A005493 -id:A005493 - cp's OEIS frontend

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.

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

A045379 Expansion of e.g.f.: exp(4*z + exp(z) - 1).

Original entry on oeis.org

1, 5, 26, 141, 799, 4736, 29371, 190497, 1291020, 9131275, 67310847, 516369838, 4116416797, 34051164985, 291871399682, 2588914083065, 23733360653955, 224592570163192, 2191466128865567, 22024934452712437, 227771488390279260

Offset: 0

N. J. A. Sloane

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..500
R. Jakimczuk, Successive Derivatives and Integer Sequences, J. Int. Seq. 14 (2011) # 11.7.3.
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
I. Mezo, The r-Bell numbers, J. Int. Seq. 14 (2011) # 11.1.1.

Cf. A000110, A005493, A005494, A108087, A196834.

Equals the row sums of triangle A143496. - Wolfdieter Lang, Sep 29 2011

A045379:= func< n | (&+[Binomial(n,j)*4^(n-j)*Bell(j): j in [0..n]]) >;
[A045379(n): n in [0..30]]; // G. C. Greubel, Dec 01 2022

a[0]= 1; a[n_]:= a[n]= 4*a[n-1] +Sum[Binomial[n-1, k]*a[k], {k,0,n-1}]; Array[a, 21, 0] (* Amiram Eldar, Jul 03 2020 *)

def A045379(n): return sum( 4^(n-j)*bell_number(j)*binomial(n,j) for j in range(n+1))
[A045379(n) for n in range(31)] # G. C. Greubel, Dec 01 2022

a(n) = exp(-1)*Sum_{k>=0} ((k+4)^n)/k!. - Gerald McGarvey, Jun 03 2004
A recursive formula to compute some integer sequences (including A000110, A005493, A005494 and the present sequence). Define G(n, m), where n, m >= 0, as follows: G(0, m) = 1; G(n, m) = G(n-1, m) * (m+1) + G(n-1, m+1), where n > 0. Then G(n, 0) = A000110(n+1); G(n, 1) = A005493(n+1); G(n, 2) = A005494(n+1); G(n, 3) = A045379(n+1). - Alexey Andreev (ava12(AT)nm.ru), Jan 05 2006
Define f_1(x), f_2(x), ... such that f_1(x)=x^3*e^x, f_{n+1}(x) = (d/dx)(x*f_n(x)), for n=2,3,.... Then a(n-1) = e^(-1)*f_n(1). - Milan Janjic, May 30 2008
Let A be the upper Hessenberg matrix of order n defined by: A[i,i-1]=-1, A[i,j]=binomial(j-1,i-1), (i <= j), and A[i,j]=0 otherwise. Then, for n >= 1, a(n) = (-1)^(n)*charpoly(A,-4). - Milan Janjic, Jul 08 2010
G.f.: 1/U(0) where U(k) = 1 - x*(k+5) - x^2*(k+1)/U(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 11 2012
a(n) ~ exp(n/LambertW(n) - n - 1) * n^(n + 4) / LambertW(n)^(n + 9/2). - Vaclav Kotesovec, Jun 10 2020
a(0) = 1; a(n) = 4 * a(n-1) + Sum_{k=0..n-1} binomial(n-1,k) * a(k). - Ilya Gutkovskiy, Jul 02 2020
a(n) = Sum_{j=0..n} binomial(n, j)*4^(n-j)*A000110(j). - G. C. Greubel, Dec 01 2022

A276719 Number A(n,k) of set partitions of [n] such that for each block b the smallest integer interval containing b has at most k elements; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 5, 5, 1, 0, 1, 1, 2, 5, 10, 8, 1, 0, 1, 1, 2, 5, 15, 20, 13, 1, 0, 1, 1, 2, 5, 15, 37, 42, 21, 1, 0, 1, 1, 2, 5, 15, 52, 87, 87, 34, 1, 0, 1, 1, 2, 5, 15, 52, 151, 208, 179, 55, 1, 0, 1, 1, 2, 5, 15, 52, 203, 409, 515, 370, 89, 1, 0

Offset: 0

Alois P. Heinz, Sep 16 2016

The sequence of column k satisfies a linear recurrence with constant coefficients of order 2^(k-1) for k>0.

