A001118 -id:A001118 - cp's OEIS frontend

A000918 a(n) = 2^n - 2.

-1, 0, 2, 6, 14, 30, 62, 126, 254, 510, 1022, 2046, 4094, 8190, 16382, 32766, 65534, 131070, 262142, 524286, 1048574, 2097150, 4194302, 8388606, 16777214, 33554430, 67108862, 134217726, 268435454, 536870910, 1073741822, 2147483646, 4294967294, 8589934590, 17179869182, 34359738366, 68719476734, 137438953470

Offset: 0

N. J. A. Sloane

For n > 1, a(n) is the expected number of tosses of a fair coin to get n-1 consecutive heads. - Pratik Poddar, Feb 04 2011
For n > 2, Sum_{k=1..a(n)} (-1)^binomial(n, k) = A064405(a(n)) + 1 = 0. - Benoit Cloitre, Oct 18 2002
For n > 0, the number of nonempty proper subsets of an n-element set. - Ross La Haye, Feb 07 2004
Numbers j such that abs( Sum_{k=0..j} (-1)^binomial(j, k)*binomial(j + k, j - k) ) = 1. - Benoit Cloitre, Jul 03 2004
For n > 2 this formula also counts edge-rooted forests in a cycle of length n. - Woong Kook (andrewk(AT)math.uri.edu), Sep 08 2004
For n >= 1, conjectured to be the number of integers from 0 to (10^n)-1 that lack 0, 1, 2, 3, 4, 5, 6 and 7 as a digit. - Alexandre Wajnberg, Apr 25 2005
Beginning with a(2) = 2, these are the partial sums of the subsequence of A000079 = 2^n beginning with A000079(1) = 2. Hence for n >= 2 a(n) is the smallest possible sum of exactly one prime, one semiprime, one triprime, ... and one product of exactly n-1 primes. A060389 (partial sums of the primorials, A002110, beginning with A002110(1)=2) is the analog when all the almost primes must also be squarefree. - Rick L. Shepherd, May 20 2005
From the second term on (n >= 1), the binary representation of these numbers is a 0 preceded by (n - 1) 1's. This pattern (0)111...1110 is the "opposite" of the binary 2^n+1: 1000...0001 (cf. A000051). - Alexandre Wajnberg, May 31 2005
The numbers 2^n - 2 (n >= 2) give the positions of 0's in A110146. Also numbers k such that k^(k + 1) = 0 mod (k + 2). - Zak Seidov, Feb 20 2006
Partial sums of A155559. - Zerinvary Lajos, Oct 03 2007
Number of surjections from an n-element set onto a two-element set, with n >= 2. - Mohamed Bouhamida, Dec 15 2007
It appears that these are the numbers n such that 3*A135013(n) = n*(n + 1), thus answering Problem 2 on the Mathematical Olympiad Foundation of Japan, Final Round Problems, Feb 11 1993 (see link Japanese Mathematical Olympiad).
Let P(A) be the power set of an n-element set A and R be a relation on P(A) such that for all x, y of P(A), xRy if x is a proper subset of y or y is a proper subset of x and x and y are disjoint. Then a(n+1) = |R|. - Ross La Haye, Mar 19 2009
The permutohedron Pi_n has 2^n - 2 facets [Pashkovich]. - Jonathan Vos Post, Dec 17 2009
First differences of A005803. - Reinhard Zumkeller, Oct 12 2011
For n >= 1, a(n + 1) is the smallest even number with bit sum n. Cf. A069532. - Jason Kimberley, Nov 01 2011
a(n) is the number of branches of a complete binary tree of n levels. - Denis Lorrain, Dec 16 2011
For n>=1, a(n) is the number of length-n words on alphabet {1,2,3} such that the gap(w)=1. For a word w the gap g(w) is the number of parts missing between the minimal and maximal elements of w. Generally for words on alphabet {1,2,...,m} with g(w)=g>0 the e.g.f. is Sum_{k=g+2..m} (m - k + 1)*binomial((k - 2),g)*(exp(x) - 1)^(k - g). a(3)=6 because we have: 113, 131, 133, 311, 313, 331. Cf. A240506. See the Heubach/Mansour reference. - Geoffrey Critzer, Apr 13 2014
For n > 0, a(n) is the minimal number of internal nodes of a red-black tree of height 2*n-2. See the Oct 02 2015 comment under A027383. - Herbert Eberle, Oct 02 2015
Conjecture: For n>0, a(n) is the least m such that A007814(A000108(m)) = n-1. - L. Edson Jeffery, Nov 27 2015
Actually this follows from the procedure for determining the multiplicity of prime p in C(n) given in A000108 by Franklin T. Adams-Watters: For p=2, the multiplicity is the number of 1 digits minus 1 in the binary representation of n+1. Obviously, the smallest k achieving "number of 1 digits" = k is 2^k-1. Therefore C(2^k-2) is divisible by 2^(k-1) for k > 0 and there is no smaller m for which 2^(k-1) divides C(m) proving the conjecture. - Peter Schorn, Feb 16 2020
For n >= 0, a(n) is the largest number you can write in bijective base-2 (a.k.a. the dyadic system, A007931) with n digits. - Harald Korneliussen, May 18 2019
The terms of this sequence are also the sum of the terms in each row of Pascal's triangle other than the ones. - Harvey P. Dale, Apr 19 2020
For n > 1, binomial(a(n),k) is odd if and only if k is even. - Charlie Marion, Dec 22 2020
For n >= 2, a(n+1) is the number of n X n arrays of 0's and 1's with every 2 X 2 square having density exactly 2. - David desJardins, Oct 27 2022
For n >= 1, a(n+1) is the number of roots of unity in the unique degree-n unramified extension of the 2-adic field Q_2. Note that for each p, the unique degree-n unramified extension of Q_p is the splitting field of x^(p^n) - x, hence containing p^n - 1 roots of unity for p > 2 and 2*(2^n - 1) for p = 2. - Jianing Song, Nov 08 2022

