine O. Munagi, Extended set partitions with successions, European J. Combin. 29(5) (2008), 1298--1308.

Cf. A000110, A052889, A105479, A105480, A105481, A175757.

Row sums of A270236, A270701, A270702, A286416, A319298, A319375.

[n*Bell(n): n in [0..25]]; // Vincenzo Librandi, Mar 15 2014

with(combinat): a:=n->sum(numbcomb (n,0)*bell(n), j=1..n): seq(a(n), n=0..21); # Zerinvary Lajos, Apr 25 2007
with(combinat): a:=n->sum(bell(n), j=1..n): seq(a(n), n=0..21); # Zerinvary Lajos, Apr 25 2007
a:=n->sum(sum(Stirling2(n, k), j=1..n), k=1..n): seq(a(n), n=0..21); # Zerinvary Lajos, Jun 28 2007

a[n_] := n!*Coefficient[Series[x E^(E^x+x-1), {x, 0, n}], x, n]
Table[Sum[BellB[n, 1], {i, 1, n}], {n, 0, 21}] (* Zerinvary Lajos, Jul 16 2009 *)
Table[n*BellB[n], {n, 0, 20}] (* Vaclav Kotesovec, Mar 13 2014 *)

a(n)=local(t); if(n<0,0,t=exp(x+O(x^n)); n!*polcoeff(x*t*exp(t-1),n))

[bell_number(n)*n for n in range(22) ] # Zerinvary Lajos, Mar 14 2009

E.g.f: x*exp(x)*exp(exp(x)-1).
Sum_{k=1..n} n*binomial(n-1, k-1)*Bell(n-k), n >= 2. - Zerinvary Lajos, Nov 22 2006
a(n) ~ n^(n+1) * exp(n/LambertW(n)-1-n) / (sqrt(1+LambertW(n)) * LambertW(n)^n). - Vaclav Kotesovec, Mar 13 2014
a(n) = Sum_{k=1..n} k * A175757(n,k). - Alois P. Heinz, Mar 03 2020
a(n) = Sum_{j=0..n} n * Stirling2(n,j). - Detlef Meya, Apr 11 2024

A124427 Sum of the sizes of the blocks containing the element 1 in all set partitions of {1,2,...,n}.

Original entry on oeis.org

0, 1, 3, 9, 30, 112, 463, 2095, 10279, 54267, 306298, 1838320, 11677867, 78207601, 550277003, 4055549053, 31224520322, 250547144156, 2090779592827, 18110124715919, 162546260131455, 1509352980864191, 14478981877739094, 143299752100925452, 1461455003961745247

Offset: 0

Emeric Deutsch, Nov 10 2006

nonn

			a(3)=9 because the 5 (=A000110(3)) set partitions of {1,2,3} are 123, 12|3, 13|2, 1|23 and 1|2|3 and 3+2+2+1+1=9.

Alois P. Heinz, Table of n, a(n) for n = 0..575

Cf. A000110, A056857.
Column p=1 of A270236 or of A270702.
Main diagonal of A270701.

with(combinat): seq(add(k*binomial(n-1,k-1)*bell(n-k),k=1..n),n=0..30);

Table[Sum[Binomial[n-1,k-1] * BellB[n-k] * k, {k,1,n}], {n,0,22}] (* Geoffrey Critzer, Jun 14 2013 *)
Flatten[{0, Table[(n-1)*BellB[n-1] + BellB[n], {n, 1, 20}]}] (* Vaclav Kotesovec, Mar 19 2016, after Vladeta Jovovic *)

a(n) = Sum(k*binomial(n-1,k-1)*B(n-k), k=1..n) = Sum(k*A056857(n,k), k=1..n), where B(q) are the Bell numbers (A000110).
a(n) = (n-1)*B(n-1)+B(n). - Vladeta Jovovic, Nov 10 2006
a(n) ~ Bell(n) * (LambertW(n) + 1). - Vaclav Kotesovec, Jul 28 2021

a(0)=0 prepended by Alois P. Heinz, Mar 17 2016

A270703 Total sum of the sizes of all blocks with maximal element n in all set partitions of {1,2,...,2n-1}.

Original entry on oeis.org

1, 4, 41, 670, 15717, 492112, 19610565, 961547874, 56562256041, 3914022281500, 313638627550657, 28730918805512678, 2976543225606178893, 345587228510915829224, 44615408909143456529309, 6361213086726610526079402, 995709801367376369056571089

Offset: 1

Alois P. Heinz, Mar 21 2016

nonn

Also total sum of the sizes of all blocks with minimal element n in all set partitions of {1,2,...,2n-1}.

			a(2) = 4 = 0+2+1+0+1 = sum of the sizes of all blocks with maximal element 2 in all set partitions of {1,2,3}: 123, 12|3, 13|2, 1|23, 1|2|3.

