A052849 -id:A052849 - cp's OEIS frontend

A000629 Number of necklaces of partitions of n+1 labeled beads.

1, 2, 6, 26, 150, 1082, 9366, 94586, 1091670, 14174522, 204495126, 3245265146, 56183135190, 1053716696762, 21282685940886, 460566381955706, 10631309363962710, 260741534058271802, 6771069326513690646, 185603174638656822266, 5355375592488768406230

Offset: 0

N. J. A. Sloane, Don Knuth, Nick Singer (nsinger(AT)eos.hitc.com)

Also the number of logically distinct strings of first order quantifiers in which n variables occur (C. S. Peirce, c. 1903). - Stephen Pollard (spollard(AT)truman.edu), Jun 07 2002
Stirling transform of A052849(n) = [2, 4, 12, 48, 240, ...] is a(n) = [2, 6, 26, 150, 1082, ...]. - Michael Somos, Mar 04 2004
Stirling transform of A000142(n-1) = [1, 1, 2, 6, 24, ...] is a(n-1) = [1, 2, 6, 26, ...]. - Michael Somos, Mar 04 2004
Stirling transform of (-1)^n * A024167(n-1) = [0, 1, -1, 5, -14, 94, ...] is a(n-2) = [0, 1, 2, 6, 26, ...]. - Michael Somos, Mar 04 2004
The asymptotic expansion of 2*log(n) - (2^1*log(1) + 2^2*log(2) + ... + 2^n*log(n))/2^n is (a(1)/1)/n + (a(2)/2)/n^2 + (a(3)/3)/n^3 + ... - Michael Somos, Aug 22 2004
This is the sequence of cumulants of the probability distribution of the number of tails before the first head in a sequence of fair coin tosses. - Michael Hardy (hardy(AT)math.umn.edu), May 01 2005
Appears to be row sums of A154921. - Mats Granvik, Jan 18 2009
This is the number of cyclically ordered partitions of n+1 labeled points. The ordered version is A000670. - Michael Somos, Jan 08 2011
A000670(n+1) = p(n+1) where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0, 1, ..., n. - Michael Somos, Apr 27 2012
Row sums of A154921 as conjectured above by Granvik. a(n) gives the number of outcomes of a race between n horses H1,...,Hn, where if a horse falls it is not ranked. For example, when n = 2 the 6 outcomes are a dead heat, H1 wins H2 second, H2 wins H1 second, H1 wins H2 falls, H2 wins H1 falls or both fall. - Peter Bala, May 15 2012
Also the number of disjoint areas of a Venn diagram for n multisets. - Aurelian Radoaca, Jun 27 2016
Also the number of ways of ordering n nonnegative integers, allowing for the possibility of ties, and also comparing the smallest integers with 0. Each comparison with 0 gives two possibilities, x > 0 or x=0. As such, without comparison with 0, we get A000670, the number of ways of ordering n nonnegative integers, allowing for the possibility of ties, or the number of ways n competitors can rank in a competition, allowing for the possibility of ties. For instance, for 2 nonnegative integers x,y, there are the following 6 ways of ordering them: x = y = 0, x = y > 0, x > y = 0, x > y > 0, y > x = 0, y > x > 0. - Aurelian Radoaca, Jul 09 2016
Also the number of ordered set partitions of subsets of {1,...,n}. Also the number of chains of distinct nonempty subsets of {1,...,n}. - Gus Wiseman, Feb 01 2019
Number of combinations of a Simplex lock having n buttons.
Row sums of the unsigned cumulant expansion polynomials A127671 and logarithmic polynomials A263634. - Tom Copeland, Jun 04 2021
Also the number of vertices in the axis-aligned polytope consisting of all vectors x in R^n where, for all k in {1,...,n}, the k-th smallest coordinate of x lies in the interval [0, k]. - Adam P. Goucher, Jan 18 2023
Number of idempotent Boolean relation matrices whose complement is also idempotent.  See Rosenblatt link. - Geoffrey Critzer, Feb 26 2023

			a(2)=6: the necklace representatives on 1,2,3 are ({123}), ({12},{3}), ({13},{2}), ({23},{1}), ({1},{2},{3}), ({1},{3},{2})
G.f. = 1 + 2*x + 6*x^2 + 26*x^3 + 150*x^4 + 1082*x^5 + 9366*x^6 + 94586*x^7 + ...
From _Gus Wiseman_, Feb 01 2019: (Start)
The a(3) = 26 ordered set partitions of subsets of {1,2,3} are:
  {}  {{1}}  {{2}}  {{3}}  {{12}}    {{13}}    {{23}}    {{123}}
                           {{1}{2}}  {{1}{3}}  {{2}{3}}  {{1}{23}}
                           {{2}{1}}  {{3}{1}}  {{3}{2}}  {{12}{3}}
                                                         {{13}{2}}
                                                         {{2}{13}}
                                                         {{23}{1}}
                                                         {{3}{12}}
                                                         {{1}{2}{3}}
                                                         {{1}{3}{2}}
                                                         {{2}{1}{3}}
                                                         {{2}{3}{1}}
                                                         {{3}{1}{2}}
                                                         {{3}{2}{1}}
(End)

R. Austin, R. K. Guy, and R. Nowakowski, unpublished notes, circa 1987.
N. G. de Bruijn, Asymptotic Methods in Analysis, Dover, 1981, p. 36.
Eric Hammer, The Calculations of Peirce's 4.453, Transactions of the Charles S. Peirce Society, Vol. 31 (1995), pp. 829-839.
D. E. Knuth, personal communication.
J. D. E. Konhauser et al., Which Way Did the Bicycle Go?, MAA 1996, p. 174.
Charles Sanders Peirce, Collected Papers, eds. C. Hartshorne and P. Weiss, Harvard University Press, Cambridge, Vol. 4, 1933, pp. 364-365. (CP 4.453 in the electronic edition of The Collected Papers of Charles Sanders Peirce.)
Dawidson Razafimahatolotra, Number of Preorders to Compute Probability of Conflict of an Unstable Effectivity Function, Preprint, Paris School of Economics, University of Paris I, Nov 23 2007.