			A(3,2) = 3: 12|3, 1|23, 1|2|3.
A(4,3) = 10: 123|4, 12|34, 12|3|4, 13|24, 13|2|4, 1|234, 1|23|4, 1|24|3, 1|2|34, 1|2|3|4.
A(5,4) = 37: 1234|5, 123|45, 123|4|5, 124|35, 124|3|5, 12|345, 12|34|5, 12|35|4, 12|3|45, 12|3|4|5, 134|25, 134|2|5, 13|245, 13|24|5, 13|25|4, 13|2|45, 13|2|4|5, 14|235, 14|23|5, 1|2345, 1|234|5, 1|235|4, 1|23|45, 1|23|4|5, 14|25|3, 14|2|35, 14|2|3|5, 1|245|3, 1|24|35, 1|24|3|5, 1|25|34, 1|2|345, 1|2|34|5, 1|25|3|4, 1|2|35|4, 1|2|3|45, 1|2|3|4|5.
Square array A(n,k) begins:
  1, 1,  1,   1,   1,    1,    1,    1,    1, ...
  0, 1,  1,   1,   1,    1,    1,    1,    1, ...
  0, 1,  2,   2,   2,    2,    2,    2,    2, ...
  0, 1,  3,   5,   5,    5,    5,    5,    5, ...
  0, 1,  5,  10,  15,   15,   15,   15,   15, ...
  0, 1,  8,  20,  37,   52,   52,   52,   52, ...
  0, 1, 13,  42,  87,  151,  203,  203,  203, ...
  0, 1, 21,  87, 208,  409,  674,  877,  877, ...
  0, 1, 34, 179, 515, 1100, 2066, 3263, 4140, ...

Alois P. Heinz, Antidiagonals n = 0..40, flattened
Pierpaolo Natalini, Paolo Emilio Ricci, New Bell-Sheffer Polynomial Sets, Axioms 2018, 7(4), 71.
Wikipedia, Partition of a set

Columns k=0..10 give: A000007, A000012, A000045(n+1), A129847, A276720, A276721, A276722, A276723, A276724, A276725, A276726.
Main diagonal gives A000110.
A(n+1,n) gives A005493(n-1) for n>0.
Cf. A276727, A276837.

b:= proc(n, m, l) option remember; `if`(n=0, 1,
      add(b(n-1, max(m, j), [subsop(1=NULL, l)[],
      `if`(j<=m, 0, j)]), j={l[], m+1} minus {0}))
    end:
A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, b(n, 0, [0$(k-1)]))):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, m_, l_List] := b[n, m, l] = If[n == 0, 1, Sum[b[n - 1, Max[m, j], Append[ReplacePart[l, 1 -> Nothing], If[j <= m, 0, j]]], {j, Append[l, m + 1] ~Complement~ {0}}]]; A[n_, k_] := If[n == 0, 1, If[k < 2, k, b[n, 0, Array[0&, k-1]]]]; Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jan 06 2017, after Alois P. Heinz *)

A(n,k) = Sum_{i=0..k} A276727(n,i).

A108087 Array, read by antidiagonals, where A(n,k) = exp(-1)*Sum_{i>=0} (i+k)^n/i!.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 5, 3, 1, 15, 15, 10, 4, 1, 52, 52, 37, 17, 5, 1, 203, 203, 151, 77, 26, 6, 1, 877, 877, 674, 372, 141, 37, 7, 1, 4140, 4140, 3263, 1915, 799, 235, 50, 8, 1, 21147, 21147, 17007, 10481, 4736, 1540, 365, 65, 9, 1, 115975, 115975, 94828, 60814, 29371, 10427, 2727, 537, 82, 10, 1

Offset: 0

Gerald McGarvey, Jun 05 2005

The column for k=0 is A000110 (Bell or exponential numbers). The column for k=1 is A000110 starting at offset 1. The column for k=2 is A005493 (Sum_{k=0..n} k*Stirling2(n,k).). The column for k=3 is A005494 (E.g.f.: exp(3*z+exp(z)-1).). The column for k=4 is A045379 (E.g.f.: exp(4*z+exp(z)-1).). The row for n=0 is 1's sequence, the row for n=1 is the natural numbers. The row for n=2 is A002522 (n^2 + 1.). The row for n=3 is A005491 (n^3 + 3n + 1.). The row for n=4 is A005492.
Number of ways of placing n labeled balls into n+k boxes, where k of the boxes are labeled and the rest are indistinguishable. - Bradley Austin (artax(AT)cruzio.com), Apr 24 2006
The column for k = -1 (not shown) is A000296 (Number of partitions of an n-set into blocks of size >1. Also number of cyclically spaced (or feasible) partitions.). - Gerald McGarvey, Oct 08 2006
Equals antidiagonals of an array in which (n+1)-th column is the binomial transform of n-th column, with leftmost column = the Bell sequence, A000110. - Gary W. Adamson, Apr 16 2009
Number of partitions of [n+k] where at least k blocks contain their own index element.  A(2,2) = 10: 134|2, 13|24, 13|2|4, 14|23, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4. - Alois P. Heinz, Jan 07 2022