Alois P. Heinz, Table of n, a(n) for n = 1..200
Wikipedia, Partition of a set

Cf. A000110, A270701, A270702.

b:= proc(n, m, t) option remember; `if`(n=0, [1, 0], add(
     `if`(t=1 and j<>m+1, 0, (p->p+`if`(j=-t or t=1 and j=m+1,
      [0, p[1]], 0))(b(n-1, max(m, j), `if`(t=1 and j=m+1, -j,
     `if`(t<0, t, `if`(t>0, t-1, 0)))))), j=1..m+1))
    end:
a:= n-> b(2*n-1, 0, n)[2]:
seq(a(n), n=1..20);

b[n_, m_, t_] := b[n, m, t] = If[n==0, {1, 0}, Sum[If[t==1 && j != m+1, 0, Function[p, p+If[j == -t || t == 1 && j == m+1, {0, p[[1]]}, 0]][b[n-1, Max[m, j], If[t == 1 && j == m+1, -j, If[t<0, t, If[t>0, t-1, 0]]]]]], {j, 1, m+1}]]; a[n_] := b[2*n-1, 0, n][[2]]; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Feb 15 2017, translated from Maple *)

a(n) = A270701(2n-1,n) = A270702(2n-1,n).

A270756 Total sum of the sizes of all blocks with maximal element 2 in all set partitions of {1,2,...,n}.

Original entry on oeis.org

3, 4, 9, 25, 82, 307, 1283, 5894, 29427, 158269, 910520, 5570737, 36071631, 246188196, 1764757189, 13246059237, 103825154098, 847806545767, 7196895817375, 63389642645486, 578318132627495, 5456455370760825, 53165437331978992, 534262881004973981

Offset: 2

Alois P. Heinz, Mar 22 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..577
Wikipedia, Partition of a set

Column k=2 of A270701.

A diagonal of A270702.

b[n_, m_, t_] := b[n, m, t] = If[n == 0, {1, 0}, Sum[If[t == 1 && j != m + 1, 0, Function[p, p + If[j == -t || t == 1 && j == m + 1, {0, p[[1]]}, 0]][b[n - 1, Max[m, j], If[t == 1 && j == m + 1, -j, If[t < 0, t, If[t > 0, t - 1, 0]]]]]], {j, 1, m + 1}]];
a[n_] := b[n, 0, Max[0, 1 + n - 2]][[2]];
Array[a, 24, 2] (* Jean-François Alcover, May 26 2018, from Maple code for A270701 *)

A270757 Total sum of the sizes of all blocks with maximal element 3 in all set partitions of {1,2,...,n}.

Original entry on oeis.org

9, 16, 41, 127, 456, 1845, 8257, 40338, 212983, 1205911, 7275802, 46534535, 314117861, 2229489144, 16584674293, 128934314027, 1044976711816, 8809644039105, 77101357474077, 699264675713410, 6561367477780443, 63603478257343891, 636087039930484642

Offset: 3

Alois P. Heinz, Mar 22 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..577
Wikipedia, Partition of a set

Column k=3 of A270701.

A diagonal of A270702.

b[n_, m_, t_] := b[n, m, t] = If[n == 0, {1, 0}, Sum[If[t == 1 && j != m + 1, 0, Function[p, p + If[j == -t || t == 1 && j == m + 1, {0, p[[1]]}, 0]][b[n - 1, Max[m, j], If[t == 1 && j == m + 1, -j, If[t < 0, t, If[t > 0, t - 1, 0]]]]]], {j, 1, m + 1}]];
a[n_] := b[n, 0, Max[0, 1 + n - 3]][[2]];
Array[a, 24, 3] (* Jean-François Alcover, May 26 2018, from Maple code for A270701 *)

A270758 Total sum of the sizes of all blocks with maximal element 4 in all set partitions of {1,2,...,n}.

Original entry on oeis.org

30, 67, 195, 670, 2623, 11437, 54692, 283625, 1581303, 9413380, 59497049, 397402597, 2794008798, 20606565063, 158955946879, 1279119138486, 10712907438835, 93190762200361, 840437752639132, 7844783028326405, 75673025264120531, 753330825335964276

Offset: 4

Alois P. Heinz, Mar 22 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..577
Wikipedia, Partition of a set

Column k=4 of A270701.

A diagonal of A270702.

b[n_, m_, t_] := b[n, m, t] = If[n == 0, {1, 0}, Sum[If[t == 1 && j != m + 1, 0, Function[p, p + If[j == -t || t == 1 && j == m + 1, {0, p[[1]]}, 0]][b[n - 1, Max[m, j], If[t == 1 && j == m + 1, -j, If[t < 0, t, If[t > 0, t - 1, 0]]]]]], {j, 1, m + 1}]];
a[n_] := b[n, 0, Max[0, 1 + n - 4]][[2]];
Array[a, 24, 4] (* Jean-François Alcover, May 26 2018, from Maple code for A270701 *)

A270759 Total sum of the sizes of all blocks with maximal element 5 in all set partitions of {1,2,...,n}.