Seiichi Manyama, Table of n, a(n) for n = 0..424 (terms 0..100 from T. D. Noe)
R. Austin, R. K. Guy, & R. Nowakowski, Unpublished notes, 1987
Paul Barry, Eulerian polynomials as moments, via exponential Riordan arrays (2011), arXiv preprint arXiv:1105.3043 [math.CO], 2011. J. Int. Seq. 14 (2011) # 11.9.5.
Paul Barry, Eulerian-Dowling Polynomials as Moments, Using Riordan Arrays, arXiv:1702.04007 [math.CO], 2017.
Paul Barry, On a transformation of Riordan moment sequences, arXiv:1802.03443 [math.CO], 2018.
Paul Barry, Series reversion with Jacobi and Thron continued fractions, arXiv:2107.14278 [math.NT], 2021.
Arthur T. Benjamin, Combinatorics and campus security, The UMAP Journal 17.2 (summer 1996), pp. 111-116.
Zhanar Berikkyzy, Pamela E. Harris, Anna Pun, Catherine Yan, and Chenchen Zhao, Combinatorial Identities for Vacillating Tableaux, arXiv:2308.14183 [math.CO], 2023. See pp. 27, 29.
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
Mircea I. Cirnu, Determinantal formulas for sum of generalized arithmetic-geometric series, Boletin de la Asociacion Matematica Venezolana, Vol. XVIII, No. 1 (2011), p. 13.
Colin Defant, Troupes, Cumulants, and Stack-Sorting, arXiv:2004.11367 [math.CO], 2020.
G. H. E. Duchamp, N. Hoang-Nghia, and A. Tanasa, A word Hopf algebra based on the selection/quotient principle, Séminaire Lotharingien de Combinatoire 68 (2013), Article B68c.
Thomas Fink, Recursively divisible numbers, arXiv:1912.07979 [math.NT], 2019.
Olivier Golinelli, Remote control system of a binary tree of switches - II. balancing for a perfect binary tree, arXiv:2405.16968 [cs.DM], 2024. See p. 17.
W. S. Gray and M. Thitsa, System Interconnections and Combinatorial Integer Sequences, in: System Theory (SSST), 2013 45th Southeastern Symposium on, Date of Conference: 11-11 Mar 2013, Digital Object Identifier: 10.1109/SSST.2013.6524939.
Robin Houston, Adam P. Goucher, and Nathaniel Johnston, A New Formula for the Determinant and Bounds on Its Tensor and Waring Ranks, arXiv:2301.06586 [math.CO], 2023.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 99
H. K. Kim, D. S. Krotov and J. Y. Lee, Matrices uniquely determined by their lonesums, Linear Algebra and its Applications, 438 (2013) no 7, 3107-3123.
Germain Kreweras, Une dualité élémentaire souvent utile dans les problèmes combinatoires, Mathématiques et Sciences Humaines 3 (1963): 31-41.
Qipeng Kuang, Václav Kůla, Ondřej Kuželka, Yuanhong Wang, and Yuyi Wang, Weighted First Order Model Counting for Two-variable Logic with Axioms on Two Relations, arXiv:2508.11515 [cs.LO], 2025. See pp. 20-21.
Rajesh Kumar Mohapatra and Tzung-Pei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics (2022) Vol. 10, No. 7, 1161.
Konstantin Nestmann and Carsten Timm, Time-convolutionless master equation: Perturbative expansions to arbitrary order and application to quantum dots, arXiv:1903.05132 [cond-mat.mes-hall], 2019.
Aurelian Radoaca, Properties of Multisets Compared to Sets
J. Randon-Furling and S. Redner, Residence Time Near an Absorbing Set, arXiv:1806.09028 [cond-mat.stat-mech], 2018.
D. Rosenblatt, On the graphs of finite Boolean relation matrices, Journal of Research of the National Bureau of Standards, 67B No. 4, 1963.
John K. Sikora, On Calculating the Coefficients of a Polynomial Generated Sequence Using the Worpitzky Number Triangles, arXiv:1806.00887 [math.NT], 2018.
S. L. Snover and N. J. A. Sloane, Correspondence, 1991
J. F. Steffensen, On a class of polynomials and their application to actuarial problems, Skandinavisk Aktuarietidskrift, Vol. 11, pp. 75-97, 1928.
M. Thitsa and W. S. Gray, On the Radius of Convergence of Interconnected Analytic Nonlinear Input-Output Systems, SIAM Journal on Control and Optimization, Vol. 50, No. 5, pp. 2786-2813. - From _N. J. A. Sloane_, Dec 26 2012
Eric Weisstein's World of Mathematics, Geometric Distribution
Eric Weisstein's World of Mathematics, Stirling Number of the Second Kind
Herbert S. Wilf, The Redheffer matrix of a partially ordered set, The Electronic Journal of Combinatorics 11(2) (2004), #R10
Herbert S. Wilf, The Redheffer matrix of a partially ordered set, arXiv:math/0408263 [math.CO], 2004.

Same as A076726 except for a(0). Cf. A008965, A052861, A008277.
Binomial transform of A000670, also double of A000670. - Joe Keane (jgk(AT)jgk.org)
A002050(n) = a(n) - 1.
A000629, A000670, A002050, A052856, A076726 are all more-or-less the same sequence. - N. J. A. Sloane, Jul 04 2012
Row sums of A028246.
A diagonal of the triangular array in A241168.
Row sums of unsigned A127671 and A263634.

spec := [ B, {B=Cycle(Set(Z,card>=1))}, labeled ]; [seq(combstruct[count](spec, size=n), n=0..20)];
a:=n->add(Stirling2(n+1,k)*(k-1)!,k=1..n+1); # Mike Zabrocki, Feb 05 2005

a[ 0 ] = 1; a[ n_ ] := (a[ n ] = 1 + Sum[ Binomial[ n, k ] a[ n-k ], {k, 1, n} ])
Table[ PolyLog[n, 1/2], {n, 0, -18, -1}] (* Robert G. Wilson v, Aug 05 2010 *)
a[ n_] := If[ n<0, 0, PolyLog[ -n, 1/2]]; (* Michael Somos, Mar 07 2011 *)
Table[Sum[(-1)^(n-k) StirlingS2[n,k]k! 2^k,{k,0,n}],{n,0,20}] (* Harvey P. Dale, Oct 21 2011 *)
Join[{1}, Rest[t=30; Range[0, t]! CoefficientList[Series[2/(2 - Exp[x]), {x, 0, t}], x]]] (* Vincenzo Librandi, Jan 02 2016 *)

{a(n) = if( n<0, 0, n! * polcoeff(subst( (1 + y) / (1 - y), y, exp(x + x * O(x^n)) - 1), n))} /* Michael Somos, Mar 04 2004 */

