A000296 -id:A000296 - cp's OEIS frontend

A178867 Irregular triangle read by rows: multinomial coefficients, version 3; alternatively, row n gives coefficients of the n-th complete exponential Bell polynomial B_n(x_1, x_2, ...) with monomials sorted into some unknown order.

1, 1, 1, 1, 3, 1, 1, 6, 4, 3, 1, 1, 10, 10, 15, 5, 10, 1, 1, 15, 20, 45, 15, 60, 6, 15, 15, 10, 1, 1, 21, 35, 105, 35, 210, 21, 105, 105, 70, 7, 105, 35, 21, 1, 1, 28, 56, 210, 70, 560, 56, 420, 420, 280, 28, 840, 280, 168, 8, 280, 210, 105, 56, 35, 28, 1

Offset: 1

Johannes W. Meijer and Manuel Nepveu (Manuel.Nepveu(AT)tno.nl), Jun 21 2010, Jun 24 2010

"Exponential" here means in contrast to "ordinary", cf. A263633 (see Comtet). "Standard order" means as produced by Maple's "sort" command. - N. J. A. Sloane, Oct 28 2015
From Petros Hadjicostas, May 27 2020: (Start)
According to the Maple help files for the "sort" command, polynomials in multiple variables are "sorted in total degree with ties broken by lexicographic order (this is called graded lexicographic order)."
Thus, for example, x_1^2*x_3 = x_1*x_1*x_3 > x_1*x_2*x_2 = x_1*x_2^2, while x_1^2*x_4 = x_1*x_1*x_4 > x_1*x_2*x_3.
It appears that the authors' n-th row does give the coefficients of the n-th complete exponential Bell polynomial. Starting with row 6, however, it is unknown in what order the monomials of the n-th complete exponential Bell polynomial follow. It is definitely not the standard order nor any other known order. (End)
This version of the multinomial coefficients was discovered while calculating the probability that two 23 year old chessplayers would play each other on their birthday during a Dutch Chess Championship. This unique encounter took place on Apr 05 2008. Its probability is 1 in 50000 years, see the Meijer-Nepveu article.
The third version of the multinomial coefficients can be constructed with the basic multinomial coefficients A178866; see the formulas below. These multinomial coefficients appear in the columns of the multinomial coefficient matrix MCM[n,m] (n >= 1 and m >= 1).
We observe that the sum of the C(m,n) coefficients follow the A000296(n) sequence. The missing C(m, n=1) corresponds to A000296(1) = 0.
The number of multinomial coefficients in a triangle row leads to the partition numbers A000041. The row sums lead to the Bell numbers A000110.

			The first few complete exponential Bell polynomials are:
(1) x[1];
(2) x[1]^2 + x[2];
(3) x[1]^3 + 3*x[1]*x[2] + x[3];
(4) x[1]^4 + 6*x[1]^2*x[2] + 4*x[1]*x[3] + 3*x[2]^2 + x[4];
(5) x[1]^5 + 10*x[1]^3*x[2] + 10*x[1]^2*x[3] + 15*x[1]*x[2]^2 + 5*x[1]*x[4] + 10*x[2]*x[3] + x[5];
(6) x[1]^6 + 15*x[1]^4*x[2] + 20*x[1]^3*x[3] + 45*x[1]^2*x[2]^2 + 15*x[1]^2*x[4] + 60*x[1]*x[2]*x[3] + 6*x[1]*x[5] + 15*x[2]^3 + 15*x[2]*x[4] + 10*x[3]^2 + x[6].
(7) x[1]^7 + 21*x[1]^5*x[2] + 35*x[1]^4*x[3] + 105*x[1]^3*x[2]^2 + 35*x[1]^3*x[4] + 210*x[1]^2*x[2]*x[3] + 21*x[1]^2*x[5] + 105*x[1]*x[2]^3 + 105*x[1]*x[2]*x[4] + 70*x[1]*x[3]^2 + 7*x[1]*x[6] + 105*x[2]^2*x[3] + 35*x[3]*x[4] + 21*x[2]*x[5] + x[7].
...
The first few rows of the triangle are
  1;
  1,  1;
  1,  3,  1;
  1,  6,  4,   3,  1;
  1, 10, 10,  15,  5,  10,  1;
  1, 15, 20,  45, 15,  60,  6,  15,  15, 10, 1;
  1, 21, 35, 105, 35, 210, 21, 105, 105, 70, 7, 105, 35, 21, 1;
  ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 134, 307-310.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, Chapter 2, Section 8 and table on page 49.