			Array A(n,k) begins:
   1,   1,   1,    1,    1,     1,     1,     1,     1,      1, ... A000012;
   1,   2,   3,    4,    5,     6,     7,     8,     9,     10, ... A000027;
   2,   5,  10,   17,   26,    37,    50,    65,    82,    101, ... A002522;
   5,  15,  37,   77,  141,   235,   365,   537,   757,   1031, ... A005491;
  15,  52, 151,  372,  799,  1540,  2727,  4516,  7087,  10644, ... A005492;
  52, 203, 674, 1915, 4736, 10427, 20878, 38699, 67340, 111211, ... ;
Antidiagonal triangle, T(n, k), begins as:
     1;
     1,    1;
     2,    2,    1;
     5,    5,    3,    1;
    15,   15,   10,    4,   1;
    52,   52,   37,   17,   5,   1;
   203,  203,  151,   77,  26,   6,  1;
   877,  877,  674,  372, 141,  37,  7,  1;
  4140, 4140, 3263, 1915, 799, 235, 50,  8,  1;

F. Ruskey, Combinatorial Generation, preprint, 2001.

Alois P. Heinz, Antidiagonals n = 0..140, flattened
I. Mezo, The r-Bell numbers, J. Int. Seq. 14 (2011) # 11.1.1, Figure 1.
J. Riordan, Letter, Oct 31 1977, The array is on the second page.
F. Ruskey, Combinatorial Generation, 2003.
F. Ruskey, Lexicographic Algorithms [Broken link]

Cf. A000012, A000027, A000110, A002522, A005490, A005491.
Cf. A005492, A005493, A005494, A045379, A086659, A124824.
Main diagonal gives A134980.
Antidiagonal sums give A347420.

A108087:= func< n,k | (&+[Binomial(n-k,j)*k^j*Bell(n-k-j): j in [0..n-k]]) >;
[A108087(n,k): k in [0..n], n in [0..13]]; // G. C. Greubel, Dec 02 2022

with(combinat):
A:= (n, k)-> add(binomial(n, i) * k^i * bell(n-i), i=0..n):
seq(seq(A(d-k, k), k=0..d), d=0..12);  # Alois P. Heinz, Jul 18 2012

Unprotect[Power]; 0^0 = 1; A[n_, k_] := Sum[Binomial[n, i] * k^i * BellB[n - i], {i, 0, n}]; Table[Table[A[d - k, k], {k, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Nov 05 2015, after Alois P. Heinz *)

f(n,k)=round (suminf(i=0,(i+k)^n/i!)/exp(1));
g(n,k)=for(k=0,k,print1(f(n,k),",")) \\ prints k+1 terms of n-th row

def A108087(n,k): return sum( k^j*bell_number(n-k-j)*binomial(n-k,j) for j in range(n-k+1))
flatten([[A108087(n,k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Dec 02 2022

For n> 1, A(n, k) = k^n + sum_{i=0..n-2} A086659(n, i)*k^i. (A086659 is set partitions of n containing k-1 blocks of length 1, with e.g.f: exp(x*y)*(exp(exp(x)-1-x)-1).)
A(n, k) = k * A(n-1, k) + A(n-1, k+1), A(0, k) = 1. - Bradley Austin (artax(AT)cruzio.com), Apr 24 2006
A(n,k) = Sum_{i=0..n} C(n,i) * k^i * Bell(n-i). - Alois P. Heinz, Jul 18 2012
Sum_{k=0..n-1} A(n-k,k) = A005490(n). - Alois P. Heinz, Jan 05 2022
From G. C. Greubel, Dec 02 2022: (Start)
T(n, n) = A000012(n).
T(n, n-1) = A000027(n).
T(n, n-2) = A002522(n-1).
T(n, n-3) = A005491(n-2).
T(n, n-4) = A005492(n+1).
T(2*n, n) = A134980(n).
T(2*n, n+1) = A124824(n), n >= 1.
Sum_{k=0..n} T(n, k) = A347420(n). (End)

A087650 a(n) = Sum_{k=0..n} (-1)^(n-k)*Bell(k).