{a(n)=polcoeff(sum(m=0, n, 2^m*m!*x^m/prod(k=1, m, 1+k*x+x*O(x^n))), n)} \\ Paul D. Hanna, Jul 20 2011

from math import comb
from functools import lru_cache
@lru_cache(maxsize=None)
def A000629(n): return 1+sum(comb(n,j)*A000629(j) for j in range(n)) if n else 1 # Chai Wah Wu, Sep 25 2023

a(n) = 2*A000670(n) - 0^n. - Michael Somos, Jan 08 2011
O.g.f.: Sum_{n>=0} 2^n*n!*x^n / Product_{k=0..n} (1+k*x). - Paul D. Hanna, Jul 20 2011
E.g.f.: exp(x) / (2 - exp(x)) = d/dx log(1 / (2 - exp(x))).
a(n) = Sum_{k>=1} k^n/2^k.
a(n) = 1 + Sum_{j=0..n-1} C(n, j)*a(j).
a(n) = round(n!/log(2)^(n+1)) (just for n <= 15). - Henry Bottomley, Jul 04 2000
a(n) is asymptotic to n!/log(2)^(n+1). - Benoit Cloitre, Oct 20 2002
a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling2(n, k)*k!*2^k. - Vladeta Jovovic, Sep 29 2003
a(n) = Sum_{k=1..n} A008292(n, k)*2^k; A008292: triangle of Eulerian numbers. - Philippe Deléham, Jun 05 2004
a(1) = 1, a(n) = 2*Sum_{k=1..n-1} k!*A008277(n-1, k) for n>1 or a(n) = Sum_{k=1..n} (k-1)!*A008277(n, k). - Mike Zabrocki, Feb 05 2005
a(n) = Sum_{k=0..n} Stirling2(n+1, k+1)*k!. - Paul Barry, Apr 20 2005
A000629 = binomial transform of this sequence. a(n) = sum of terms in n-th row of A028246. - Gary W. Adamson, May 30 2005
a(n) = 2*(-1)^n * n!*Laguerre(n,P((.),2)), umbrally, where P(j,t) are the polynomials in A131758. - Tom Copeland, Sep 28 2007
a(n) = 2^n*A(n,1/2); A(n,x) the Eulerian polynomials. - Peter Luschny, Aug 03 2010
a(n) = (-1)^n*b(n), where b(n) = -2*Sum_{k=0..n-1} binomial(n,k)*b(k), b(0)=1. - Vladimir Kruchinin, Jan 29 2011
Row sums of A028246. Let f(x) = x+x^2. Then a(n+1) = (f(x)*d/dx)^n f(x) evaluated at x = 1. - Peter Bala, Oct 06 2011
O.g.f.: 1+2*x/(U(0)-2*x) where U(k)=1+3*x+3*x*k-2*x*(k+2)*(1+x+x*k)/U(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 14 2011
E.g.f.: exp(x)/(2 - exp(x)) = 2/(2-Q(0))-1; Q(k)=1+x/(2*k+1-x*(2*k+1)/(x+(2*k+2)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Nov 14 2011
G.f.: 1 / (1 - 2*x / (1 - 1*x / (1 - 4*x / (1 - 2*x / (1 - 6*x / ...))))). - Michael Somos, Apr 27 2012
PSUM transform of A162509. BINOMIAL transform is A007047. - Michael Somos, Apr 27 2012
G.f.: 1/G(0) where G(k) = 1 - x*(2*k+2)/( 1 - x*(k+1)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 23 2013
E.g.f.: 1/E(0) where E(k) = 1 - x/(k+1)/(1 - 1/(1 + 1/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Mar 27 2013
G.f.: T(0)/(1-2*x), where T(k) = 1 - 2*x^2*(k+1)^2/(2*x^2*(k+1)^2 - (1 - 2*x - 3*x*k)*(1 - 5*x - 3*x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 29 2013
a(n) = log(2)*integral_{x>=0} (ceiling(x))^n * 2^(-x) dx. - Peter Bala, Feb 06 2015

a(19) from Michael Somos, Mar 07 2011

A006252 Expansion of e.g.f. 1/(1 - log(1+x)).

Original entry on oeis.org

1, 1, 1, 2, 4, 14, 38, 216, 600, 6240, 9552, 319296, -519312, 28108560, -176474352, 3998454144, -43985078784, 837126163584, -12437000028288, 237195036797184, -4235955315745536, 85886259443020800, -1746536474655406080, 38320721602434017280, -864056965711935974400

Offset: 0

N. J. A. Sloane

sign

From Michael Somos, Mar 04 2004: (Start)
Stirling transform of a(n+1)=[1,2,4,14,38,...] is A000255(n)=[1,3,11,53,309,...].
Stirling transform of 2*a(n)=[2,2,4,8,28,...] is A052849(n)=[2,4,12,48,240,...].
Stirling transform of a(n)=[1,1,2,4,14,38,216,...] is A000142(n)=[1,2,6,24,120,...].
Stirling transform of a(n-1)=[1,1,1,2,4,14,38,...] is A000522(n-1)=[1,2,5,16,65,...].
Stirling transform of a(n-1)=[0,1,1,2,4,14,38,...] is A007526(n-1)=[0,1,4,15,64,...].
(End)
For n > 0: a(n) = sum of n-th row in triangle A048594. - Reinhard Zumkeller, Mar 02 2014
Coefficients in a factorial series representation of the exponential integral: exp(z)*E_1(z) = Sum_{n >= 0} (-1)^n*a(n)/(z)n, where (z)_n denotes the rising factorial z*(z + 1)*...*(z + n) and E_1(z) = Integrate{t = z..inf} exp(-t)/t dt. See Weninger, equation 6.4. - Peter Bala, Feb 12 2019

G. Pólya, Induction and Analogy in Mathematics. Princeton Univ. Press, 1954, p. 9.
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..400
Beáta Bényi and Daniel Yaqubi, Mixed coloured permutations, arXiv:1903.07450 [math.CO], 2019.
Takao Komatsu and Amalia Pizarro-Madariaga, Harmonic numbers associated with inversion numbers in terms of determinants, Turkish Journal of Mathematics (2019) Vol. 43, 340-354.
E. J. Weniger, Summation of divergent power series by means of factorial series arXiv:1005.0466v1 [math.NA], 2010.

Column k=1 of A320080.