Kevin Brown, Generalized Birthday Problem (N Items in M Bins), 1994-2010.
Paul E. Gunnells, Generalized Catalan numbers from hypergraphs, arXiv:2102.05121 [math.CO], 2021. Mentions this sequence p. 27.
J. W. Meijer and M. Nepveu, Euler's ship on the Pentagonal Sea, Acta Nova, 4(1) (2008), 176-187.
Jin Wang, Nonlinear Inverse Relations for Bell Polynomials via the Lagrange Inversion Formula, J. Int. Seq., Vol. 22 (2019), Article 19.3.8.

Cf. A036040 (version 1 of multinomial coefficients), A080575 (version 2).

Cf. A000041, A000110, A000296, A178866, A263633.

with(combinat): nmax:=11; A178866(1):=1: T:=1: for n from 1 to nmax do y(n):=numbpart(n): P(n):=sort(partition(n)): k:=0: for r from 1 to y(n) do if P(n)[r,1]>1 then k:=k+1; B(k):=P(n)[r]: fi; od: A002865(n):=k; for k from 1 to A002865(n) do s:=0: j:=1: while sA002865(n))], `>`): for k from 1 to A002865(n) do M3[n,k]:=a[k] od: for k from 1 to A002865(n) do T:=T+1: A178866(T):= M3[n,k]: od: od:
mmax:=nmax: n:=1: for m from 1 to mmax do MCM[n,m]:= A178866(n) od: n:=2: r:=1: for i from 2 to nmax do p:=A002865(i): r:=r+1: while p>0 do for m from 1 to mmax do MCM[n,m]:=A178866(n)*binomial(m,r) od: p:=p-1: n:=n+1: od: od: T:=0: for m from 1 to mmax do for n from 1 to numbpart(m) do T:=T+1; A178867(T):= MCM[n,m]; od: od; seq(A178867(n), n=1..T);
# To produce the complete exponential Bell polynomials in standard order, from N. J. A. Sloane, Oct 28 2015
M:=12;
EE:=add(x[i]*t^i/i!,i=1..M);
t1:=exp(EE);
t2:=series(t1,t,M);
Q:=k->sort(expand(k!*coeff(t2,t,k)));
for k from 1 to 8 do lprint(k,Q(k)); od;
# To produce the coefficients of the complete exponential Bell polynomials in standard order:
triangle := proc(numrows) local E, s, Q;
E := add(x[i]*t^i/i!, i=1..numrows);
s := series(exp(E), t, numrows+1);
Q := k -> sort(expand(k!*coeff(s, t, k)));
seq(print(coeffs(Q(k))), k=1..numrows) end:
triangle(6); # Peter Luschny, May 27 2020

G.f.: Exp(Sum_{i >= 1} x_i*t^i/i!) = 1 + Sum_{n >= 1} B_n(x_1, x_2,...)*t^n/n!. [Comtet, p. 134, Eq. [3b]. - N. J. A. Sloane, Oct 28 2015]
For m >= 1, the formulas for the first few matrix columns are:
MCM[1,m] = A178866(1)*C(m,0) = 1*C(m,0);
MCM[2,m] = A178866(2)*C(m,2) = 1*C(m,2);
MCM[3,m] = A178866(3)*C(m,3) = 1*C(m,3);
MCM[4,m] = A178866(4)*C(m,4) = 3*C(m,4) and
MCM[5,m] = A178866(5)*C(m,4) = 1*C(m,4);
MCM[6,m] = A178866(6)*C(m,5) = 10*C(m,5) and
MCM[7,m] = A178866(7)*C(m,5) = 1*C(m,5);
MCM[8,m] = A178866(8)*C(m,6) = 15*C(m,6) and
MCM[9,m] = A178866(9)*C(m,6) = 15*C(m,6) and
MCM[10,m] = A178866(10)*C(m,6) = 10*C(m,6) and
MCM[11,m] = A178866(11)*C(m,6) = 1*C(m,6).

Alternative definition as coefficients of complete Bell polynomials added by N. J. A. Sloane, Oct 28 2015

Various sections and name edited by Petros Hadjicostas, May 28 2020

A006343 Arkons: number of elementary maps with n-1 nodes.