Original entry on oeis.org

1, 0, 2, 3, 12, 40, 163, 714, 3426, 17721, 98254, 580316, 3633281, 24011156, 166888166, 1216070379, 9264071768, 73600798036, 608476008123, 5224266196934, 46499892038438, 428369924118313, 4078345814329010, 40073660040755336

Offset: 0

Vladeta Jovovic, Sep 23 2003

nonn

a(n) is the number of set partitions of [n] that contain exactly one singleton block and all other blocks contain an entry > this singleton. For example, a(3)=3 counts 124/3, 134/2, 1/234 but not 123/4. - David Callan, Aug 27 2014

Partial sums are A173109. - Vladimir Reshetnikov, Oct 29 2015

			G.f. = 1 + 2*x^2 + 3*x^3 + 12*x^4 + 40*x^5 + 163*x^6 + 714*x^7 + ...

Cf. A000110, A000296, A005001, A005493, A173109.

f[n_] := Sum[ StirlingS2[n, k], {k, 1, n}]; Table[(-1)^n + Sum[(-1)^(n - k)*f[k], {k, 0, n}], {n, 0, 23}] (* Robert G. Wilson v *)
Needs["DiscreteMath`Combinatorica`"]; Table[ Sum[(-1)^(n - k)*BellB[k], {k, 0, n}], {n, 0, 23}] (* Robert G. Wilson v *)

makelist(sum((-1)^(n-k)*belln(k),k,0,n),n,0,40); /* Emanuele Munarini, Sep 27 2012 */

vector(30, n, n--; sum(k=0, n, (-1)^(n-k)*polcoeff(sum(i=0, k, prod( j=1, i, x / (1 - j*x)), x^k * O(x)), k))) \\ Altug Alkan, Oct 30 2015

def A087650_list(len): # After the formula of David Callan.
    if len == 1: return [1]
    if len == 2: return [1,0]
    R = []; A = [1]; p = -1
    for i in (0..len-1):
        A.append(A[0] - A[i])
        A[i] = A[0]
        for k in range(i, 0, -1):
            A[k-1] += A[k]
        p = -p
        R.append(A[i+1] + p)
    return R
A087650_list(24) # Peter Luschny, Aug 28 2014

E.g.f.: exp(-x)*((exp(x)-1)*exp(exp(x)-1)+1).
a(n) = (-1)^n + Bell(n) - A000296(n), with Bell(n) = A000110(n). - Wolfdieter Lang, Dec 01 2003
a(n) = A000296(n+1) + (-1)^n. - David Callan, Aug 27 2014
G.f.: 1/(1+x)/W(0), where W(k) = 1 - x/(1 - x*(k+1)/W(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 10 2014
a(0) = 1; a(n) = Sum_{k=1..n-1} binomial(n,k) * a(k-1). - Ilya Gutkovskiy, Mar 04 2021

A095149 Triangle read by rows: Aitken's array (A011971) but with a leading diagonal before it given by the Bell numbers (A000110), 1, 1, 2, 5, 15, 52, ...

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 5, 2, 3, 5, 15, 5, 7, 10, 15, 52, 15, 20, 27, 37, 52, 203, 52, 67, 87, 114, 151, 203, 877, 203, 255, 322, 409, 523, 674, 877, 4140, 877, 1080, 1335, 1657, 2066, 2589, 3263, 4140, 21147, 4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147

Offset: 0

Gary W. Adamson, May 30 2004

Or, prefix Aitken's array (A011971) with a leading diagonal of 0's and take the differences of each row to get the new triangle.
With offset 1, triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n} in which k is the largest entry in the block containing 1 (1 <= k <= n). - Emeric Deutsch, Oct 29 2006
Row term sums = the Bell numbers starting with A000110(1): 1, 2, 5, 15, ...
The k-th term in the n-th row is the number of permutations of length n starting with k and avoiding the dashed pattern 23-1. Equivalently, the number of permutations of length n ending with k and avoiding 1-32. - Andrew Baxter, Jun 13 2011
From Gus Wiseman, Aug 11 2020: (Start)
Conjecture: Also the number of divisors d with distinct prime multiplicities of the superprimorial A006939(n) that are of the form d = m * 2^k where m is odd. For example, row n = 4 counts the following divisors:
  1     2     4    8     16
  3     18    12   24    48
  5     50    20   40    80
  7     54    28   56    112
  9     1350  108  72    144
  25          540  200   400
  27          756  360   432
  45               504   720
  63               600   1008
  75               1400  1200
  135                    2160
  175                    2800
  189                    3024
  675                    10800
  4725                   75600
Equivalently, T(n,k) is the number of length-n vectors 0 <= v_i <= i whose nonzero values are distinct and such that v_n = k.
Crossrefs:
A008278 is the version counted by omega A001221.
A336420 is the version counted by Omega A001222.
A006939 lists superprimorials or Chernoff numbers.
A008302 counts divisors of superprimorials by Omega.
A022915 counts permutations of prime indices of superprimorials.
A098859 counts partitions with distinct multiplicities.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
Cf. A000005, A000142, A027423, A076954, A124010, A146291, A181818, A336417, A336419, A336421, A336499, A336942.
(End)