Cf. A007840.

a006252 0 = 1
a006252 n = sum $ a048594_row n  -- Reinhard Zumkeller, Mar 02 2014

With[{nn=30},CoefficientList[Series[1/(1-Log[1+x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 12 2016 *)

a(n)=if(n<0,0,n!*polcoeff(1/(1-log(1+x+x*O(x^n))),n))

{a(n)=local(CF=1+x*O(x^n)); for(k=0, n-1, CF=1/((n-k+1)-(n-k)*x+(n-k+1)^2*x*CF)); n!*polcoeff(1+x/(1-x+x*CF), n, x)} /* Paul D. Hanna, Dec 31 2011 */

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, (-1)^(j-1)*(j-1)!*binomial(i, j)*v[i-j+1])); v; \\ Seiichi Manyama, May 22 2022

def A006252_list(len):
    f, R, C = 1, [1], [1]+[0]*len
    for n in (1..len):
        f *= n
        for k in range(n, 0, -1):
            C[k] = -C[k-1]*((k-1)/(k) if k>1 else 1)
        C[0] = -sum(C[k] for k in (1..n))
        R.append(C[0]*f)
    return R
print(A006252_list(24)) # Peter Luschny, Feb 21 2016

a(n) = Sum_{k=0..n} k!*stirling1(n, k). - Vladeta Jovovic, Sep 08 2002
a(n) = D^n(1/(1-x)) evaluated at x = 0, where D is the operator exp(-x)*d/dx. Row sums of A048594. Cf. A007840. - Peter Bala, Nov 25 2011
E.g.f.: 1/(1-log(1+x)) = 1 + x/(1-x + x/(2-x + 4*x/(3-2*x + 9*x/(4-3*x + 16*x/(5-4*x + 25*x/(6-5*x +...)))))), a continued fraction. - Paul D. Hanna, Dec 31 2011
a(n)/n! ~ -(-1)^n / (n * (log(n))^2) * (1 - 2*(1 + gamma)/log(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 01 2018
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, May 22 2022

A007047 Number of chains in power set of n-set.

Original entry on oeis.org

1, 3, 11, 51, 299, 2163, 18731, 189171, 2183339, 28349043, 408990251, 6490530291, 112366270379, 2107433393523, 42565371881771, 921132763911411, 21262618727925419, 521483068116543603, 13542138653027381291, 371206349277313644531

Offset: 0

N. J. A. Sloane, Roger B. Nelsen

Stirling transform of A052849(n-1) = [1,2,4,12,48,...] is a(n-1) =[1,3,11,51,299,...]. - Michael Somos, Mar 04 2004
It is interesting to note that a chain in the power set of a set X can be thought of as a fuzzy subset of X and conversely. Chains originating with empty set are fuzzy subsets with empty core and those chains not ending with the whole set are with support strictly contained in X. - Venkat Murali (v.murali(AT)ru.ac.za), May 18 2005
Equals the binomial transform of A000629: (1, 2, 6, 26, 150, 1082, ...) and the double binomial transform of A000670: (1, 1, 3, 13, 75, 541, ...). - Gary W. Adamson, Aug 04 2009
Row sums of A038719. - Peter Bala, Jul 09 2014
Also the number of restricted barred preferential arrangements of an n-set having two bars, where one fixed section is a free section and the other two sections are restricted sections. - Sithembele Nkonkobe, Jun 16 2015
The number of all predictable outcomes of a race between a given number registered competitors, where clean finishes, dead heats (ties), disqualifications, cancellations and their combinations are all counted. (11 outcomes for two competitors, 51 for three, 299 for four, etc.. Example for two competitors shown below.) - Gergely Földvári, Jul 28 2024

			If there are two registered competitors, A and B, in a race, the total number of predictable outcomes counting all possibilities of clean finishes (f), dead heats (t), disqualifications (d), cancellations (c) and their combinations is 11 (eleven). Here is the breakdown: AfBf, BfAf, AtBt, AfBd, AfBc, BfAd, BfAc, AdBd, AcBc, AdBc, AcBd. - _Gergely Földvári_, Jul 28 2024

V. Murali, Counting fuzzy subsets of a finite set, preprint, Rhodes University, Grahamstown 6140, South Africa, 2003.
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..424 (first 101 terms from T. D. Noe)
Paul Barry, General Eulerian Polynomials as Moments Using Exponential Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.9.6.
Lisa Berry, Stefan Forcey, Maria Ronco, and Patrick Showers, Species substitution, graph suspension, and graded hopf algebras of painted tree polytopes.
Gergely Földvári, the total number of all predictable outcomes of a race between "k" number of registered competitors: Race Outcome Formula
Jun Ma, S.-M. Ma, and Y.-N. Yeh, Recurrence relations for binomial-Eulerian polynomials, arXiv preprint arXiv:1711.09016 [math.CO], 2017.
T. Manneville and V. Pilaud, Compatibility fans for graphical nested complexes, arXiv preprint arXiv:1501.07152 [math.CO], 2015-2016.
Rajesh Kumar Mohapatra and Tzung-Pei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics (2022) Vol. 10, No. 7, 1161.
V. Murali, Number of fuzzy subsets of a finite set, fuzzy systems research group, Universities of Rhodes and Fort Hare. [broken link]
V. Murali and B. B. Makamba, Finite Fuzzy Sets, International Journal of General Systems, Vol. 34, No. 1 (2005), pp. 61-75.
R. B. Nelsen, Letter to N. J. A. Sloane, Aug. 1991
R. B. Nelsen and H. Schmidt, Jr., Chains in power sets, Math. Mag., 64 (1991), 23-31.
S. Nkonkobe and V. Murali, A study of a family of generating functions of Nelsen-Schmidt type and some identities on restricted barred preferential arrangements, arXiv:1503.06172 [math.CO], Apr 2015.
Index entries for sequences related to posets

Cf. A000629, A000629, A000670. Row sums of A038719.

a007047 = sum . a038719_row  -- Reinhard Zumkeller, Feb 05 2014

P := proc(n,x) option remember; if n = 0 then 1 else
   (n*x+2*(1-x))*P(n-1,x)+x*(1-x)*diff(P(n-1,x),x);
   expand(%) fi end:
A007047 := n -> 2^n*subs(x=1/2, P(n,x)):
seq(A007047(n), n=0..19);  # Peter Luschny, Mar 07 2014
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 4,
      add(b(n-j)*binomial(n, j), j=1..n))
    end:
a:= n-> `if`(n=0, 1, b(n)-1):
seq(a(n), n=0..21);  # Alois P. Heinz, Feb 07 2020

Table[LerchPhi[1/2, -n, 2]/2, {n, 0, 10}] (* Vladimir Reshetnikov, Feb 16 2011 *)
Table[2*PolyLog[-n, 1/2] - 1 , {n, 0, 19}] (* Jean-François Alcover, Aug 14 2013 *)
With[{nn=20},CoefficientList[Series[Exp[2x]/(2-Exp[x]),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Dec 08 2015 *)
Table[-(-1)^k HurwitzLerchPhi[2, -k, -1], {k, 0, 30}] (* Federico Provvedi,Sep 05 2020 *)
a[n_]:= a[n] = 2^n + Sum[Binomial[n, k]*a[k], {k, 0, n-1}]; Table[a[n], {n, 0, 20}] (* Rajesh Kumar Mohapatra, Jul 02 2025 *)
a[0] = 1; a[n_]:= a[n] = 2^n + Sum[Binomial[n, k]*a[n-k], {k, 1, n}]; Table[a[n], {n, 0, 20}] (* Rajesh Kumar Mohapatra, Jul 02 2025 *)

a(n)=if(n<0,0,n!*polcoeff(subst((y+1)^2/(1-y),y,exp(x+x*O(x^n))-1),n));

my(x='x+O('x^66)); Vec(serlaplace(exp(2*x)/(2-exp(x)))) \\ Joerg Arndt, Aug 14 2013

