A057814 -id:A057814 - cp's OEIS frontend

A000296 Set partitions without singletons: number of partitions of an n-set into blocks of size > 1. Also number of cyclically spaced (or feasible) partitions.

1, 0, 1, 1, 4, 11, 41, 162, 715, 3425, 17722, 98253, 580317, 3633280, 24011157, 166888165, 1216070380, 9264071767, 73600798037, 608476008122, 5224266196935, 46499892038437, 428369924118314, 4078345814329009, 40073660040755337, 405885209254049952, 4232705122975949401

Offset: 0

N. J. A. Sloane

a(n+2) = p(n+1) where p(x) is the unique degree-n polynomial such that p(k) = A000110(k) for k = 0, 1, ..., n. - Michael Somos, Oct 07 2003
Number of complete rhyming schemes.
Whereas the Bell number B(n) (A000110(n)) is the number of terms in the polynomial that expresses the n-th moment of a probability distribution as a function of the first n cumulants, these numbers give the number of terms in the corresponding expansion of the central moment as a function of the first n cumulants. - Michael Hardy (hardy(AT)math.umn.edu), Jan 26 2005
a(n) is the number of permutations on [n] for which the left-to-right maxima coincide with the descents (entries followed by a smaller number). For example, a(4) counts 2143, 3142, 3241, 4123. - David Callan, Jul 20 2005
From Gus Wiseman, Feb 10 2019: (Start)
Also the number of stable partitions of an n-cycle, where a stable partition of a graph is a set partition of the vertex set such that no edge has both ends in the same block. A bijective proof is given in David Callan's article. For example, the a(5) = 11 stable partitions are:
  {{1},{2},{3},{4},{5}}
  {{1},{2},{3,5},{4}}
  {{1},{2,4},{3},{5}}
  {{1},{2,5},{3},{4}}
  {{1,3},{2},{4},{5}}
  {{1,4},{2},{3},{5}}
  {{1},{2,4},{3,5}}
  {{1,3},{2,4},{5}}
  {{1,3},{2,5},{4}}
  {{1,4},{2},{3,5}}
  {{1,4},{2,5},{3}}
(End)
Also number of partitions of {1, 2, ..., n-1} with singletons. E.g., a(4) = 4: {1|2|3, 12|3, 13|2, 1|23}. Also number of cyclical adjacencies partitions of {1, 2, ..., n-1}. E.g., a(4) = 4: {12|3, 13|2, 1|23, 123}. The two partitions can be mapped by a Kreweras bijection. - Yuchun Ji, Feb 22 2021
Also the k-th central moment of a Poisson random variable with mean 1. a(n) = E[(X-1)^n, X~Poisson(1)]. - Thomas Dybdahl Ahle, Dec 14 2022

			a(4) = card({{{1, 2}, {3, 4}}, {{1, 4}, {2, 3}}, {{1, 3}, {2, 4}}, {{1, 2, 3, 4}}}) = 4.