Original entry on oeis.org

1, 0, 1, 1, 4, 10, 34, 112, 398, 1443, 5387, 20482, 79177, 310102, 1228187, 4910413, 19792582, 80343445, 328159601, 1347699906, 5561774999, 23052871229, 95926831442, 400587408251, 1678251696379, 7051768702245, 29710764875014

Offset: 0

N. J. A. Sloane

K. Appel and W. Haken, Every planar map is four colorable. With the collaboration of J. Koch. Contemporary Mathematics, 98. American Mathematical Society, Providence, RI, 1989. xvi+741 pp. ISBN: 0-8218-5103-9.
F. R. Bernhart, Topics in Graph Theory Related to the Five Color Conjecture. Ph.D. Dissertation, Kansas State Univ., 1974.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999), 73-112.
F. R. Bernhart, Fundamental chromatic numbers, Unpublished. (Annotated scanned copy)
F. R. Bernhart and N. J. A. Sloane, Correspondence, 1977.
F. R. Bernhart and N. J. A. Sloane, Emails, April-May 1994.
G. D. Birkhoff and D. C. Lewis, Chromatic polynomials, Trans. Amer. Math. Soc. 60, (1946). 355-451.

Cf. A000108, A000934, A007318.

a006343 0 = 1
a006343 n = sum $ zipWith div
   (zipWith (*) (map (a007318 n) ks)
                (map (\k -> a007318 (2*n - 3*k - 4) (n - 2*k - 2)) ks))
   (map (toInteger . (n - 1 -)) ks)
   where ks = [0 .. (n - 2) `div` 2]
-- Reinhard Zumkeller, Aug 23 2012

A006343:=n->add(binomial(n,k)*binomial(2*n-3*k-4,n-2*k-2)/(n-k-1), k=0..(n-2)/2): (1, seq(A006343(n), n=1..30)); # Wesley Ivan Hurt, Jun 22 2015

a[n_] := Sum[ Binomial[n, k]*Binomial[2*n-3*k-4, n-2*k-2]/(n-k-1), {k, 0, (n-2)/2}]; a[0] = 1; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Dec 14 2012, from formula *)

a(n-1) = Sum (n-k-1)^(-1)*binomial(n, k)*binomial(2*n-3*k-4, n-2*k-2); k = 0..[ (n-2)/2 ], n >= 3.
From Peter Bala, Jun 22 2015: (Start)
O.g.f. A(x) equals 1/x * series reversion ( x/(1 + x^2*C(x)) ), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for A000108.
A(x) is an algebraic function satisfying x^3*A^3(x) - (x - 1)*A^2(x) + (x - 2)*A(x) + 1 = 0. (End)
a(n) ~ sqrt(s*(1 - s + 3*r^2*s^2) / (1 - r + 3*r^3*s)) / (2*sqrt(Pi) * n^(3/2) * r^(n - 1/2)), where r = 0.2229935155751761877673240243525445951244491757706... and s = 1.116796494086474135831052534637944909439048671327... are real roots of the system of equations 1 + (r-2)*s + r^3*s^3 = (r-1)*s^2, r + 2*s + 3*r^3*s^2 = 2 + 2*r*s. - Vaclav Kotesovec, Nov 27 2017
Conjecture: D-finite with recurrence: -(n+3)*(n-1)*a(n) +(11*n^2-2*n-45)*a(n-1) -(37*n+29)*(n-3)*a(n-2) +(29*n^2-125*n+78)*a(n-3) +(61*n-106)*(n-3)*a(n-4) -155*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Feb 20 2020

Erroneously duplicated term 4 removed per Frank Bernhart's report by Max Alekseyev, Feb 11 2010

A011968 Apply (1+Shift) to Bell numbers.