E.g.f.: exp(2*x)/(2-exp(x)).
a(n) = Sum_{k>=1} (k+1)^n/2^k = 2*A000629(n)-1. - Benoit Cloitre, Sep 08 2002
a(n) = one less than sum of quotients with numerator 4 times (n!)((k_1 + k_2 + ... + k_n)!) and with denominator (k_1!k_2!...k_n!)(1!^k_1 2!^k_2...n!^k_n) where the sum is taken over all partitions 1*k_1 + 2*k_2 + ... + n*k_n = n. E.g. a(1) = 3 because the membership value of x to {x} is either 1, alpha with 0 < alpha < 1 or 0. a(2) = 11 since the membership values x and y to {x, y} are 1 >= alpha >= beta >= 0 for {empty set, x, y} in that order or {empty set, y, x} exercising all possible > or = . - Venkat Murali (v.murali(AT)ru.ac.za), May 18 2005
G.f.: 1/Q(0), where Q(k) = 1 - 3*x*(k+1) - 2*x^2*(k+1)*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 02 2013
a(n) ~ 2*n! / (log(2))^(n+1). - Vaclav Kotesovec, Sep 27 2017
a(n) = 4 * A000670(n) - 1 for n > 0. - Alois P. Heinz, Feb 07 2020
a(n) = -(-1)^n Phi(2,-n,-1), where Phi(z,s,a) is the Lerch zeta function. - Federico Provvedi, Sep 05 2020
a(n) = 1 + 2 * Sum_{k=0..n-1} (-1)^(n-k-1) * binomial(n,k) * a(k). - Ilya Gutkovskiy, Apr 28 2021
From Rajesh Kumar Mohapatra, Jul 02 2025: (Start)
a(n) = 2*A000629(n) - 1.
a(n) = 2^n + Sum_{k=0..n-1} binomial(n,k) * a(k).
a(n) = 2^n + Sum_{k=1..n} binomial(n,k) * a(n-k), a(0) = 1. (End)

A002050 Number of simplices in barycentric subdivision of n-simplex.

Original entry on oeis.org

0, 1, 5, 25, 149, 1081, 9365, 94585, 1091669, 14174521, 204495125, 3245265145, 56183135189, 1053716696761, 21282685940885, 460566381955705, 10631309363962709, 260741534058271801, 6771069326513690645

Offset: 0

N. J. A. Sloane

Stirling transform of A052849(n)=[1,4,12,48,240,...] is a(n)=[1,5,25,149,1081,...]. - Michael Somos, Mar 04 2004
Stirling transform of A000142(n-1)=[0,1,2,6,24,...] is a(n-1)=[0,1,5,25,149,...]. - Michael Somos, Mar 04 2004
Stirling transform of 2*A005359(n-1)=[1,0,4,0,48,0,...] is a(n-1)=[1,1,5,25,149,...]. - Michael Somos, Mar 04 2004
"Stirling-Bernoulli transform" of A000225. - Paul Barry, Apr 20 2005
a(n) is the number of nonempty words that can be formed from an alphabet of nonempty subsets of [n] so that the letters in each word are pairwise disjoint. - Geoffrey Critzer, Apr 12 2009
Row sums of A053440. - Peter Bala, Jul 12 2014
Conjecture: Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is eventually periodic with the period dividing phi(k) = A000010(k). For example, modulo 9 we obtain the sequence [0, 1, 5, 7, 5, 1, 5, 4, 5, 7, 5, 1, 5, 4, 5, 7, 5, 1, 5, 4, 5, 7, 5, ...], with an apparent period of 6 = phi(9) beginning at a(5). - Peter Bala, Aug 03 2023

R. Austin, R. K. Guy, and R. Nowakowski, unpublished notes, circa 1987.
Ulrike Sattler, Decidable classes of formal power series with nice closure properties, Diplomarbeit im Fach Informatik, Univ. Erlangen - Nuernberg, Jul 27 1994
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).

T. D. Noe, Table of n, a(n) for n = 0..100
R. Austin, R. K. Guy, & R. Nowakowski, Unpublished notes, 1987
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 28.
R. K. Guy, Letter to N. J. A. Sloane, Nov 21 1974
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 149
D. S. Kluk & N. J. A. Sloane, Correspondence, 1979
G. J. Simmons, A combinatorial problem associated with a family of combination locks, Math. Mag., 37 (1964), 127-132 (but there are errors).
G. J. Simmons, A combinatorial problem associated with a family of combination locks, Math. Mag., 37 (1964), 127-132 [Annotated, corrected, scanned copy]
G. J. Simmons, Letter to N. J. Sloane, May 29 1974
J. F. Steffensen, On a class of polynomials and their application to actuarial problems, Skandinavisk Aktuarietidskrift, Vol. 11, pp. 75-97, 1928.

A000629, A000670, A002050, A052856, A076726 are all more-or-less the same sequence. - N. J. A. Sloane, Jul 04 2012

A diagonal of the triangle in A241168. Row sums of A053440.

Table[Sum[Binomial[n, i]*Sum[StirlingS2[i, k]*k!, {k, 1, i}], {i, 1, n}], {n, 0, 20}] (* Geoffrey Critzer, Apr 12 2009 *)
With[{nn=20},CoefficientList[Series[(Exp[2x]-Exp[x])/(2-Exp[x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 28 2013 *)
a[0] = 0; a[n_] := 2*Sum[k!*StirlingS2[n, k], {k, 2, n}] + 1; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Sep 27 2013, after Vladimir Kruchinin *)

a(n)=if(n<0,0,n!*polcoeff(subst((y+y^2)/(1-y),y,exp(x+x*O(x^n))-1),n));

E.g.f.: (exp(2x)-exp(x))/(2-exp(x)).
a(n) = A000629(n) - 1.
a(n) = Sum_{k=0..n} (-1)^(n-k)k!*S2(n, k)(2^k-1). - Paul Barry, Apr 20 2005
a(n) = Sum_{k=1...n} binomial(n,k)*A000670(k). - Geoffrey Critzer, Apr 12 2009
a(n) ~ n!/log(2)^(n+1). - Vaclav Kotesovec, Jul 29 2013
a(n) = 1 + 2*Sum_{k=2..n} k!*Stirling2(n,k), n > 0, a(0)=1. - Vladimir Kruchinin, Sep 27 2013
G.f.: T(0)/(1-2*x) - 1/(1-x), where T(k) = 1 - 2*x^2*(k+1)^2/(2*x^2*(k+1)^2 - (1 - 2*x - 3*x*k)*(1 - 5*x - 3*x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 29 2013
G.f.: Sum_{j>=1} j!*x^j / Product_{k=0..j} (1 - (k + 1)*x). - Ilya Gutkovskiy, Apr 04 2019

More terms from James Sellers, Aug 22 2000

A245334 A factorial-like triangle read by rows: T(0,0) = 1; T(n+1,0) = T(n,0)+1; T(n+1,k+1) = T(n,0)*T(n,k), k=0..n.