Martin Gardner in Sci. Amer. May 1977.
D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.5 (p. 436).
G. Pólya and G. Szegő, Problems and Theorems in Analysis, Springer-Verlag, NY, 2 vols., 1972, Vol. 1, p. 228.
J. Riordan, A budget of rhyme scheme counts, pp. 455-465 of Second International Conference on Combinatorial Mathematics, New York, April 4-7, 1978. Edited by Allan Gewirtz and Louis V. Quintas. Annals New York Academy of Sciences, 319, 1979.
J. Shallit, A Triangle for the Bell numbers, in V. E. Hoggatt, Jr. and M. Bicknell-Johnson, A Collection of Manuscripts Related to the Fibonacci Sequence, 1980, pp. 69-71.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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..575 (first 101 terms from T. D. Noe)
Yasemin Alp and E. Gokcen Kocer, Exponential Almost-Riordan Arrays, Results Math. (2024) Vol. 79, 173.
Joerg Arndt and N. J. A. Sloane, Counting Words that are in "Standard Order".
E. Bach, Random bisection and evolutionary walks, J. Applied Probability, v. 38, pp. 582-596, 2001.
M. Bauer and O. Golinelli, Random incidence matrices: Moments of the spectral density, arXiv:cond-mat/0007127 [cond-mat.stat-mech], 2000-2001. See Sect. 3.2; J. Stat. Phys. 103, 301-307 (2001).
H. D. Becker, Solution to problem E 461, American Math Monthly 48 (1941), 701-702.
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 & N. J. A. Sloane, Correspondence, 1977.
J. R. Britnell and M. Wildon, Bell numbers, partition moves and the eigenvalues of the random-to-top shuffle in types A, B and D, arXiv 1507.04803 [math.CO], 2015.
David Callan, On conjugates for set partitions and integer compositions [math.CO].
Pascal Caron, Jean-Gabriel Luque, Ludovic Mignot, and Bruno Patrou, State complexity of catenation combined with a boolean operation: a unified approach, arXiv preprint arXiv:1505.03474 [cs.FL], 2015.
Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. II. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 3.
Éva Czabarka, Péter L. Erdős, Virginia Johnson, Anne Kupczok and László A. Székely, Asymptotically normal distribution of some tree families relevant for phylogenetics, and of partitions without singletons, arXiv preprint arXiv:1108.6015 [math.CO], 2011.
Gesualdo Delfino and Jacopo Viti, Potts q-color field theory and scaling random cluster model, arXiv preprint arXiv:1104.4323 [hep-th], 2011.
E. A. Enneking and J. C. Ahuja, Generalized Bell numbers, Fib. Quart., 14 (1976), 67-73.
Steven R. Finch, Moments of sums, April 23, 2004. [Cached copy, with permission of the author]
Robert C. Griffiths, P. A. Jenkins, and S. Lessard, A coalescent dual process for a Wright-Fisher diffusion with recombination and its application to haplotype partitioning, arXiv preprint arXiv:1604.04145 [q-bio.PE], 2016.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 16.
V. P. Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. Southern Calif., 2012.
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
Peter Luschny, Set partitions.
Gregorio Malajovich, Complexity of sparse polynomial solving: homotopy on toric varieties and the condition metric, arXiv preprint arXiv:1606.03410 [math.NA], 2016.
Toufik Mansour and Mark Shattuck, A recurrence related to the Bell numbers, INTEGERS 11 (2011), #A67.
T. Mansour and A. O. Munagi, Set partitions with circular successions, European Journal of Combinatorics, 42 (2014), 207-216.
I. Mezo, Periodicity of the last digits of some combinatorial sequences, J. Integer Seq. 17, Article 14.1.1, 2014.
Scott Morrison, Noah Snyder, and Dylan P. Thurston, Towards the quantum exceptional series, arXiv:2402.03637 [math.QA], 2024. See p. 39.
E. Norton, Symplectic Reflection Algebras in Positive Characteristic as Ore Extensions, arXiv preprint arXiv:1302.5411 [math.RA], 2013.
Rosa Orellana, Nancy Wallace, and Mike Zabrocki, Quasipartition and planar quasipartition algebras, Sém. Lotharingien Comb., Proc. 36th Conf. Formal Power Series Alg. Comb. (2024) Vol. 91B, Art. No. 50. See p. 7.
Aleksandar Petojević, Marjana Gorjanac Ranitović, and Milinko Mandić, New equivalents for Kurepa's hypothesis for left factorial, Univ. Novi Sad (2023).
Tilman Piesk, Table showing non-singleton partitions for n = 1..6.
R. A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
Jocelyn Quaintance and Harris Kwong, A combinatorial interpretation of the Catalan and Bell number difference tables, Integers, 13 (2013), #A29.
D. Reidenbach and J. C. Schneider, Morphically Primitive Words. In Pierre Arnoux, Nicolas Bedaride and Julien Cassaigne, editors, Proc. 6th International Conference on Words, WORDS 2007, pages 262-272. 2007. [Different from the paper with the same name, referenced below.]
Daniel Reidenbach and Johannes C. Schneider, Morphically primitive words, (2009). See Table 1.
Daniel Reidenbach and Johannes C. Schneider, Morphically primitive words, Theoretical Computer Science, (2009), 140 (21-23), pp. 2148-2161.
J. Riordan, Cached copy of paper.
Jeffrey Shallit, A Triangle for the Bell numbers, in V. E. Hoggatt, Jr. and M. Bicknell-Johnson, A Collection of Manuscripts Related to the Fibonacci Sequence, 1980, pp. 69-71.
Index entries for related partition-counting sequences

Cf. A000110, A005493, A006505, A057814, A057837.
A diagonal of triangle in A106436.
Row sums of the triangle of associated Stirling numbers of second kind A008299. - Philippe Deléham, Feb 10 2005
Row sums of the triangle of basic multinomial coefficients A178866. - Johannes W. Meijer, Jun 21 2010
Row sums of A105794. - Peter Bala, Jan 14 2015
Row sums of A261139, main diagonal of A261137.
Column k=0 of A216963.
Column k=0 of A124323.
Cf. A000126, A001610, A066982, A169985, A240936.