			a(4) = 14 because the 14 = 6 + 4 + 4 rationals (in lowest terms) for n = 3 are (see the Jun 14 2017 formula above): 1/2, 1, 3/2, 2, 5/2, 3; 1/4, 3/4, 5/4, 7/4; 1/8, 3/8, 5/8, 7/8. - _Wolfdieter Lang_, Jun 14 2017

H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 212.
Ralph P. Grimaldi, Discrete and Combinatorial Mathematics: An Applied Introduction, Fifth Edition, Addison-Wesley, 2004, p. 134. - Mohammad K. Azarian, Oct 27 2011
S. Heubach and T. Mansour, Combinatorics of Compositions and Words, Chapman and Hall, 2009 page 86, Exercise 3.16.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 33.
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).
A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911, p. 31.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Andrei Asinowski, Cyril Banderier, and Benjamin Hackl, Flip-sort and combinatorial aspects of pop-stack sorting, arXiv:2003.04912 [math.CO], 2020.
O. Bagdasar, On some functions involving the lcm and gcd of integer tuples, Scientific Publications of the State University of Novi Pazar, Appl. Maths. Inform. and Mech., Vol. 6, 2 (2014), 91--100.
S. Bilotta, E. Grazzini, and E. Pergola, Enumeration of Two Particular Sets of Minimal Permutations, J. Int. Seq. 18 (2015) 15.10.2
R. B. Campbell, The effect of inbreeding constraints and offspring distribution on time to the most recent common ancestor, Journal of Theoretical Biology, Volume 382, 7 October 2015, Pages 74-80.
Adam M. Goyt and Lara K. Pudwell, Avoiding colored partitions of two elements in the pattern sense, arXiv preprint arXiv:1203.3786 [math.CO], 2012. - From _N. J. A. Sloane_, Sep 17 2012
M. A. Hill, M. J. Hopkins and D. C. Ravenel, On the non-existence of elements of Kervaire invariant one. - _N. J. A. Sloane_, Sep 27 2010
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 77
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Milan Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
Japanese Mathematical Olympiad 1993, Final Round - Problem 2, Feb 11 1993.
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.
T. Manneville, V. Pilaud, Compatibility fans for graphical nested complexes, arXiv preprint arXiv:1501.07152 [math.CO], 2015.
Kanstantsin Pashkovich, Symmetry in Extended Formulations of the Permutahedron [sic], arXiv:0912.3446 [math.CO], 2009-2013. [_Jonathan Vos Post_, Dec 17 2009]
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260.
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260. [Annotated scanned copy]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Pratik Poddar, Consecutive Heads Puzzle, Oct 2009.
H. P. Robinson, Letter to N. J. A. Sloane, Sep 1975
A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911. [Annotated scans of pages 30-33 only]
Eric Weisstein's World of Mathematics, Sphere Line Picking
Index entries for linear recurrences with constant coefficients, signature (3,-2).
Index to sequences related to Olympiads.