Original entry on oeis.org

1, 2, 1, 3, 4, 2, 4, 9, 12, 6, 5, 16, 36, 48, 24, 6, 25, 80, 180, 240, 120, 7, 36, 150, 480, 1080, 1440, 720, 8, 49, 252, 1050, 3360, 7560, 10080, 5040, 9, 64, 392, 2016, 8400, 26880, 60480, 80640, 40320, 10, 81, 576, 3528, 18144, 75600, 241920, 544320

Offset: 0

Reinhard Zumkeller, Aug 30 2014

row(0) = {1}; row(n+1) = row(n) multiplied by n and prepended with (n+1);
A111063(n+1) = sum of n-th row;
T(2*n,n) = A002690(n), central terms;
T(n,0) = n + 1;
T(n,1) = A000290(n), n > 0;
T(n,2) = A011379(n-1), n > 1;
T(n,3) = A047927(n), n > 2;
T(n,4) = A192849(n-1), n > 3;
T(n,5) = A000142(5) * A027810(n-5), n > 4;
T(n,6) = A000142(6) * A027818(n-6), n > 5;
T(n,7) = A000142(7) * A056001(n-7), n > 6;
T(n,8) = A000142(8) * A056003(n-8), n > 7;
T(n,9) = A000142(9) * A056114(n-9), n > 8;
T(n,n-10) = 11 * A051431(n-10), n > 9;
T(n,n-9) = 10 * A049398(n-9), n > 8;
T(n,n-8) = 9 * A049389(n-8), n > 7;
T(n,n-7) = 8 * A049388(n-7), n > 6;
T(n,n-6) = 7 * A001730(n), n > 5;
T(n,n-5) = 6 * A001725(n), n > 5;
T(n,n-4) = 5 * A001720(n), n > 4;
T(n,n-3) = 4 * A001715(n), n > 2;
T(n,n-2) = A070960(n), n > 1;
T(n,n-1) = A052849(n), n > 0;
T(n,n) = A000142(n);
T(n,k) = A137948(n,k) * A007318(n,k), 0 <= k <= n.

			.  0:   1;
.  1:   2,  1;
.  2:   3,  4,   2;
.  3:   4,  9,  12,    6;
.  4:   5, 16,  36,   48,    24;
.  5:   6, 25,  80,  180,   240,   120;
.  6:   7, 36, 150,  480,  1080,  1440,    720;
.  7:   8, 49, 252, 1050,  3360,  7560,  10080,   5040;
.  8:   9, 64, 392, 2016,  8400, 26880,  60480,  80640,  40320;
.  9:  10, 81, 576, 3528, 18144, 75600, 241920, 544320, 725760, 362880.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Cf. A111063 (row sums), A240993 (row products), A002690 (central terms).
Cf. A000290, A011379, A027810, A027818, A047927, A056001, A056003, A056114, A192849.
Cf. A000142, A001715, A001720, A001725, A001730, A049388, A049389, A049398, A051431, A052849, A070960.
Cf. A007318, A137948.

a245334 n k = a245334_tabl !! n !! k
a245334_row n = a245334_tabl !! n
a245334_tabl = iterate (\row@(h:_) -> (h + 1) : map (* h) row) [1]

Table[(n!)/((n - k)!)*(n + 1 - k), {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Sep 10 2017 *)

T(n,k) = n!*(n+1-k)/(n-k)!. - Werner Schulte, Sep 09 2017

A062119 a(n) = n! * (n-1).

Original entry on oeis.org

0, 2, 12, 72, 480, 3600, 30240, 282240, 2903040, 32659200, 399168000, 5269017600, 74724249600, 1133317785600, 18307441152000, 313841848320000, 5690998849536000, 108840352997376000, 2189611807358976000, 46225138155356160000, 1021818843434188800000

Offset: 1

Olivier Gérard, Jun 13 2001

For n > 0, a(n) = number of permutations of length n+1 that have 2 predetermined elements nonadjacent; e.g., for n=2, the permutations with, say, 1 and 2 nonadjacent are 132 and 231, therefore a(2)=2. - Jon Perry, Jun 08 2003
Number of multiplications performed when computing the determinant of an n X n matrix by definition. - Mats Granvik, Sep 12 2008
Sum of the length of all cycles (excluding fixed points) in all permutations of [n]. - Olivier Gérard, Oct 23 2012
Number of permutations of n distinct objects (ABC...) 1 (one) times >>("-", A, AB, ABC, ABCD, ABCDE, ..., ABCDEFGHIJK, infinity) and one after the other to resemble motif: A (1) AB (1-1), AAB (2-1), AAAB (3-1), AAAAB (4-1), AAAAAB (5-1), AAAAAAB (6-1), AAAAAAAB (7-1), AAAAAAAAB (8-1) etc.,>> "1(one) fixed point". Example:motif: AAAB (or BBBA) 12 * one (1) fixed point etc. Let: AAAB ................ 'A'BCD 1. 'A'BDC 2. 'A'CBD 3. ACDB 'A'DBC 4. 'A'DCB B'A'CD 5. B'A'DC 6. BCAD 7. BCDA BD'A'C 8. BDCA C'A'BD 9. C'A'DB CB'A'D 10. CBDA CDAB CDBA D'A'BC 11. DACB DB'A'C 12. DBCA DCAB DCBA. - Zerinvary Lajos, Nov 27 2009 (does anybody understand what this is supposed to say? - Joerg Arndt, Jan 10 2015)
a(n) is the number of ways to arrange n books on two bookshelves so that each shelf receives at least one book. - Geoffrey Critzer, Feb 21 2010
a(n) = number whose factorial base representation (A007623) begins with digit {n-1} and is followed by n-1 zeros. Viewed in that base, this sequence looks like this: 0, 10, 200, 3000, 40000, 500000, 6000000, 70000000, 800000000, 9000000000, A0000000000, B00000000000, ... (where "digits" A and B stand for placeholder values 10 and 11 respectively). - Antti Karttunen, May 07 2015