[1,0] cat [ n le 1 select 1 else Bell(n)-Self(n-1) : n in [1..40]]; // Vincenzo Librandi, Jun 22 2015

spec := [ B, {B=Set(Set(Z,card>1))}, labeled ]; [seq(combstruct[count](spec, size=n), n=0..30)];
with(combinat): A000296 :=n->(-1)^n + add((-1)^(j-1)*bell(n-j),j=1..n): seq(A000295(n),n=0..30); # Emeric Deutsch, Oct 29 2006
f:=exp(exp(x)-1-x): fser:=series(f, x=0, 31): 1, seq(n!*coeff(fser, x^n), n=1..23); # Zerinvary Lajos, Nov 22 2006
G:={P=Set(Set(Atom,card>=2))}: combstruct[gfsolve](G,unlabeled,x): seq(combstruct[count]([P,G,labeled], size=i), i=0..23); # Zerinvary Lajos, Dec 16 2007
# [a(0),a(1),..,a(n)]
A000296_list := proc(n)
local A, R, i, k;
if n = 0 then return 1 fi;
A := array(0..n-1);
A[0] := 1; R := 1;
for i from 0 to n-2 do
   A[i+1] := A[0] - A[i];
   A[i] := A[0];
   for k from i by -1 to 1 do
      A[k-1] := A[k-1] + A[k] od;
   R := R,A[i+1];
od;
R,A[0]-A[i] end:
A000296_list(100);  # Peter Luschny, Apr 09 2011

nn = 25; Range[0, nn]! CoefficientList[Series[Exp[Exp[x] - 1 - x], {x, 0, nn}], x]
(* Second program: *)
a[n_] := a[n] = If[n==0, 1, Sum[Binomial[n-1, i]*a[n-i-1], {i, 1, n-1}]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 06 2016, after Vladimir Kruchinin *)
spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:= Join@@Function[s,Prepend[#,s]&/@spsu[ Select[foo,Complement[#, Complement[set,s]]=={}&], Complement[set,s]]]/@Cases[foo,{i,_}];
Table[Length[spsu[Select[Subsets[Range[n]],Select[Partition[Range[n],2,1,1], Function[ed,Complement[ed,#]=={}]]=={}&],Range[n]]],{n,8}] (* Gus Wiseman, Feb 10 2019 *)
s = 1; Join[{1}, Table[s = BellB[n] - s, {n, 0, 25}]] (* Vaclav Kotesovec, Jun 20 2022 *)

a(n):=if n=0 then 1 else sum(binomial(n-1,i)*a(n-i-1),i,1,n-1); /* Vladimir Kruchinin, Feb 22 2015 */

a(n) = if(n<2, n==0, subst( polinterpolate( Vec( serlaplace( exp( exp( x+O(x^n)/x )-1 ) ) ) ), x, n) )

from itertools import accumulate, islice
def A000296_gen():
    yield from (1,0)
    blist, a, b = (1,), 0, 1
    while True:
        blist = list(accumulate(blist, initial = (b:=blist[-1])))
        yield (a := b-a)
A000296_list = list(islice(A000296_gen(),20)) # Chai Wah Wu, Jun 22 2022