Original entry on oeis.org

1, 2, 3, 7, 20, 67, 255, 1080, 5017, 25287, 137122, 794545, 4892167, 31858034, 218543759, 1573857867, 11863100692, 93345011951, 764941675963, 6514819011216, 57556900440429, 526593974392123, 4981585554604074, 48658721593531669, 490110875149889635

Offset: 0

N. J. A. Sloane

nonn

Number of set partitions of n+2 with at least one singleton and the smallest element in any singleton is exactly n. The maximum number of singletons is therefore 3. Alternatively, number of set partitions of n+2 with at least one singleton and the largest element in any singleton is exactly 3 (or n+2 if n+2 < 3). For example, a(3)=7 counts the following set partitions of [5]: {1245, 3}, {12, 3, 45}, {124, 3, 5}, {15, 24, 3}, {125, 3, 4}, {14, 25, 3}, {12, 3, 4, 5}. - Olivier Gérard, Oct 29 2007

Let V(N)={v(1),v(2),...,v(N)} denote an ordered set of increasing positive integers containing a pair of adjacent elements that differ by at least 2, that is, v(i),v(i+1) with v(i+1)-v(i) > 1. Then for n > 0, a(n) is the number of partitions of V(n+1) into blocks of nonconsecutive integers. - Augustine O. Munagi, Jul 17 2008

			a(3)=7 because the set {1,3,4,5} has 7 different partitions into blocks of nonconsecutive integers: 14/35, 135/4, 1/35/4, 13/4/5, 14/3/5, 15/3/4, 1/3/4/5.

Olivier Gérard and Karol Penson, A budget of set partitions statistics, in preparation, unpublished as of Sep 22 2011

Chai Wah Wu, Table of n, a(n) for n = 0..500 n = 0..200 from Vincenzo Librandi.
Cohn, Martin; Even, Shimon; Menger, Karl, Jr.; Hooper, Philip K.; On the Number of Partitionings of a Set of n Distinct Objects, Amer. Math. Monthly 69 (1962), no. 8, 782--785. MR1531841.
Cohn, Martin; Even, Shimon; Menger, Karl, Jr.; Hooper, Philip K.; On the Number of Partitionings of a Set of n Distinct Objects, Amer. Math. Monthly 69 (1962), no. 8, 782--785. MR1531841. [Annotated scanned copy]
Augustine O. Munagi, Extended set partitions with successions, European J. Combin. 29(5) (2008), 1298--1308.
Jocelyn Quaintance and Harris Kwong, A combinatorial interpretation of the Catalan and Bell number difference tables, Integers, 13 (2013), #A29.

Cf. A000110, A000296, A011969, A011970.
A diagonal of A011971 and A106436. - N. J. A. Sloane, Jul 31 2012
Cf. A000569, A240936, A321750.

with(combinat): seq(`if`(n>0,bell(n)+bell(n-1),1),n=0..21); # Augustine O. Munagi, Jul 17 2008

# requires python 3.2 or higher. Otherwise use def'n of accumulate in python docs.
from itertools import accumulate
A011968_list, blist, b = [1,2], [1], 1
for _ in range(10**2):
    blist = list(accumulate([b]+blist))
    A011968_list.append(b+blist[-1])
    b = blist[-1] # Chai Wah Wu, Sep 02 2014, updated Chai Wah Wu, Sep 20 2014

For n >= 1, a(n+1) = exp(-1)*Sum_{k>=0} ((k+1)/k!)*k^n. - Benoit Cloitre, Mar 09 2008
For n >= 1, a(n) = Bell(n) + Bell(n-1). - Augustine O. Munagi, Jul 17 2008
G.f.: G(0) where G(k) = 1 - 2*x*(k+1)/((2*k+1)*(2*x*k-1) - x*(2*k+1)*(2*k+3)*(2*x*k-1)/(x*(2*k+3) - 2*(k+1)*(2*x*k+x-1)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 19 2012
G.f.: 1 + x*E(0) where E(k) = 1 + 1/(1-x*k-x)/(1-x/(x+1/E(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 20 2013
G.f.: 1 + Sum_{k>=0} ( 1+1/(1-x-x*k) )*x^(k+1)/Product_{i=0..k} (1-x*i). - Sergei N. Gladkovskii, Jan 20 2013
a(n) ~ Bell(n) * (1 + LambertW(n)/n). - Vaclav Kotesovec, Jul 28 2021

A324011 Number of set partitions of {1, ..., n} with no singletons or cyclical adjacencies (successive elements in the same block, where 1 is a successor of n).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 5, 14, 66, 307, 1554, 8415, 48530, 296582, 1913561, 12988776, 92467629, 688528288, 5349409512, 43270425827, 363680219762, 3170394634443, 28619600156344, 267129951788160, 2574517930001445, 25587989366964056, 261961602231869825

Offset: 0

Gus Wiseman, Feb 12 2019

nonn

These set partitions are fixed points under Callan's bijection phi on set partitions.