Row sums of triangle A026998.
Cf. A000919, A001117, A001118, A095121, A110146.
Cf. A033484, A000225, A000325, A095151, A125128.
Cf. A058809 (3^n-3, n>0).

a000918 = (subtract 2) . (2 ^)
a000918_list = iterate ((subtract 2) . (* 2) . (+ 2)) (- 1)
-- Reinhard Zumkeller, Apr 23 2013

[2^n - 2: n in [0..40]]; // Vincenzo Librandi, Jun 23 2011

Maple
```
seq(2^n-2,n=0..20) ;
```

Table[2^n - 2, {n, 0, 29}] (* Alonso del Arte, Dec 01 2012 *)

a(n)=2^n-2 \\ Charles R Greathouse IV, Jun 16 2011

def A000918(n): return (1<Chai Wah Wu, Jun 10 2025

a(n) = 2*A000225(n-1).
G.f.: 1/(1 - 2*x) - 2/(1 - x), e.g.f.: (e^x - 1)^2 - 1. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 07 2001
For n >= 1, a(n) = A008970(n + 1, 2). - Philippe Deléham, Feb 21 2004
G.f.: (3*x - 1)/((2*x - 1)*(x - 1)). - Simon Plouffe in his 1992 dissertation for the sequence without the leading -1
a(n) = 2*a(n - 1) + 2. - Alexandre Wajnberg, Apr 25 2005
a(n) = A000079(n) - 2. - Omar E. Pol, Dec 16 2008
a(n) = A058896(n)/A052548(n). - Reinhard Zumkeller, Feb 14 2009
a(n) = A164874(n - 1, n - 1) for n > 1. - Reinhard Zumkeller, Aug 29 2009
a(n) = A173787(n,1); a(n) = A028399(2*n)/A052548(n), n > 0. - Reinhard Zumkeller, Feb 28 2010
a(n + 1) = A027383(2*n - 1). - Jason Kimberley, Nov 02 2011
G.f.: U(0) - 1, where U(k) = 1 + x/(2^k + 2^k/(x - 1 - x^2*2^(k + 1)/(x*2^(k + 1) - (k + 1)/U(k + 1) ))); (continued fraction, 3rd kind, 4-step). - Sergei N. Gladkovskii, Dec 01 2012
a(n+1) is the sum of row n in triangle A051601. - Reinhard Zumkeller, Aug 05 2013
a(n+1) = A127330(n,0). - Reinhard Zumkeller, Nov 16 2013
a(n) = Sum_{k=1..n-1} binomial(n, k) for n > 0. - Dan McCandless, Nov 14 2015
From Miquel Cerda, Aug 16 2016: (Start)
a(n) = A000225(n) - 1.
a(n) = A125128(n-1) - A000325(n).
a(n) = A095151(n) - A125128(n) - 1. (End)
a(n+1) = 2*(n + Sum_{j=1..n-1} (n-j)*2^(j-1)), n >= 1. This is the number of the rationals k/2, k = 1..2*n for n >= 1 and (2*k+1)/2^j for j = 2..n, n >= 2, and 2*k+1 < n-(j-1). See the example for n = 3 below. Motivated by the proposal A287012 by Mark Rickert. - Wolfdieter Lang, Jun 14 2017

Maple programs fixed by Vaclav Kotesovec, Dec 13 2014

A019538 Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= k <= n).

Original entry on oeis.org

1, 1, 2, 1, 6, 6, 1, 14, 36, 24, 1, 30, 150, 240, 120, 1, 62, 540, 1560, 1800, 720, 1, 126, 1806, 8400, 16800, 15120, 5040, 1, 254, 5796, 40824, 126000, 191520, 141120, 40320, 1, 510, 18150, 186480, 834120, 1905120, 2328480, 1451520, 362880, 1, 1022, 55980, 818520, 5103000, 16435440, 29635200, 30240000, 16329600, 3628800