Harry J. Smith, Table of n, a(n) for n = 1..100
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Ugbene Ifeanyichukwu Jeff, Ogundele Olaniyi Suraju, and Ndubuisi Rich Ugochukwu, Digraph of the full transformation semigroup, J. Disc. Math. Sci. Crypt. (2021).
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
S. R. Schmitt Determinants [From _Mats Granvik_, Sep 12 2008]
Index entries for sequences related to factorial base representation

Column 2 of A257503 (apart from initial zero. Equally, row 2 of A257505).
Cf. A001286 (same sequence divided by 2).
Cf. A000142, A007623, A018931, A052849.
Cf. A001563. - Zerinvary Lajos, Aug 27 2008
Cf. sequences with formula (n + k)*n! listed in A282466.
Cf. A001113, A001620, A068985, A091725, A099285.

a062119 n = (n - 1) * a000142 n  -- Reinhard Zumkeller, Aug 27 2012

G(x):=x^2/(1-x)^2: f[0]:=G(x): for n from 1 to 19 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=1..19); # Zerinvary Lajos, Apr 01 2009

a[n_]:=1*(n+2)!-2*(n+1)!; (* or *) a[n_]:=n!*(n-1); (* Vladimir Joseph Stephan Orlovsky, Dec 05 2008 *)
Table[n!(n-1),{n,20}] (* Harvey P. Dale, Aug 29 2021 *)

{ f=1; for (n=1, 100, f*=n; write("b062119.txt", n, " ", f*(n - 1)) ) } \\ Harry J. Smith, Aug 02 2009

a(n) = n!*(n-1); \\ Altug Alkan, May 04 2018

(define (A062119 n) (* (- n 1) (A000142 n))) ;; Antti Karttunen, May 07 2015

a(n) = n! * (n-1).
E.g.f.: x^2/(1-x)^2. - Geoffrey Critzer, Feb 21 2010
a(n) = 2 * A001286(n).
a(n) = A001563(n) - A000142(n). - Antti Karttunen, May 07 2015, hinted by crossref left by Lajos.
From Amiram Eldar, Jul 11 2020: (Start)
Sum_{n>=2} 1/a(n) = Ei(1) + 2 - e - gamma = A091725 + 2 - A001113 - A001620.
Sum_{n>=2} (-1)^n/a(n) = gamma - Ei(-1) - 1/e = A001620 + A099285 - A068985. (End)

Last term a(19) corrected by Harry J. Smith, Aug 02 2009

A098558 Expansion of e.g.f. (1+x)/(1-x).

Original entry on oeis.org

1, 2, 4, 12, 48, 240, 1440, 10080, 80640, 725760, 7257600, 79833600, 958003200, 12454041600, 174356582400, 2615348736000, 41845579776000, 711374856192000, 12804747411456000, 243290200817664000, 4865804016353280000, 102181884343418880000, 2248001455555215360000

Offset: 0

Paul Barry, Sep 14 2004

Essentially the same as A052849: a(0)=0 and a(n) = A052849(n) for n>=1.
Equals row sums (unsigned) of triangle A158471. - Gary W. Adamson, Mar 20 2009
Also the number of graceful labelings of the star graph on n+1 nodes. - Eric W. Weisstein, Mar 31 2020

G. C. Greubel, Table of n, a(n) for n = 0..445
Eric Weisstein's World of Mathematics, Graceful Labeling
Eric Weisstein's World of Mathematics, Star Graph

Cf. A000007, A000142, A008290.
Row sums of A008518 and of A128564.
Cf. A158471.

[1] cat [2*Factorial(n): n in [1..30]]; // G. C. Greubel, Jan 17 2018

Join[{1}, 2*Range[30]!] (* G. C. Greubel, Jan 17 2018 *)
With[{nn=30},CoefficientList[Series[(1+x)/(1-x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 05 2021 *)
a[n_] := Hypergeometric2F1Regularized[1, -n, 2 - n, -1];
Table[a[n], {n, 0, 22}]  (* Peter Luschny, Apr 26 2024 *)

concat([1], vector(30, n, 2*n!)) \\ G. C. Greubel, Jan 17 2018

CF = ComplexBallField(100)
def a(n):
    return Integer(CF(-1).hypergeometric([1, -n], [2 - n], regularized=True))
print([a(n) for n in range(23)]) # Peter Luschny, Apr 26 2024

a(n) = 2*n! - 0^n.
a(n) = Sum_{k=0..n} (k+1) * A008290(n,k). - Alois P. Heinz, Mar 11 2022
Sum_{n>=0} 1/a(n) = (e+1)/2. - Amiram Eldar, Feb 02 2023
a(n) = HypergeomRegularized([1, -n], [2 - n], -1). - Peter Luschny, Apr 26 2024

A276955 Square array A(row,col): A(row,1) = A273670(row-1), and for col > 1, A(row,col) = A153880(A(row,col-1)); Dispersion of factorial base left shift A153880.

Original entry on oeis.org

1, 2, 3, 6, 8, 4, 24, 30, 12, 5, 120, 144, 48, 14, 7, 720, 840, 240, 54, 26, 9, 5040, 5760, 1440, 264, 126, 32, 10, 40320, 45360, 10080, 1560, 744, 150, 36, 11, 362880, 403200, 80640, 10800, 5160, 864, 168, 38, 13, 3628800, 3991680, 725760, 85680, 41040, 5880, 960, 174, 50, 15, 39916800, 43545600, 7257600, 766080, 367920, 46080, 6480, 984, 246, 56, 16

Offset: 1

Antti Karttunen, Sep 22 2016

The square array A(row,col) is read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

When viewed in factorial base (A007623) the terms on each row start all with the same prefix, but with an increasing number of zeros appended to the end. For example, for row 8 (A001344 from a(1)=11 onward), the terms written in factorial base look as: 121, 1210, 12100, 121000, ...

			The top left {1..9} x {1..18} corner of the array:
   1,  2,   6,   24,   120,    720,    5040,    40320,    362880
   3,  8,  30,  144,   840,   5760,   45360,   403200,   3991680
   4, 12,  48,  240,  1440,  10080,   80640,   725760,   7257600
   5, 14,  54,  264,  1560,  10800,   85680,   766080,   7620480
   7, 26, 126,  744,  5160,  41040,  367920,  3669120,  40279680
   9, 32, 150,  864,  5880,  46080,  408240,  4032000,  43908480
  10, 36, 168,  960,  6480,  50400,  443520,  4354560,  47174400
  11, 38, 174,  984,  6600,  51120,  448560,  4394880,  47537280
  13, 50, 246, 1464, 10200,  81360,  730800,  7297920,  80196480
  15, 56, 270, 1584, 10920,  86400,  771120,  7660800,  83825280
  16, 60, 288, 1680, 11520,  90720,  806400,  7983360,  87091200
  17, 62, 294, 1704, 11640,  91440,  811440,  8023680,  87454080
  18, 72, 360, 2160, 15120, 120960, 1088640, 10886400, 119750400
  19, 74, 366, 2184, 15240, 121680, 1093680, 10926720, 120113280
  20, 78, 384, 2280, 15840, 126000, 1128960, 11249280, 123379200
  21, 80, 390, 2304, 15960, 126720, 1134000, 11289600, 123742080
  22, 84, 408, 2400, 16560, 131040, 1169280, 11612160, 127008000
  23, 86, 414, 2424, 16680, 131760, 1174320, 11652480, 127370880