E.g.f.: exp(exp(x) - 1 - x).
B(n) = a(n) + a(n+1), where B = A000110 = Bell numbers [Becker].
Inverse binomial transform of Bell numbers (A000110).
a(n)= Sum_{k>=-1} (k^n/(k+1)!)/exp(1). - Vladeta Jovovic and Karol A. Penson, Feb 02 2003
a(n) = Sum_{k=0..n} ((-1)^(n-k))*binomial(n, k)*Bell(k) = (-1)^n + Bell(n) - A087650(n), with Bell(n) = A000110(n). - Wolfdieter Lang, Dec 01 2003
O.g.f.: A(x) = 1/(1-0*x-1*x^2/(1-1*x-2*x^2/(1-2*x-3*x^2/(1-... -(n-1)*x-n*x^2/(1- ...))))) (continued fraction). - Paul D. Hanna, Jan 17 2006
a(n) = Sum_{k = 0..n} {(-1)^(n-k) * Sum_{j = 0..k}((-1)^j * binomial(k,j) * (1-j)^n)/ k!} = sum over row n of A105794. - Tom Copeland, Jun 05 2006
a(n) = (-1)^n + Sum_{j=1..n} (-1)^(j-1)*B(n-j), where B(q) are the Bell numbers (A000110). - Emeric Deutsch, Oct 29 2006
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 >= 2, a(n) = (-1)^(n)charpoly(A,1). - Milan Janjic, Jul 08 2010
From Sergei N. Gladkovskii, Sep 20 2012, Oct 11 2012, Dec 19 2012, Jan 15 2013, May 13 2013, Jul 20 2013, Oct 19 2013, Jan 25 2014: (Start)
Continued fractions:
G.f.: (2/E(0) - 1)/x where E(k) = 1 + 1/(1 + 2*x/(1 - 2*(k+1)*x/E(k+1))).
G.f.: 1/U(0) where U(k) = 1 - x*k - x^2*(k+1)/U(k+1).
G.f.: G(0)/(1+2*x) where G(k) = 1 - 2*x*(k+1)/((2*k+1)*(2*x*k-x-1) - x*(2*k+1)*(2*k+3)*(2*x*k-x-1)/(x*(2*k+3) - 2*(k+1)*(2*x*k-1)/G(k+1))).
G.f.: (G(0) - 1)/(x-1) where G(k) = 1 - 1/(1+x-k*x)/(1-x/(x-1/G(k+1))).
G.f.: 1 + x^2/(1+x)/Q(0) where Q(k) = 1-x-x/(1-x*(2*k+1)/(1-x-x/(1-x*(2*k+2)/Q(k+1)))).
G.f.: 1/(x*Q(0)) where Q(k) = 1 + 1/(x + x^2/(1 - x - (k+1)/Q(k+1))).
G.f.: -(1+(2*x+1)/G(0))/x where G(k) = x*k - x - 1 - (k+1)*x^2/G(k+1).
G.f.: T(0) where T(k) = 1 - x^2*(k+1)/( x^2*(k+1) - (1-x*k)*(1-x-x*k)/T(k+1)).
G.f.: (1 + x * Sum_{k>=0} (x^k / Product_{p=0..k}(1 - p*x))) / (1 + x). (End)
a(n) = Sum_{i=1..n-1} binomial(n-1,i)*a(n-i-1), a(0)=1. - Vladimir Kruchinin, Feb 22 2015
G.f. A(x) satisfies: A(x) = (1/(1 + x)) * (1 + x * A(x/(1 - x)) / (1 - x)). - Ilya Gutkovskiy, May 21 2021
a(n) ~ exp(n/LambertW(n) - n - 1) * n^(n-1) / (sqrt(1 + LambertW(n)) * LambertW(n)^(n-1)). - Vaclav Kotesovec, Jun 28 2022

More terms, new description from Christian G. Bower, Nov 15 1999

a(23) corrected by Sean A. Irvine, Jun 22 2015

A006505 Number of partitions of an n-set into boxes of size >2.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 11, 36, 92, 491, 2557, 11353, 60105, 362506, 2169246, 13580815, 91927435, 650078097, 4762023647, 36508923530, 292117087090, 2424048335917, 20847410586719, 185754044235873, 1711253808769653, 16272637428430152

Offset: 0

N. J. A. Sloane

J. Riordan, A budget of rhyme scheme counts, pp. 455 - 465 of Second International Conference on Combinatorial Mathematics, New York, April 4-7, 1978. Edited by Allan Gewirtz and Louis V. Quintas. Annals New York Academy of Sciences, 319, 1979.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..579 (terms 0..250 from Alois P. Heinz)
E. A. Enneking and J. C. Ahuja, Generalized Bell numbers, Fib. Quart., 14 (1976), 67-73.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 102
Vladimir Victorovich Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
I. Mezo, Periodicity of the last digits of some combinatorial sequences, arXiv preprint arXiv:1308.1637 [math.CO], 2013.
J. Riordan, Cached copy of paper [With permission]

Column k=2 of A293024.
Cf. A000110, A000296, A057814, A057837, A059022.
Cf. A293038.