			The a(4) = 1, a(6) = 5, and a(7) = 14 set partitions:
  {{13}{24}}  {{135}{246}}    {{13}{246}{57}}
              {{13}{25}{46}}  {{13}{257}{46}}
              {{14}{25}{36}}  {{135}{26}{47}}
              {{14}{26}{35}}  {{135}{27}{46}}
              {{15}{24}{36}}  {{136}{24}{57}}
                              {{136}{25}{47}}
                              {{14}{257}{36}}
                              {{14}{26}{357}}
                              {{146}{25}{37}}
                              {{146}{27}{35}}
                              {{15}{246}{37}}
                              {{15}{247}{36}}
                              {{16}{24}{357}}
                              {{16}{247}{35}}

David Callan, On conjugates for set partitions and integer compositions, arXiv:math/0508052 [math.CO], 2005.

Cf. A000110, A000126, A000296 (singletons allowed, or adjacencies allowed), A001610, A124323, A169985, A261139, A324012, A324014, A324015.

Table[Select[sps[Range[n]],And[Count[#,{_}]==0,Total[If[First[#]==1&&Last[#]==n,1,0]+Count[Subtract@@@Partition[#,2,1],-1]&/@#]==0]&]//Length,{n,0,10}]

a(11)-a(26) from Alois P. Heinz, Feb 12 2019

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

A261139 S'_t(n) is the number of set partitions of {1,2,...,t} into exactly n parts such that no part contains both 1 and t or both i and i+1 for some i with 1 <= i < t; triangle S'_t(n), t >= 0, 0 <= n <= t, read by rows.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 2, 1, 0, 0, 0, 5, 5, 1, 0, 0, 1, 10, 20, 9, 1, 0, 0, 0, 21, 70, 56, 14, 1, 0, 0, 1, 42, 231, 294, 126, 20, 1, 0, 0, 0, 85, 735, 1407, 924, 246, 27, 1, 0, 0, 1, 170, 2290, 6363, 6027, 2400, 435, 35, 1

Offset: 0

Mark Wildon, Aug 10 2015

S'A261137%20may%20be%20defined%20by%20B'_t(n)%20=%20Sum">t(n) is the number of sequences of t non-identity top-to-random shuffles of a deck of n cards that move each card at some time, and overall leave the deck invariant. (See link below.) A261137 may be defined by B'_t(n) = Sum{m=0..n} S'_t(m).

			Triangle starts:
  1;
  0, 0;
  0, 0, 1;
  0, 0, 0,  1;
  0, 0, 1,  2,   1;
  0, 0, 0,  5,   5,    1;
  0, 0, 1, 10,  20,    9,   1;
  0, 0, 0, 21,  70,   56,  14,   1;
  0, 0, 1, 42, 231,  294, 126,  20,  1;
  0, 0, 0, 85, 735, 1407, 924, 246, 27,  1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
John R. Britnell and Mark Wildon, Bell numbers, partition moves and the eigenvalues of the random-to-top shuffle in Dynkin Types A, B and D, arXiv:1507.04803 [math.CO], 2015.
D. E. Knuth and O. P. Lossers, Partitions of a circular set, Problem 11151 in Amer. Math. Monthly 114 (3), (2007), p 265, E_4.
Sophie Morier-Genoud, Counting Coxeter's friezes over a finite field via moduli spaces, arXiv:1907.12790 [math.CO], 2019.
Augustine O. Munagi, Two Applications of the Bijection on Fibonacci Set Partitions, Fibonacci Quart. 55 (2017), no. 5, 144-148. See c(n,k) p. 145 giving shifted triangle.

Columns n=3,4 give: A000975, A243869.
Row sums give A000296.
Cf. A261137.
The same as A105794, except for the first two columns.

g:= proc(t, l, h) option remember; `if`(t=0, `if`(l=1, 0, x^h),
       add(`if`(j=l, 0, g(t-1, j, max(h,j))), j=1..h+1))
    end:
S:= t-> (p-> seq(coeff(p, x, i), i=0..t))(g(t, 0$2)):
seq(S(t), t=0..12);  # Alois P. Heinz, Aug 10 2015