Original entry on oeis.org

112, 299, 979, 3702, 15717, 73561, 374718, 2057641, 12088759, 75528808, 499336559, 3478563389, 25443377280, 194791872127, 1556720156567, 12955521377878, 112041717014289, 1004979994978317, 9333443375249734, 89610138696275685, 888152337277216747

Offset: 5

Alois P. Heinz, Mar 22 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..577
Wikipedia, Partition of a set

Column k=5 of A270701.

A diagonal of A270702.

b[n_, m_, t_] := b[n, m, t] = If[n == 0, {1, 0}, Sum[If[t == 1 && j != m + 1, 0, Function[p, p + If[j == -t || t == 1 && j == m + 1, {0, p[[1]]}, 0]][b[n - 1, Max[m, j], If[t == 1 && j == m + 1, -j, If[t < 0, t, If[t > 0, t - 1, 0]]]]]], {j, 1, m + 1}]];
a[n_] := b[n, 0, Max[0, 1 + n - 5]][[2]];
Array[a, 24, 5] (* Jean-François Alcover, May 26 2018, from Maple code for A270701 *)

A270760 Total sum of the sizes of all blocks with maximal element 6 in all set partitions of {1,2,...,n}.

Original entry on oeis.org

463, 1429, 5204, 21485, 98367, 492112, 2661473, 15433189, 95330022, 623920659, 4307488855, 31251896082, 237507413011, 1885386460081, 15594381406204, 134098780567817, 1196511239506523, 11057997444651072, 105684135069638365, 1043003293054453121

Offset: 6

Alois P. Heinz, Mar 22 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..577
Wikipedia, Partition of a set

Column k=6 of A270701.

A diagonal of A270702.

b[n_, m_, t_] := b[n, m, t] = If[n == 0, {1, 0}, Sum[If[t == 1 && j != m + 1, 0, Function[p, p + If[j == -t || t == 1 && j == m + 1, {0, p[[1]]}, 0]][b[n - 1, Max[m, j], If[t == 1 && j == m + 1, -j, If[t < 0, t, If[t > 0, t - 1, 0]]]]]], {j, 1, m + 1}]];
a[n_] := b[n, 0, Max[0, 1 + n - 6]][[2]];
Array[a, 24, 6] (* Jean-François Alcover, May 26 2018, from Maple code for A270701 *)

A270761 Total sum of the sizes of all blocks with maximal element 7 in all set partitions of {1,2,...,n}.

Original entry on oeis.org

2095, 7307, 29278, 131007, 643401, 3426532, 19610565, 119762455, 776018428, 5310053713, 38218929257, 288361428110, 2274040468083, 18695782169431, 159876844021430, 1419201089763907, 13053572286094533, 124202259586825404, 1220668798203165121

Offset: 7

Alois P. Heinz, Mar 22 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..577
Wikipedia, Partition of a set

Column k=7 of A270701.

A diagonal of A270702.

b[n_, m_, t_] := b[n, m, t] = If[n == 0, {1, 0}, Sum[If[t == 1 && j != m + 1, 0, Function[p, p + If[j == -t || t == 1 && j == m + 1, {0, p[[1]]}, 0]][b[n - 1, Max[m, j], If[t == 1 && j == m + 1, -j, If[t < 0, t, If[t > 0, t - 1, 0]]]]]], {j, 1, m + 1}]];
a[n_] := b[n, 0, Max[0, 1 + n - 7]][[2]];
Array[a, 24, 7] (* Jean-François Alcover, May 26 2018, from Maple code for A270701 *)

cp's OEIS Frontend

A270701 Total sum T(n,k) of the sizes of all blocks with maximal element k in all set partitions of {1,2,...,n}; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

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A070071 a(n) = n*B(n), where B(n) are the Bell numbers, A000110.

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A124427 Sum of the sizes of the blocks containing the element 1 in all set partitions of {1,2,...,n}.

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A270703 Total sum of the sizes of all blocks with maximal element n in all set partitions of {1,2,...,2n-1}.

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A270756 Total sum of the sizes of all blocks with maximal element 2 in all set partitions of {1,2,...,n}.

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A270757 Total sum of the sizes of all blocks with maximal element 3 in all set partitions of {1,2,...,n}.

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A270758 Total sum of the sizes of all blocks with maximal element 4 in all set partitions of {1,2,...,n}.

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A270759 Total sum of the sizes of all blocks with maximal element 5 in all set partitions of {1,2,...,n}.

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