Copy ZL := [ B,{B=Set(Set(Z, card>=3))}, labeled ]: [seq(combstruct[count](ZL, size=n), n=0..25)]; # Zerinvary Lajos, Mar 13 2007
G:={P=Set(Set(Atom,card>=3))}:combstruct[gfsolve](G,unlabeled,x):seq(combstruct[count]([P,G,labeled],size=i),i=0..25); # Zerinvary Lajos, Dec 16 2007
g:=proc(n) option remember; if n=0 then RETURN(1); fi; if n<=2 then RETURN(0); fi; if n<=5 then RETURN(x); fi; expand(x*add(binomial(n-1,i)*g(i),i=0..n-3)); end; [seq(subs(x=1,g(n)),n=0..60)]; # N. J. A. Sloane, Jul 20 2011

a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[ Exp @ x - 1 - x - x^2 / 2], {x, 0, n}]] (* Michael Somos, Jul 20 2011 *)
a[0] = 1; a[n_] := n!*Sum[Sum[k!*(-1)^(m-k)*Binomial[m, k]*Sum[StirlingS2[i+k, k]* Binomial[m-k, n-m-i]*2^(-n+m+i)/(i+k)!, {i, 0, n-m}], {k, 0, m}]/m!, {m, 1, n}]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 03 2015, after Vladimir Kruchinin *)
Table[Sum[(-1)^j * Binomial[n, j] * BellB[n-j] * 2^((j-1)/2) * HypergeometricU[(1 - j)/2, 3/2, 1/2], {j, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Feb 09 2020 *)

{a(n) = if( n<0, 0, n! * polcoeff( exp( exp( x + x * O(x^n)) - 1 - x - x^2 / 2), n))} /* Michael Somos, Jul 20 2011 */

E.g.f.: exp ( exp x - 1 - x - (1/2)*x^2 ).
a(n) = Sum_{k=1..[n/3]} A059022(n,k), n>=3. - R. J. Mathar, Nov 08 2008
a(n) = n! * sum(m=1..n, sum(k=0..m, k!*(-1)^(m-k) *binomial(m,k) *sum(i=0..n-m, stirling2(i+k,k) *binomial(m-k,n-m-i) *2^(-n+m+i)/ (i+k)!))/m!); a(0)=1. - Vladimir Kruchinin, Feb 01 2011
Define polynomials g_n by g_0=1, g_1=g_2=0, g_3=g_4=g_5=x; g(n) = x*Sum_{i=0..n-3} binomial(n-1,i)*g_i; then a(n) = g_n(1). [Riordan]
a(0) = 1; a(n) = Sum_{k=0..n-3} binomial(n-1,k+2) * a(n-k-3). - Seiichi Manyama, Sep 22 2023

More terms from Christian G. Bower, Nov 09 2000

Edited by N. J. A. Sloane, Jul 20 2011

A057837 Number of partitions of a set of n elements where the partitions are of size > 3.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 1, 1, 36, 127, 337, 793, 7525, 48764, 238954, 997790, 6401435, 49107697, 345482807, 2150694855, 14656830110, 116678887407, 978172378669, 7886661080873, 63905475745765, 553437891603452, 5122279358273976, 48331088541366296, 458771027309344261

Offset: 0

Steven C. Fairgrieve (fsteven(AT)math.wvu.edu), Nov 06 2000

Seiichi Manyama, Table of n, a(n) for n = 0..583 (terms 0..250 from Alois P. Heinz)
E. A. Enneking and J. C. Ahuja, Generalized Bell numbers, Fib. Quart., 14 (1976), 67-73.
I. Mezo, Periodicity of the last digits of some combinatorial sequences, arXiv preprint arXiv:1308.1637 [math.CO], 2013 and J. Int. Seq. 17 (2014) #14.1.1.

Column k=3 of A293024.
Row sums of A059023.
Cf. A000110, A000296, A006505, A057814.
Cf. A293039.

G:={P=Set(Set(Atom,card>=4))}:combstruct[gfsolve](G,unlabeled,x):seq(combstruct[count]([P,G,labeled],size=i),i=0..26); # Zerinvary Lajos, Dec 16 2007

With[{nn=30},CoefficientList[Series[Exp[Exp[x]-1-x-x^2/2-x^3/6],{x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, Jun 28 2012 *)

E.g.f.: exp(exp(x)-1-x-x^2/2-x^3/6).

a(0) = 1; a(n) = Sum_{k=4..n} binomial(n-1,k-1) * a(n-k). - Ilya Gutkovskiy, Feb 09 2020

Corrected and extended by Christian G. Bower and James Sellers, Nov 09 2000

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.