			Triangle starts:
   1;
   1,  1;
   2,  1,  2;
   5,  2,  3,  5;
  15,  5,  7, 10, 15;
  52, 15, 20, 27, 37, 52;
From _Gus Wiseman_, Aug 11 2020: (Start)
Row n = 3 counts the following set partitions (described in Emeric Deutsch's comment above):
  {1}{234}      {12}{34}    {123}{4}    {1234}
  {1}{2}{34}    {12}{3}{4}  {13}{24}    {124}{3}
  {1}{23}{4}                {13}{2}{4}  {134}{2}
  {1}{24}{3}                            {14}{23}
  {1}{2}{3}{4}                          {14}{2}{3}
(End)

Alois P. Heinz, Rows n = 0..150, flattened (first 51 rows from Chai Wah Wu)
Andrew M. Baxter and Lara K. Pudwell, Enumeration schemes for dashed patterns, arXiv:1108.2642 [math.CO], 2011.
Anders Claesson, Generalized pattern avoidance, Europ. J. Combin., 22 7 (2001), 961-971. See Proposition 3.
A. Bernini, M. Bouvel and L. Ferrari, Some statistics on permutations avoiding generalized patterns, PU.M.A. Vol. 18 (2007), No. 3-4, pp. 223-237 (see transposed array p. 227).

Cf. A000110, A005493, A008278, A011971, A188919, A271466.

T(2n,n) gives A020556.

with(combinat): T:=proc(n,k) if k=1 then bell(n-1) elif k=2 and n>=2 then bell(n-2) elif k<=n then add(binomial(k-2,i)*bell(n-2-i),i=0..k-2) else 0 fi end: matrix(8,8,T): for n from 1 to 11 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form
Q[1]:=t*s: for n from 2 to 11 do Q[n]:=expand(t^n*subs(t=1,Q[n-1])+s*diff(Q[n-1],s)-Q[n-1]+s*Q[n-1]) od: for n from 1 to 11 do P[n]:=sort(subs(s=1,Q[n])) od: for n from 1 to 11 do seq(coeff(P[n],t,k),k=1..n) od; # yields sequence in triangular form - Emeric Deutsch, Oct 29 2006
A011971 := proc(n,k) option remember ; if k = 0 then if n=0 then 1; else A011971(n-1,n-1) ; fi ; else A011971(n,k-1)+A011971(n-1,k-1) ; fi ; end: A000110 := proc(n) option remember; if n<=1 then 1 ; else add( binomial(n-1,i)*A000110(n-1-i),i=0..n-1) ; fi ; end: A095149 := proc(n,k) option remember ; if k = 0 then A000110(n) ; else A011971(n-1,k-1) ; fi ; end: for n from 0 to 11 do for k from 0 to n do printf("%d, ",A095149(n,k)) ; od ; od ; # R. J. Mathar, Feb 05 2007
# alternative Maple program:
b:= proc(n, m, k) option remember; `if`(n=0, 1, add(
      b(n-1, max(j, m), max(k-1, -1)), j=`if`(k=0, m+1, 1..m+1)))
    end:
T:= (n, k)-> b(n, 0, n-k):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Dec 20 2018

nmax = 10; t[n_, 1] = t[n_, n_] = BellB[n-1]; t[n_, 2] = BellB[n-2]; t[n_, k_] /; n >= k >= 3 := t[n, k] = t[n, k-1] + t[n-1, k-1]; Flatten[ Table[ t[n, k], {n, 1, nmax}, {k, 1, n}]] (* Jean-François Alcover, Nov 15 2011, from formula with offset 1 *)

# requires Python 3.2 or higher.
from itertools import accumulate
A095149_list, blist = [1,1,1], [1]
for _ in range(2*10**2):
    b = blist[-1]
    blist = list(accumulate([b]+blist))
    A095149_list += [blist[-1]]+ blist
# Chai Wah Wu, Sep 02 2014, updated Chai Wah Wu, Sep 20 2014

With offset 1, T(n,1) = T(n,n) = T(n+1,2) = B(n-1) = A000110(n-1) (the Bell numbers). T(n,k) = T(n,k-1) + T(n-1,k-1) for n >= k >= 3. T(n,n-1) = B(n-1) - B(n-2) = A005493(n-3) for n >= 3 (B(q) are the Bell numbers A000110). T(n,k) = A011971(n-2,k-2) for n >= k >= 2. In other words, deleting the first row and first column we obtain Aitken's array A011971 (also called Bell or Pierce triangle; offset in A011971 is 0). - Emeric Deutsch, Oct 29 2006