StirPrimedGF[n_, x_] := x^n/(1 + x)*Product[1/(1 - j*x), {j, 1, n - 1}]; T[0, 0] = 1; T[, 0] = T[, 1] = 0; T[n_, k_] := SeriesCoefficient[ StirPrimedGF[k, x], {x, 0, n}]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* script completed by Jean-François Alcover, Jan 31 2016 *)

a(n,k)=if(k==0, n==0, sum(j=0, k, binomial(k, j) * (-1)^(k-j) * ((j-1)^n + (-1)^n * (j-1))) / k!);
for(n=0, 10, for(k=0, n, print1( a(n, k), ", "); ); print(); ); \\ Andrew Howroyd, Apr 08 2017

G.f. for column n > 1: x^n/((1+x)*Product_{j=1..n-1} (1-j*x)).
S'_t(n) ~ (n-1)^t/n! as t tends to infinity.
Recurrence: S't(n) = S'{t-1}(n-1) + (n-1)*S'_{t-1}(n) for n >= 3.
S't(n) = (1/n!) * Sum{j=0..n} (-1)^(n-j) * binomial(n, j) * ((j-1)^t + (-1)^t * (j-1)) for t>0. - Andrew Howroyd, Apr 08 2017
Sum_{n=0..t} (n-1)*S'{t-1}(n) + n*S'{t-2}(n) = A000296(t) for t >= 3. - Yuchun Ji, Feb 23 2021
T(m, k) = Sum_{i=k..m} Stirling2(i-1, k-1)*(-1)^(i+m), for k >= 2. (See Peter Bala's original formula at A105794 dated Jul 10 2013.) - Igor Victorovich Statsenko, May 31 2024
T(m, k) = (Sum_{i=0..m} Stirling2(i, k)*binomial(m,i)*(-1)^(m-i))*I(m,k), where I(m,k) = (1-Sum_{i=0..m} Stirling1(k, i))^(m+k) for k >= 0. (See Peter Bala's original formula at A105794 dated Jul 10 2013.) - Igor Victorovich Statsenko, Jun 01 2024

A346738 Expansion of e.g.f.: exp(exp(x) - 3*x - 1).

Original entry on oeis.org

1, -2, 5, -13, 36, -101, 293, -848, 2523, -7365, 22402, -64395, 205285, -541802, 2057617, -3403993, 28685420, 43885023, 824532745, 4878097904, 44263112047, 357891860463, 3169228222338, 28506399763969, 266822555964441, 2573194635922990, 25606751525353741

Offset: 0

Ilya Gutkovskiy, Jul 31 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 0..579

Cf. A000110, A000296, A005494, A126617, A193683, A290219, A346739, A346740.

R:=PowerSeriesRing(Rationals(), 50);
Coefficients(R!(Laplace( Exp(Exp(x)-3*x-1) ))) // G. C. Greubel, Jun 12 2024

nmax = 26; CoefficientList[Series[Exp[Exp[x] - 3 x - 1], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] (-3)^(n - k) BellB[k], {k, 0, n}], {n, 0, 26}]
a[0] = 1; a[n_] := a[n] = -3 a[n - 1] + Sum[Binomial[n - 1, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 26}]

[factorial(n)*( exp(exp(x)-3*x-1) ).series(x, n+1).list()[n] for n in (0..30)] # G. C. Greubel, Jun 12 2024

G.f. A(x) satisfies: A(x) = (1 - x + x * A(x/(1 - x))) / ((1 - x) * (1 + 3*x)).
a(n) = Sum_{k=0..n} binomial(n,k) * (-3)^(n-k) * Bell(k).
a(n) = exp(-1) * Sum_{k>=0} (k - 3)^n / k!.
a(0) = 1; a(n) = -3 * a(n-1) + Sum_{k=0..n-1} binomial(n-1,k) * a(k).

A290219 a(n) = n! * [x^n] exp(exp(x) - n*x - 1).

Original entry on oeis.org

1, 0, 2, -13, 127, -1573, 23711, -421356, 8626668, -199971255, 5177291275, -148078588667, 4636966634653, -157786054331852, 5797411243015250, -228749440644895405, 9646951350227609155, -433035586385769361001, 20614401475233006857035, -1037331650810058231498688

Offset: 0

Ilya Gutkovskiy, Oct 06 2017

sign

The n-th term of the n-th inverse binomial transform of A000110.

Seiichi Manyama, Table of n, a(n) for n = 0..386
N. J. A. Sloane, Transforms

Cf. A000110, A000296, A126617, A134980, A346738, A346739, A346740.