A059024 Triangle of Stirling numbers of order 5.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 126, 1, 462, 1, 1254, 1, 3003, 1, 6721, 1, 14443, 126126, 1, 30251, 1009008, 1, 62322, 5309304, 1, 127024, 23075052, 1, 257108, 89791416, 1, 518092, 325355316, 488864376, 1, 1041029, 1122632043, 6844101264, 1, 2088043

Offset: 5

Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 14 2000

The number of partitions of the set N, |N|=n, into k blocks, all of cardinality greater than or equal to 5. This is the 5-associated Stirling number of the second kind.
This is entered as a triangular array. The entries S_5(n,k) are zero for 5k>n, so these values are omitted. Initial entry in sequence is S_5(5,1).
Rows are of lengths 1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,...

			There are 126 ways of partitioning a set N of cardinality 10 into 2 blocks each of cardinality at least 5, so S_5(10,2) = 126.
Triangle begins:
1;
1;
1;
1;
1;
1,    126;
1,    462;
1,   1254;
1,   3003;
1,   6721;
1,  14443,    126126;
1,  30251,   1009008;
1,  62322,   5309304;
1, 127024,  23075052;
1, 257108,  89791416;
1, 518092, 325355316, 488864376;
...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 222.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 76.

Alois P. Heinz, Rows n = 5..320, flattened
A. E. Fekete, Apropos two notes on notation, Amer. Math. Monthly, 101 (1994), 771-778.

Row sums give A057814.

Cf. A008299, A059022, A059023, A059025.

T:= proc(n,k) option remember; `if`(k<1 or k>n/5, 0,
      `if`(k=1, 1, k*T(n-1, k)+binomial(n-1, 4)*T(n-5, k-1)))
    end:
seq(seq(T(n, k), k=1..n/5), n=5..25);  # Alois P. Heinz, Aug 18 2017

S5[n_ /; 5 <= n <= 9, 1] = 1; S5[n_, k_] /; 1 <= k <= Floor[n/5] := S5[n, k] = k*S5[n-1, k] + Binomial[n-1, 4]*S5[n-5, k-1]; S5[, ] = 0; Flatten[ Table[ S5[n, k], {n, 5, 25}, {k, 1, Floor[n/5]}]] (* Jean-François Alcover, Feb 21 2012 *)

S_r(n+1, k) = k*S_r(n, k) + binomial(n, r-1)*S_r(n-r+1, k-1); for this sequence, r=5.
G.f.: Sum_{n>=0, k>=0} S_r(n,k)*u^k*t^n/n! = exp(u(e^t-sum(t^i/i!, i=0..r-1))).
T(n,k) = Sum_{j=0..min(n/4,k)} (-1)^j*n!/(24^j*j!*(n-4j)!)*S_4(n-4j,k-j), where S_4 are the 4-associated Stirling numbers of the second kind A059023. - Fabián Pereyra, Feb 21 2022

A293040 E.g.f.: exp(1 + x + x^2/2! + x^3/3! + x^4/4! - exp(x)).

Original entry on oeis.org

1, 0, 0, 0, 0, -1, -1, -1, -1, -1, 125, 461, 1253, 3002, 6720, -111684, -978758, -5246983, -22948029, -89534309, 164027151, 5722510249, 55413784239, 393256686307, 2377996545081, 7807749195198, -46231762188586, -1125536160278906, -12849721017510166

Offset: 0

Seiichi Manyama, Sep 28 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..593

Column k=4 of A293051.
Cf. A000587 (k=0), A293037 (k=1), A293038 (k=2), A293039 (k=3), this sequence (k=4).
Cf. A057814.

seq(factorial(n)*coeftayl(exp(1+x+x^2/2!+x^3/3!+x^4/4!-exp(x)), x = 0, n),n=0..50); # Muniru A Asiru, Oct 06 2017

my(x='x+O('x^66)); Vec(serlaplace(exp(-exp(x)+1+x+x^2/2+x^3/6+x^4/24)))

a(0) = 1; a(n) = -Sum_{k=5..n} binomial(n-1,k-1) * a(n-k). - Ilya Gutkovskiy, Nov 20 2020

A339027 E.g.f.: exp(2 * (exp(x) - 1 - x - x^2 / 2 - x^3 / 6 - x^4 / 24)).