Offset: 1

N. J. A. Sloane and Manfred Goebel (goebel(AT)informatik.uni-tuebingen.de), Dec 11 1996

Number of ways n labeled objects can be distributed into k nonempty parcels. Also number of special terms in n variables with maximal degree k.
In older terminology these are called differences of 0. - Michael Somos, Oct 08 2003
Number of surjections (onto functions) from an n-element set to a k-element set.
Also coefficients (in ascending order) of so-called ordered Bell polynomials.
(k-1)!*Stirling2(n,k-1) is the number of chain topologies on an n-set having k open sets [Stephen].
Number of set compositions (ordered set partitions) of n items into k parts. Number of k dimensional 'faces' of the n dimensional permutohedron (see Simion, p. 162). - Mitch Harris, Jan 16 2007
Correction of comment before: Number of (n-k)-dimensional 'faces' of the permutohedron of order n (an (n-1)-dimensional polytope). - Tilman Piesk, Oct 29 2014
This array is related to the reciprocal of an e.g.f. as sketched in A133314. For example, the coefficient of the fourth-order term in the Taylor series expansion of 1/(a(0) + a(1) x + a(2) x^2/2! + a(3) x^3/3! + ...) is a(0)^(-5) * {24 a(1)^4 - 36 a(1)^2 a(2) a(0) + [8 a(1) a(3) + 6 a(2)^2] a(0)^2 - a(4) a(0)^3}. The unsigned coefficients characterize the P3 permutohedron depicted on page 10 in the Loday link with 24 vertices (0-D faces), 36 edges (1-D faces), 6 squares (2-D faces), 8 hexagons (2-D faces) and 1 3-D permutohedron. Summing coefficients over like dimensions gives A019538 and A090582. Compare to A133437 for the associahedron. - Tom Copeland, Sep 29 2008, Oct 07 2008
Further to the comments of Tom Copeland above, the permutohedron of type A_3 can be taken as the truncated octahedron. Its dual is the tetrakis hexahedron, a simplicial polyhedron, with f-vector (1,14,36,24) giving the fourth row of this triangle. See the Wikipedia entry and [Fomin and Reading p. 21]. The corresponding h-vectors of permutohedra of type A give the rows of the triangle of Eulerian numbers A008292. See A145901 and A145902 for the array of f-vectors for type B and type D permutohedra respectively. - Peter Bala, Oct 26 2008
Subtriangle of triangle in A131689. - Philippe Deléham, Nov 03 2008
Since T(n,k) counts surjective functions and surjective functions are "consistent", T(n,k) satisfies a binomial identity, namely, T(n,x+y) = Sum_{j=0..n} C(n,j)*T(j,x)*T(n-j,y). For definition of consistent functions and a generalized binomial identity, see "Toy stories and combinatorial identities" in the link section below. - Dennis P. Walsh, Feb 24 2012
T(n,k) is the number of labeled forests on n+k vertices satisfying the following two conditions: (i) each forest consists of exactly k rooted trees with roots labeled 1, 2, ..., k; (ii) every root has at least one child vertex. - Dennis P. Walsh, Feb 24 2012
The triangle is the inverse binomial transform of triangle A028246, deleting the left column and shifting up one row. - Gary W. Adamson, Mar 05 2012
See A074909 for associations among this array and the Bernoulli polynomials and their umbral compositional inverses. - Tom Copeland, Nov 14 2014
E.g.f. for the shifted signed polynomials is G(x,t) = (e^t-1)/[1+(1+x)(e^t-1)] = 1-(1+x)(e^t-1) + (1+x)^2(e^t-1)^2 - ... (see also A008292 and A074909), which has the infinitesimal generator g(x,u)d/du = [(1-x*u)(1-(1+x)u)]d/du, i.e., exp[t*g(x,u)d/du]u eval. at u=0 gives G(x,t), and dG(x,t)/dt = g(x,G(x,t)). The compositional inverse is log((1-xt)/(1-(1+x)t)). G(x,t) is a generating series associated to the generalized Hirzebruch genera. See the G. Rzadowski link for the relation of the derivatives of g(x,u) to solutions of the Riccatt differential equation, soliton solns. to the KdV equation, and the Eulerian and Bernoulli numbers. In addition A145271 connects products of derivatives of g(x,u) and the refined Eulerian numbers to the inverse of G(x,t), which gives the normalized, reverse face polynomials of the simplices (A135278, divided by n+1). See A028246 for the generator g(x,u)d/dx. - Tom Copeland, Nov 21 2014
For connections to toric varieties and Eulerian polynomials, see the Dolgachev and Lunts and the Stembridge links. - Tom Copeland, Dec 31 2015
See A008279 for a relation between the e.g.f.s enumerating the faces of permutahedra (this entry) and stellahedra. - Tom Copeland, Nov 14 2016
T(n, k) appears in a Worpitzky identity relating monomials to binomials: x^n = Sum_{k=1..n} T(n, k)*binomial(x,k), n >= 1. See eq. (11.) of the Worpitzky link on p. 209. The relation to the Eulerian numbers is given there in eqs. (14.) and (15.). See the formula below relating to A008292. See also Graham et al. eq. (6.10) (relating monomials to falling factorials) on p. 248 (2nd ed. p. 262). The Worpitzky identity given in the Graham et al. reference as eq. (6.37) (2nd ed. p. 269) is eq. (5.), p. 207, of Worpitzky. - Wolfdieter Lang, Mar 10 2017
T(n, m) is also the number of minimum clique coverings and minimum matchings in the complete bipartite graph K_{m,n}. - Eric W. Weisstein, Apr 26 2017
From the Hasan and Franco and Hasan papers: The m-permutohedra for m=1,2,3,4 are the line segment, hexagon, truncated octahedron and omnitruncated 5-cell. The first three are well-known from the study of elliptic models, brane tilings and brane brick models. The m+1 torus can be tiled by a single (m+2)-permutohedron. Relations to toric Calabi-Yau Kahler manifolds are also discussed. - Tom Copeland, May 14 2020
From Manfred Boergens, Jul 25 2021: (Start)
Number of n X k binary matrices with row sums = 1 and no zero columns. These matrices are a subset of the matrices defining A183109.
The distribution into parcels in the leading comment can be regarded as a covering of [n] by tuples (A_1,...,A_k) in P([n])^k with nonempty and disjoint A_j, with P(.) denoting the power set (corrected for clarity by Manfred Boergens, May 26 2024). For the non-disjoint case see A183109 and A218695.
For tuples with "nonempty" dropped see A089072. For tuples with "nonempty and disjoint" dropped see A092477 and A329943 (amendment by Manfred Boergens, Jun 24 2024). (End)