T(n,1) = B(n-1); T(n,2) = B(n-2) for n >= 2; T(n,k) = Sum_{i=0..k-2} binomial(k-2,i)*B(n-2-i) for 3 <= k <= n, where B(q) are the Bell numbers (A000110). Generating polynomial of row n is P[n](t) = Q[n](t,1), where Q[n](t,s) = t^n*Q[n-1](1,s) + s*dQ[n-1](t,s)/ds + (s-1) Q[n-1](t,s); Q[1](t,s) = ts. - Emeric Deutsch, Oct 29 2006

Corrected and extended by R. J. Mathar, Feb 05 2007

Entry revised by N. J. A. Sloane, Jun 01 2005, Jun 16 2007

A232472 2-Fubini numbers.

Original entry on oeis.org

2, 10, 62, 466, 4142, 42610, 498542, 6541426, 95160302, 1520385010, 26468935022, 498766780786, 10114484622062, 219641848007410, 5085371491003502, 125055112347154546, 3255163896227709422, 89416052656071565810, 2584886208925055791982, 78447137202259689678706, 2493719594804686310662382

Offset: 2

N. J. A. Sloane, Nov 27 2013

			G.f.: 2*x^2 + 10*x^3 + 62*x^4 + 466*x^5 + 4142*x^6 + 42610*x^7 + 498542*x^8 + ...

S. Alex Bradt, Jennifer Elder, Pamela E. Harris, Gordon Rojas Kirby, Eva Reutercrona, Yuxuan (Susan) Wang, and Juliet Whidden, Unit interval parking functions and the r-Fubini numbers, arXiv:2401.06937 [math.CO], 2024. See page 8.
Andrei Z. Broder, The r-Stirling numbers, Discrete Math. 49, 241-259 (1984).
Eldar Fischer, Johann A. Makowsky, and Vsevolod Rakita, MC-finiteness of restricted set partition functions, arXiv:2302.08265 [math.CO], 2023.
I. Mezo, Periodicity of the last digits of some combinatorial sequences, J. Integer Seq. 17, Article 14.1.1 (2014).

Cf. A000110, A000670, A005493, A005494, A008277, A069321, A143494, A143495, A232472, A232473, A232474.

Column k=1 of A122101.

r:=2; r_Fubini:=func;
[r_Fubini(n, r): n in [r..22]]; // Bruno Berselli, Mar 30 2016

# r-Stirling numbers of second kind (e.g., A008277, A143494, A143495):
T := (n,k,r) -> (1/(k-r)!)*add ((-1)^(k+i+r)*binomial(k-r,i)*(i+r)^(n-r),i = 0..k-r):
# r-Bell numbers (e.g. A000110, A005493, A005494):
B := (n,r) -> add(T(n,k,r),k=r..n);
SB := r -> [seq(B(n,r),n=r..30)];
SB(2);
# r-Fubini numbers (e.g., A000670, A232472, A232473, A232474):
F := (n,r) -> add((k)!*T(n,k,r),k=r..n);
SF := r -> [seq(F(n,r),n=r..30)];
SF(2);

Rest[max=20; t=Sum[n^(n - 1) x^n / n!, {n, 1, max}]; 2 Range[0, max]!CoefficientList[Series[D[1 / (1 - y (Exp[x] - 1)), y] /.y->1, {x, 0, max}], x]] (* Vincenzo Librandi Jan 03 2016 *)
Fubini[n_, r_] := Sum[k!*Sum[(-1)^(i+k+r)*(i+r)^(n-r)/(i!*(k-i-r)!), {i, 0, k-r}], {k, r, n}]; Table[Fubini[n, 2], {n, 2, 22}] (* Jean-François Alcover, Mar 30 2016 *)

Let A(x) be the g.f. A232472, B(x) the g.f. A000670, then A(x) = (1-x)*B(x) - 1. - Sergei N. Gladkovskii, Nov 29 2013
a(n) = Sum_{k>=2} T_k*k^(n-2)/2^k where T_k is the (k-1)-st triangular number (i.e., T_k = k*(k-1)/2). - Derek Orr, Jan 01 2016
a(n) = 2*A069321(n-1). - Vincenzo Librandi, Jan 03 2016, corrected by Vaclav Kotesovec, Jul 01 2018
a(n) ~ n! / (2 * (log(2))^(n+1)). - Vaclav Kotesovec, Jul 01 2018
From Peter Bala, Dec 08 2020: (Start)
a(n+2) = Sum_{k = 0..n} (k+2)!/k!*( Sum{i = 0..k} (-1)^(k-i)*binomial(k,i)*(i+2)^n ).
a(n+2) = Sum_{k = 0..n} 2^(n-k)*binomial(n,k)*( Sum_{i = 0..k} Stirling2(k,i)*(i+2)! ).
a(n) = 2*A069321(n-1) = A000670(n) - A000670(n-1).
a(n+1)= (1/2)*Sum_{k = 0..n} binomial(n,k)*A000670(k+1) for n >= 1.
E.g.f. with offset 0: 2*exp(2*z)/(2 - exp(z))^3 = 2 + 10*z + 62*z^2/2! + 466*z^3/3! + .... (End)