Original entry on oeis.org

1, 0, 0, 0, 0, 2, 2, 2, 2, 2, 506, 1850, 5018, 12014, 26886, 1066782, 8193070, 42723722, 185108514, 719359762, 10426744914, 118490840686, 976376930502, 6583701431086, 38977418758494, 377188932759354, 4671829781287922, 51479602726372402, 483303800325691922

Offset: 0

Ilya Gutkovskiy, Nov 20 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..576

Cf. A001861, A057814, A194689, A339014, A339017.

nmax = 28; CoefficientList[Series[Exp[2 (Exp[x] - 1 - x - x^2/2 - x^3/6 - x^4/24)], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 2 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 5, n}]; Table[a[n], {n, 0, 28}]

my(x='x+O('x^30)); Vec(serlaplace(exp(2 * (exp(x) - 1 - x - x^2/2 - x^3/6 - x^4/24)))) \\ Michel Marcus, Nov 20 2020

a(0) = 1; a(n) = 2 * Sum_{k=5..n} binomial(n-1,k-1) * a(n-k).

a(n) = Sum_{k=0..n} binomial(n,k) * A057814(k) * A057814(n-k).

A365896 Expansion of e.g.f. exp( Sum_{k>=0} x^(2k+5) / (2k+5)! ).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 126, 1, 792, 1, 3718, 126127, 15808, 2450449, 64702, 31593277, 489125360, 343746937, 21511079558, 3435303323, 588969369056, 5227461790937, 13006203613950, 434431246754441, 255046847579760, 21684989820772201, 128034458842091862

Offset: 0

Seiichi Manyama, Sep 22 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..599

Cf. A057814, A365897, A365898.

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=0, N\2, x^(2*k+5)/(2*k+5)!))))

a(0) = 1; a(n) = Sum_{k=0..floor((n-5)/2)} binomial(n-1,2*k+4) * a(n-2*k-5).

E.g.f.: exp( -x - x^3/6 + sinh(x) ).

A365897 Expansion of e.g.f. exp( Sum_{k>=0} x^(3k+5) / (3k+5)! ).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 126, 1, 0, 1287, 1, 126126, 10803, 1, 5513508, 87210, 488864377, 175388031, 698820, 61841343565, 5037240879, 5194678451481, 5281277511511, 139251621015, 1519441856106345, 387880753064806, 123382468421090541

Offset: 0

Seiichi Manyama, Sep 22 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..605

Cf. A057814, A365896, A365898.

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=0, N\3, x^(3*k+5)/(3*k+5)!))))

a(0) = 1; a(n) = Sum_{k=0..floor((n-5)/3)} binomial(n-1,3*k+4) * a(n-3*k-5).

A365898 Expansion of e.g.f. exp( Sum_{k>=0} x^(4k+5) / (4k+5)! ).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 126, 0, 0, 1, 2002, 126126, 0, 1, 32878, 11639628, 488864376, 1, 523754, 962159506, 164910249504, 5194672859377, 8390630, 79198593760, 44919303188760, 4895979169961881, 123378675217248882, 6434084214390, 11762691848427520

Offset: 0

Seiichi Manyama, Sep 22 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..609

Cf. A057814, A365896, A365897.

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=0, N\4, x^(4*k+5)/(4*k+5)!))))

a(0) = 1; a(n) = Sum_{k=0..floor((n-5)/4)} binomial(n-1,4*k+4) * a(n-4*k-5).

E.g.f.: exp( -x + (sinh(x) + sin(x))/2 ).

cp's OEIS Frontend

A000296 Set partitions without singletons: number of partitions of an n-set into blocks of size > 1. Also number of cyclically spaced (or feasible) partitions.

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A006505 Number of partitions of an n-set into boxes of size >2.

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A057837 Number of partitions of a set of n elements where the partitions are of size > 3.

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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!).

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A059024 Triangle of Stirling numbers of order 5.

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A293040 E.g.f.: exp(1 + x + x^2/2! + x^3/3! + x^4/4! - exp(x)).

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A339027 E.g.f.: exp(2 * (exp(x) - 1 - x - x^2 / 2 - x^3 / 6 - x^4 / 24)).

A365896 Expansion of e.g.f. exp( Sum_{k>=0} x^(2k+5) / (2k+5)! ).

A365897 Expansion of e.g.f. exp( Sum_{k>=0} x^(3k+5) / (3k+5)! ).

A365898 Expansion of e.g.f. exp( Sum_{k>=0} x^(4k+5) / (4k+5)! ).