Inverse permutation: A276956.
Transpose: A276953.
Cf. A276949 (index of column where n appears), A276951 (index of row).
Cf. A153880.
Columns 1-3: A273670, A276932, A276933.
The following lists some of the rows that have their own entries. Pattern present in the factorial base expansion of the terms on that row is given in double quotes:
Row 1: A000142 (from a(1)=1, "1" onward),
Row 2: A001048 (from a(2)=3, "11" onward),
Row 3: A052849 (from a(2)=4, "20" onward).
Row 4: A052649 (from a(1)=5, "21" onward).
Row 5: A108217 (from a(3)=7, "101" onward).
Row 6: A054119 (from a(3)=9, "111" onward).
Row 7: A052572 (from a(2)=10, "120" onward).
Row 8: A001344 (from a(1)=11, "121" onward).
Row 13: A052560 (from a(3)=18, "300" onward).
Row 16: A225658 (from a(1)=21, "311" onward).
Row 20: A276940 (from a(3) = 27, "1011" onward).
Related or similar permutations: A257505, A275848, A273666.
Cf. also arrays A276617, A276588 & A276945.

(define (A276955 n) (A276955bi (A002260 n) (A004736 n)))
(define (A276955bi row col) (if (= 1 col) (A273670 (- row 1)) (A153880 (A276955bi row (- col 1)))))

A(row,1) = A273670(row-1), and for col > 1, A(row,col) = A153880(A(row,col-1))
As a composition of other permutations:
a(n) = A275848(A257505(n)).

A090802 Triangle read by rows: a(n,k) = number of k-length walks in the Hasse diagram of a Boolean algebra of order n.

Original entry on oeis.org

1, 2, 1, 4, 4, 2, 8, 12, 12, 6, 16, 32, 48, 48, 24, 32, 80, 160, 240, 240, 120, 64, 192, 480, 960, 1440, 1440, 720, 128, 448, 1344, 3360, 6720, 10080, 10080, 5040, 256, 1024, 3584, 10752, 26880, 53760, 80640, 80640, 40320

Offset: 0

Ross La Haye, Feb 10 2004

Row sums = A010842(n); Row sums from column 1 on = A066534(n) = n*A010842(n-1) = A010842(n) - 2^n.
a(n,k) = n! = k! = A000142(n) for n = k; a(n,n-1) = 2*n! = A052849(n) for n > 1; a(n,n-2) = 2*n! = A052849(n) for n > 2; a(n,n-3) = (4/3)*n! = A082569(n) for n > 3; a(n,n-1)/a(2,1) = n!/2! = A001710(n) for n > 1; a(n,n-2)/ a(3,1) = n!/3! = A001715(n) for n > 2; a(n,n-3)/a(4,1) = n!/4! = A001720(n) for n > 3.
a(2k, k) = A052714(k+1). a(2k-1, k) = A034910(k).
a(n,0) = A000079(n); a(n,1) = A001787(n) = row sums of A003506; a(n,2) = A001815(n) = 2!*A001788(n-1); a(n,3) = A052771(n) = 3!*A001789(n); a(n,4) = A052796(n) = 4!*A003472(n); ceiling[a(n,1) / 2] = A057711(n); a(n,5) = 5!*A054849(n).
In a class of n students, the number of committees (of any size) that contain an ordered k-sized subcommittee is a(n,k). - Ross La Haye, Apr 17 2006
Antidiagonal sums [1,2,5,12,30,76,198,528,1448,4080,...] appear to be binomial transform of A000522 interleaved with itself, i.e., 1,1,2,2,5,5,16,16,65,65,... - Ross La Haye, Sep 09 2006
Let P(A) be the power set of an n-element set A. Then a(n,k) = the number of ways to add k elements of A to each element x of P(A) where the k elements are not elements of x and order of addition is important. - Ross La Haye, Nov 19 2007
The derivatives of x^n evaluated at x=2. - T. D. Noe, Apr 21 2011

			{1};
{2, 1};
{4, 4, 2};
{8, 12, 12, 6};
{16, 32, 48, 48, 24};
{32, 80, 160, 240, 240, 120};
{64, 192, 480, 960, 1440, 1440, 720};
{128, 448, 1344, 3360, 6720, 10080, 10080, 5040};
{256, 1024, 3584, 10752, 26880, 53760, 80640, 80640, 40320}
a(5,3) = 240 because P(5,3) = 60, 2^(5-3) = 4 and 60 * 4 = 240.

T. D. Noe, Rows n = 0..100, flattened
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Eric Weisstein, Walk
Eric Weisstein, Boolean Algebra
Eric Weisstein, Hasse Diagram

Cf. A000142, A001710, A001715, A001720, A001787, A001788, A001789, A001815, A003472, A010842, A052771, A052796, A052849, A054849, A057711, A066534, A082569.

Cf. A038207, A007318.

Flatten[Table[n!/(n-k)! * 2^(n-k), {n, 0, 8}, {k, 0, n}]] (* Ross La Haye, Feb 10 2004 *)

a(n, k) = 0 for n < k. a(n, k) = k!*C(n, k)*2^(n-k) = P(n, k)*2^(n-k) = (2n)!!/((n-k)!*2^k) = k!*A038207(n, k) = A068424*2^(n-k) = Sum[C(n, m)*P(n-m, k), {m, 0, n-k}] = Sum[C(n, n-m)*P(n-m, k), {m, 0, n-k}] = n!*Sum[1/(m!*(n-m-k)!), {m, 0, n-k}] = k!*Sum[C(n, m)*C(n-m, k), {m, 0, n-k}] = k!*Sum[C(n, n-m)*C(n-m, k), {m, 0, n-k}] = k!*C(n, k)*Sum[C(n-k, n-m-k), {m, 0, n-k}] = k!*C(n, k)*Sum[C(n-k, m), {m, 0, n-k}] for n >= k.
a(n, k) = 0 for n < k. a(n, k) = n*a(n-1, k-1) for n >= k >= 1.
E.g.f. (by columns): exp(2x)*x^k.

More terms from Ray Chandler, Feb 26 2004

Entry revised by Ross La Haye, Aug 18 2006

cp's OEIS Frontend

A066237 First differences give A052849.

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