			The triangle T(n, k) begins:
  n\k 1    2     3      4       5        6        7        8        9      10
  1:  1
  2:  1    2
  3:  1    6     6
  4:  1   14    36     24
  5:  1   30   150    240     120
  6:  1   62   540   1560    1800      720
  7:  1  126  1806   8400   16800    15120     5040
  8:  1  254  5796  40824  126000   191520   141120    40320
  9:  1  510 18150 186480  834120  1905120  2328480  1451520   362880
  10: 1 1022 55980 818520 5103000 16435440 29635200 30240000 16329600 3628800
  ... Reformatted and extended - _Wolfdieter Lang_, Oct 04 2014
---------------------------------------------------------------------------
T(4,1) = 1: {1234}. T(4,2) = 14: {1}{234} (4 ways), {12}{34} (6 ways), {123}{4} (4 ways). T(4,3) = 36: {12}{3}{4} (12 ways), {1}{23}{4} (12 ways), {1}{2}{34} (12 ways). T(4,4) = 1: {1}{2}{3}{4} (1 way).

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Row sums give A000670. Maximal terms in rows give A002869. Central terms T(2k-1,k) give A233734.
Diagonal is n! (A000142). 2nd diagonal is A001286. 3rd diagonal is A037960.
Cf. A000918, A000919, A001117, A001118, A008275, A008277, A008279, A048594, A059117, A059515, A074909, A084938, A089072, A092477, A183109, A218695, A329943.
Reflected version of A090582. A371568 is another version.
See also the two closely related triangles: A008277(n, k) = T(n, k)/k! (Stirling numbers of second kind) and A028246(n, k) = T(n, k)/k.
Cf. A033282 'faces' of the associahedron.
Cf. A008292, A047969, A145901, A145902. - Peter Bala, Oct 26 2008
Visible in the 3-D array in A249042.
See also A000182.

a019538 n k = a019538_tabl !! (n-1) !! (k-1)
a019538_row n = a019538_tabl !! (n-1)
a019538_tabl = iterate f [1] where
   f xs = zipWith (*) [1..] $ zipWith (+) ([0] ++ xs) (xs ++ [0])
-- Reinhard Zumkeller, Dec 15 2013

with(combinat): A019538 := (n,k)->k!*stirling2(n,k);