A033452 "STIRLING" transform of squares A000290.

Original entry on oeis.org

0, 1, 5, 22, 99, 471, 2386, 12867, 73681, 446620, 2856457, 19217243, 135610448, 1001159901, 7714225057, 61904585510, 516347066551, 4468588592739, 40058673825258, 371421499686007, 3556976106133821, 35138574378189700, 357654857584636597, 3746672593640388775

Offset: 0

N. J. A. Sloane

nonn

If an integer N is squarefree and has n+2 distinct prime factors then a(n) is the number of product signs needed to write the factorizations of N, so a(n)=A076277(N). - Floor van Lamoen, Oct 17 2002

Convolved with powers of 2 = A058681: (1, 7, 36, 171, 813, ...). Cf. triangle A180338. - Gary W. Adamson, Aug 28 2010

			G.f. = x + 5*x^2 + 22*x^3 + 99*x^4 + 471*x^5 + 2386*x^6 + 12867*x^7 + 73681*x^8 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..574

Partial sums of A005494.

Cf. A180338.

a := n -> add(Stirling2(n, j)*j^2, j=0..n): seq(a(n), n=0..20); # Zerinvary Lajos, Apr 18 2007
# second Maple program:
b:= proc(n, m) option remember;
     `if`(n=0, m^2, m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..23);  # Alois P. Heinz, Aug 04 2021

max = 20; Clear[g]; g[max + 2] = 1; g[k_] := g[k] = 2 - 1/(1 - k*x)/(1 - x/(x - 1/g[k + 1])); gf = 1/x + 1/x^2 - g[0]/x^2; CoefficientList[ Series[gf, {x, 0, max}], x] (* Jean-François Alcover, Jan 24 2013, after Sergei N. Gladkovskii *)

{a(n) = if( n<0, 0, n! * polcoeff( (exp(x + x * O(x^n)) - 1) * exp( exp(x + x * O(x^n)) - 1 + x), n))}; /* Michael Somos, Mar 28 2012 */

Representation as an infinite series: a(n) = (Sum_{k>=1} k^n*k*(k-2)/k!)/exp(1), n >= 1. This is a Dobinski-type summation formula. - Karol A. Penson, Mar 21 2002
a(n) = A005493(n) - A000110(n+1). - Floor van Lamoen and Christian Bower, Oct 16 2002. (n^2 has e.g.f.: e^x * (x^2+x), a(n) thus has e.g.f: e^(e^x-1) * ( (e^x-1)^2 + (e^x-1) ) which simplifies to e^(e^x-1) * (e^2x - e^x). A005493 has e.g.f.: e^(e^x+2x-1), A000110 has e.g.f.: e^(e^x-1), A000110(n+1) has as e.g.f.: derivative of A000110 which is e^(e^x+x-1).) [corrected by Georg Fischer, Jun 17 2020]
a(n) = Bell(n+2) - 2*Bell(n+1). - Vladeta Jovovic, Jul 28 2003
G.f.: sum{k>=0, k^2*x^k/prod[l=1..k, 1-lx]}. - Ralf Stephan, Apr 18 2004
E.g.f.: exp( exp(x) - 1 + x) * (exp(x) - 1). - Michael Somos, Mar 28 2012
a(n) = A123158(n,3). - Philippe Deléham, Oct 06 2006
G.f.: G(0)/x -1/x, where G(k) = 1 - x^2*(k+1)/( x^2*(k+1) - (2*x+x*k-1)*(3*x+x*k-1)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Feb 25 2014

A059606 Expansion of (1/2)(exp(2x)-1)*exp(exp(x)-1).

Original entry on oeis.org

0, 1, 4, 16, 68, 311, 1530, 8065, 45344, 270724, 1709526, 11376135, 79520644, 582207393, 4453142140, 35500884556, 294365897104, 2533900264547, 22604669612078, 208656457858161, 1990060882027600

Offset: 0

Vladeta Jovovic, Jan 29 2001

Starting (1, 4, 16, 68, 311, ...), = A008277 * A000217, i.e., the product of the Stirling2 triangle and triangular series. - Gary W. Adamson, Jan 31 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vladeta Jovovic, More information.