Main diagonal of A361781.

R:=PowerSeriesRing(Rationals(), 50);
A290219:= func< n | Coefficient(R!(Laplace( Exp(Exp(x)-n*x-1) )), n) >;
[A290219(n): n in [0..30]]; // G. C. Greubel, Jun 12 2024

b:= proc(n, k) option remember; `if`(n=0, 1,
      k*b(n-1, k)+ b(n-1, k+1))
    end:
a:= n-> b(n, -n):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 04 2021

Table[n! SeriesCoefficient[Exp[Exp[x] - n x - 1], {x, 0, n}], {n, 0, 19}]
Join[{1}, Table[Sum[(-n)^(n - k) Binomial[n, k] BellB[k] , {k, 0, n}], {n, 1, 19}]]

[factorial(n)*( exp(exp(x) -n*x -1) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Jun 12 2024

a(n) ~ (-1)^n * exp(exp(-1) - 1) * n^n. - Vaclav Kotesovec, Aug 04 2021

A293024 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(exp(x) - Sum_{i=0..k} x^i/i!).

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 0, 1, 5, 1, 0, 0, 1, 15, 1, 0, 0, 1, 4, 52, 1, 0, 0, 0, 1, 11, 203, 1, 0, 0, 0, 1, 1, 41, 877, 1, 0, 0, 0, 0, 1, 11, 162, 4140, 1, 0, 0, 0, 0, 1, 1, 36, 715, 21147, 1, 0, 0, 0, 0, 0, 1, 1, 92, 3425, 115975, 1, 0, 0, 0, 0, 0, 1, 1, 36, 491, 17722, 678570

Offset: 0

Seiichi Manyama, Sep 28 2017

A(n,k) is the number of set partitions of [n] into blocks of size > k.

			Square array begins:
    1,   1,  1, 1, 1, 1, 1, 1, ...
    1,   0,  0, 0, 0, 0, 0, 0, ...
    2,   1,  0, 0, 0, 0, 0, 0, ...
    5,   1,  1, 0, 0, 0, 0, 0, ...
   15,   4,  1, 1, 0, 0, 0, 0, ...
   52,  11,  1, 1, 1, 0, 0, 0, ...
  203,  41, 11, 1, 1, 1, 0, 0, ...
  877, 162, 36, 1, 1, 1, 1, 0, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
E. A. Enneking and J. C. Ahuja, Generalized Bell numbers, Fib. Quart., 14 (1976), 67-73.

Columns k=0..5 give A000110, A000296, A006505, A057837, A057814, A293025.
Rows n=0..1 give A000012, A000007.
Main diagonal gives A000007.
Cf. A182931, A282988 (as triangle), A293051, A293053.

A:= proc(n, k) option remember; `if`(n=0, 1, add(
      A(n-j, k)*binomial(n-1, j-1), j=1+k..n))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..14);  # Alois P. Heinz, Sep 28 2017

A[0, _] = 1;
A[n_, k_] /; 0 <= k <= n := A[n, k] = Sum[A[n-j, k] Binomial[n-1, j-1], {j, k+1, n}];
A[, ] = 0;
Table[A[n-k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 06 2019 *)

def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(k, n)
  ary = [1]
  (1..n).each{|i| ary << (k..i - 1).inject(0){|s, j| s + ncr(i - 1, j) * ary[-1 - j]}}
  ary
end
def A293024(n)
  a = []
  (0..n).each{|i| a << A(i, n - i)}
  ary = []
  (0..n).each{|i|
    (0..i).each{|j|
      ary << a[i - j][j]
    }
  }
  ary
end
p A293024(20)

E.g.f. of column k: Product_{i>k} exp(x^i/i!).

A(0,k) = 1, A(1,k) = A(2,k) = ... = A(k,k) = 0 and A(n,k) = Sum_{i=k..n-1} binomial(n-1,i)*A(n-1-i,k) for n > k.

cp's OEIS Frontend

A178867 Irregular triangle read by rows: multinomial coefficients, version 3; alternatively, row n gives coefficients of the n-th complete exponential Bell polynomial B_n(x_1, x_2, ...) with monomials sorted into some unknown order.

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A011968 Apply (1+Shift) to Bell numbers.

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A324011 Number of set partitions of {1, ..., n} with no singletons or cyclical adjacencies (successive elements in the same block, where 1 is a successor of n).

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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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