Table[k! StirlingS2[n, k], {n, 9}, {k, n}] // Flatten

{T(n, k) = if( k<0 || k>n, 0, sum(i=0, k, (-1)^i * binomial(k, i) * (k-i)^n))}; /* Michael Somos, Oct 08 2003 */

def T(n, k): return factorial(k)*stirling_number2(n,k) # Danny Rorabaugh, Oct 10 2015

T(n, k) = k*(T(n-1, k-1)+T(n-1, k)) with T(0, 0) = 1 [or T(1, 1) = 1]. - Henry Bottomley, Mar 02 2001
E.g.f.: (y*(exp(x)-1) - exp(x))/(y*(exp(x)-1) - 1). - Vladeta Jovovic, Jan 30 2003
Equals [0, 1, 0, 2, 0, 3, 0, 4, 0, 5, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ...] where DELTA is Deléham's operator defined in A084938.
T(n, k) = Sum_{j=0..k} (-1)^(k-j)*j^n*binomial(k, j). - Mario Catalani (mario.catalani(AT)unito.it), Nov 28 2003. See Graham et al., eq. (6.19), p. 251. For a proof see Bert Seghers, Jun 29 2013.
Sum_{k=0..n} T(n, k)(-1)^(n-k) = 1, Sum_{k=0..n} T(n, k)(-1)^k = (-1)^n. - Mario Catalani (mario.catalani(AT)unito.it), Dec 11 2003
O.g.f. for n-th row: polylog(-n, x/(1+x))/(x+x^2). - Vladeta Jovovic, Jan 30 2005
E.g.f.: 1 / (1 + t*(1-exp(x))). - Tom Copeland, Oct 13 2008
From Peter Bala, Oct 26 2008: (Start)
O.g.f. as a continued fraction: 1/(1 - x*t/(1 - (x + 1)*t/(1 - 2*x*t/(1 - 2*(x + 1)*t/(1 - ...))))) = 1 + x*t + (x + 2*x^2)*t^2 + (x + 6*x^2 + 6*x^3)*t^3 + ... .
The row polynomials R(n,x), which begin R(1,x) = x, R(2,x) = x + 2*x^2, R(3,x) = x + 6*x^2 + 6*x^3, satisfy the recurrence x*d/dx ((x + 1)*R(n,x)) = R(n+1,x). It follows that the zeros of R(n,x) are real and negative (apply Corollary 1.2 of [Liu and Wang]).
Since this is the triangle of f-vectors of the (simplicial complexes dual to the) type A permutohedra, whose h-vectors form the Eulerian number triangle A008292, the coefficients of the polynomial (x-1)^n*R(n,1/(x-1)) give the n-th row of A008292. For example, from row 3 we have x^2 + 6*x + 6 = 1 + 4*y + y^2, where y = x + 1, producing [1,4,1] as the third row of A008292. The matrix product A008292 * A007318 gives the mirror image of this triangle (see A090582).
For n,k >= 0, T(n+1,k+1) = Sum_{j=0..k} (-1)^(k-j)*binomial(k,j)*[(j+1)^(n+1) - j^(n+1)]. The matrix product of Pascal's triangle A007318 with the current array gives (essentially) A047969. This triangle is also related to triangle A047969 by means of the S-transform of [Hetyei], a linear transformation of polynomials whose value on the basis monomials x^k is given by S(x^k) = binomial(x,k). The S-transform of the shifted n-th row polynomial Q(n,x) := R(n,x)/x is S(Q(n,x)) = (x+1)^n - x^n. For example, from row 3 we obtain S(1 + 6*x + 6*x^2) = 1 + 6*x + 6*x*(x-1)/2 = 1 + 3*x + 3*x^2 = (x+1)^3 - x^3. For fixed k, the values S(Q(n,k)) give the nonzero entries in column (k-1) of the triangle A047969 (the Hilbert transform of the Eulerian numbers). (End)
E.g.f.: (exp(x)-1)^k = sum T(n,k)x^n/n!. - Vladimir Kruchinin, Aug 10 2010
T(n,k) = Sum_{i=1..k} A(n,i)*Binomial(n-i,k-i) where A(n,i) is the number of n-permutations that have i ascending runs, A008292.
From Tom Copeland, Oct 11 2011: (Start)
With e.g.f. A(x,t) = -1 + 1/(1+t*(1-exp(x))), the comp. inverse in x is B(x,t) = log(((1+t)/t) - 1/(t(1+x))).