Cf. A000110, A005493, A059604, A059605.
Cf. A035098.
Cf. A008277, A000217.

s := series(1/2*(exp(2*x)-1)*exp(exp(x)-1), x, 21): for i from 0 to 20 do printf(`%d,`,i!*coeff(s,x,i)) od:

With[{nn=20},CoefficientList[Series[((Exp[2x]-1)Exp[Exp[x]-1])/2,{x,0,nn}] ,x] Range[0,nn]!] (* Harvey P. Dale, Nov 10 2011 *)

a(n) = Sum_{i=0..n} Stirling2(n, i)*binomial(i+1, 2).
a(n) = (1/2)*(Bell(n+2)-Bell(n+1)-Bell(n)). - Vladeta Jovovic, Sep 23 2003
G.f.: Sum_{k>=1} (k*(k + 1)/2)*x^k/Product_{j=1..k} (1 - j*x). - Ilya Gutkovskiy, Jun 19 2018
a(n) ~ n^2 * Bell(n) / (2*LambertW(n)^2) * (1 - LambertW(n)/n). - Vaclav Kotesovec, Jul 28 2021

A101851 a(n) = Sum_{k=0..n} (-1)^(n-k)kStirling2(n,k).

Original entry on oeis.org

0, 1, 1, -2, -1, 11, -18, -41, 317, -680, -1767, 19911, -68264, -59643, 2076973, -11905466, 18577387, 269836343, -2819431570, 12357816867, 17355428041, -752675321800, 6318046208653, -21416130683133, -152569023028272, 3016508107668601, -23667435182395287

Offset: 0

Vladeta Jovovic, Jan 27 2005

Robert Israel, Table of n, a(n) for n = 0..580
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 28, Table 2, operad NAP.

Cf. A005493, A000587.

A101851 := proc(n) local k;
add((-1)^(n-k)*k*combinat[stirling2](n,k), k = 0..n) end:
seq(A101851(n),n = 0..26); # Peter Luschny, Apr 17 2011

Table[Sum[(-1)^(n-k) k StirlingS2[n,k],{k,0,n}],{n,0,30}] (* Harvey P. Dale, Aug 09 2013 *)
Table[(-1)^n (BellB[n, -1] + BellB[n + 1, -1]), {n, 0, 25}] (* Vladimir Reshetnikov, Oct 21 2015 *)

a(n) = sum(k=0, n, (-1)^(n-k)*k*stirling(n, k, 2)); \\ Michel Marcus, Oct 22 2015

E.g.f.: (1-exp(-x))*exp(1-exp(-x)): G.f.: Sum(k*x^k/Product(1+l*x, l = 1 .. k), k = 1 .. infinity).
a(n) = Sum_{k=0..n} (-1)^(k+1)*binomial(n,k)*A000587(k+2). - Peter Luschny, Apr 17 2011
G.f.: x*G(0)/(1+x) where G(k) = 1 + 2*x*(k+1)/((2*k+1)*(2*x*k+2*x+1) - x*(2*k+1)*(2*k+3)*(2*x*k+2*x+1)/(x*(2*k+3) + 2*(k+1)*(2*x*k+3*x+1)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 19 2012
G.f.: 1/x - G(0)/x, where G(k) = 1 - x^2*(k+1)/(x^2*(k+1) + (x*k + 1 - x)*(x*k + 1)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Feb 06 2014
a(n) = (-1)^n*(A000587(n)+A000587(n+1)). - Vladimir Reshetnikov, Oct 21 2015

cp's OEIS Frontend

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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A045379 Expansion of e.g.f.: exp(4*z + exp(z) - 1).

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A276719 Number A(n,k) of set partitions of [n] such that for each block b the smallest integer interval containing b has at most k elements; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A108087 Array, read by antidiagonals, where A(n,k) = exp(-1)*Sum_{i>=0} (i+k)^n/i!.

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A087650 a(n) = Sum_{k=0..n} (-1)^(n-k)*Bell(k).

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A095149 Triangle read by rows: Aitken's array (A011971) but with a leading diagonal before it given by the Bell numbers (A000110), 1, 1, 2, 5, 15, 52, ...

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A232472 2-Fubini numbers.

A059606 Expansion of (1/2)(exp(2x)-1)*exp(exp(x)-1).

A101851 a(n) = Sum_{k=0..n} (-1)^(n-k)kStirling2(n,k).