With h(x,t) = 1/(dB/dx)= (1+x)((1+t)(1+x)-1), the row polynomial P(n,t) is given by (h(x,t)*d/dx)^n x, eval. at x=0, A=exp(x*h(y,t)*d/dy) y, eval. at y=0, and dA/dx = h(A(x,t),t), with P(0,t)=0.
(A factor of -1/n! was removed by Copeland on Aug 25 2016.) (End)
The term linear in x of [x*h(d/dx,t)]^n 1 gives the n-th row polynomial. (See A134685.) - Tom Copeland, Nov 07 2011
Row polynomials are given by D^n(1/(1-x*t)) evaluated at x = 0, where D is the operator (1+x)*d/dx. - Peter Bala, Nov 25 2011
T(n,x+y) = Sum_{j=0..n} binomial(n,j)*T(j,x)*T(n-j,y). - Dennis P. Walsh, Feb 24 2012
Let P be a Rota-Baxter operator of weight 1 satisfying the identity P(x)*P(y) = P(P(x)*y) + P(x*P(y)) + P(x*y). Then P(1)^2 = P(1) + 2*P^2(1). More generally, Guo shows that P(1)^n = Sum_{k=1..n} T(n,k)*P^k(1). - Peter Bala, Jun 08 2012
Sum_{i=1..n} (-1)^i*T(n,i)/i = 0, for n > 1. - Leonid Bedratyuk, Aug 09 2012
T(n, k) = Sum_{j=0..k} (-1)^j*binomial(k, j)*(k-j)^n. [M. Catalani's re-indexed formula from Nov 28 2003] Proof: count the surjections of [n] onto [k] with the inclusion-exclusion principle, as an alternating sum of the number of functions from [n] to [k-j]. - Bert Seghers, Jun 29 2013
n-th row polynomial = 1/(1 + x)*( Sum_{k>=0} k^n*(x/(1 + x))^k ), valid for x in the open interval (-1/2, inf). See Tanny link. Cf. A145901. - Peter Bala, Jul 22 2014
T(n,k) = k * A141618(n,k-1) / binomial(n,k-1). - Tom Copeland, Oct 25 2014
Sum_{n>=0} n^k*a^n = Sum_{i=1..k} (a / (1 - a))^i * T(k, i)/(1-a) for |a| < 1. - David A. Corneth, Mar 09 2015
From Peter Bala, May 26 2015: (Start)
The row polynomials R(n,x) satisfy (1 + x)*R(n,x) = (-1)^n*x*R(n,-(1 + x)).
For a fixed integer k, the expansion of the function A(k,z) := exp( Sum_{n >= 1} R(n,k)*z^n/n ) has integer coefficients and satisfies the functional equation A(k,z)^(k + 1) = BINOMIAL(A(k,z))^k, where BINOMIAL(F(z))= 1/(1 - z)*F(z/(1 - z)) denotes the binomial transform of the o.g.f. F(z). Cf. A145901. For cases see A084784 (k = 1), A090352 (k = 2), A090355 (k = 3), A090357 (k = 4), A090362 (k = 5) and A084785 (k = -2 with z -> -z).
A(k,z)^(k + 1) = A(-(k + 1),-z)^k and hence BINOMIAL(A(k,z)) = A(-(k + 1),-z). (End)
From Tom Copeland, Oct 19 2016: (Start)
Let a(1) = 1 + x + B(1) = x + 1/2 and a(n) = B(n) = (B.)^n, where B(n) are the Bernoulli numbers defined by e^(B.t) = t / (e^t-1), then t / e^(a.t) = t / [(x + 1) * t + exp(B.t)] = (e^t - 1) /[ 1 + (x + 1) (e^t - 1)] = exp(p.(x)t), where (p.(x))^n = p_n(x) are the shifted, signed row polynomials of this array: p_0(x) = 0, p_1(x) = 1, p_2(x) = -(1 + 2 x), p_3(x) = 1 + 6 x + 6 x^2, ... and p_n(x) = n * b(n-1), where b(n) are the partition polynomials of A133314 evaluated with these a(n).
Sum_{n > 0} R(n,-1/2) x^n/n! = 2 * tanh(x/2), where R(n,x) = Sum_{k = 1..n} T(n,k) x^(k-1) are the shifted row polynomials of this entry, so R(n,-1/2) = 4 * (2^(n+1)-1) B(n+1)/(n+1). (Cf. A000182.)
(End)
Also the Bernoulli numbers are given by B(n) = Sum_{k =1..n} (-1)^k T(n,k) / (k+1). - Tom Copeland, Nov 06 2016
G.f. for column k: k! x^k / Product_{i=1..k} (1-i*x). - Robert A. Russell, Sep 25 2018
a(j) <= A183109(j). - Manfred Boergens, Jul 25 2021

A131689 Triangle of numbers T(n,k) = k!Stirling2(n,k) = A000142(k)A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 6, 6, 0, 1, 14, 36, 24, 0, 1, 30, 150, 240, 120, 0, 1, 62, 540, 1560, 1800, 720, 0, 1, 126, 1806, 8400, 16800, 15120, 5040, 0, 1, 254, 5796, 40824, 126000, 191520, 141120, 40320, 0, 1, 510, 18150, 186480, 834120, 1905120, 2328480, 1451520, 362880

Offset: 0

Philippe Deléham, Sep 14 2007

Triangle T(n,k), 0 <= k <= n, read by rows given by [0,1,0,2,0,3,0,4,0,5,0,6,0,7,0,...] DELTA [1,1,2,2,3,3,4,4,5,5,6,6,...] where DELTA is the operator defined in A084938; another version of A019538.
See also A019538: version with n > 0 and k > 0. - Philippe Deléham, Nov 03 2008
From Peter Bala, Jul 21 2014: (Start)
T(n,k) gives the number of (k-1)-dimensional faces in the interior of the first barycentric subdivision of the standard (n-1)-dimensional simplex. For example, the barycentric subdivision of the 1-simplex is o--o--o, with 1 interior vertex and 2 interior edges, giving T(2,1) = 1 and T(2,2) = 2.
This triangle is used when calculating the face vectors of the barycentric subdivision of a simplicial complex. Let S be an n-dimensional simplicial complex and write f_k for the number of k-dimensional faces of S, with the usual convention that f_(-1) = 1, so that F := (f_(-1), f_0, f_1,...,f_n) is the f-vector of S. If M(n) denotes the square matrix formed from the first n+1 rows and n+1 columns of the present triangle, then the vector F*M(n) is the f-vector of the first barycentric subdivision of the simplicial complex S (Brenti and Welker, Lemma 2.1). For example, the rows of Pascal's triangle A007318 (but with row and column indexing starting at -1) are the f-vectors for the standard n-simplexes. It follows that A007318*A131689, which equals A028246, is the array of f-vectors of the first barycentric subdivision of standard n-simplexes. (End)
This triangle T(n, k) appears in the o.g.f. G(n, x) = Sum_{m>=0} S(n, m)*x^m with S(n, m) = Sum_{j=0..m} j^n for n >= 1 as G(n, x) = Sum_{k=1..n} (x^k/(1 - x)^(k+2))*T(n, k). See also the Eulerian triangle A008292 with a Mar 31 2017 comment for a rewritten form. For the e.g.f. see A028246 with a Mar 13 2017 comment. - Wolfdieter Lang, Mar 31 2017
T(n,k) = the number of alignments of length k of n strings each of length 1. See Slowinski. An example is given below. Cf. A122193 (alignments of strings of length 2) and A299041 (alignments of strings of length 3). - Peter Bala, Feb 04 2018
The row polynomials R(n,x) are the Fubini polynomials. - Emanuele Munarini, Dec 05 2020
From Gus Wiseman, Feb 18 2022: (Start)
Also the number of patterns of length n with k distinct parts (or with maximum part k), where we define a pattern to be a finite sequence covering an initial interval of positive integers. For example, row n = 3 counts the following patterns:
  (1,1,1)  (1,2,2)  (1,2,3)
           (2,1,2)  (1,3,2)
           (2,2,1)  (2,1,3)
           (1,1,2)  (2,3,1)
           (1,2,1)  (3,1,2)
           (2,1,1)  (3,2,1)
(End)
Regard A048994 as a lower-triangular matrix and divide each term A048994(n,k) by n!, then this is the matrix inverse. Because Sum_{k=0..n} (A048994(n,k) * x^n / n!) = A007318(x,n), Sum_{k=0..n} (A131689(n,k) * A007318(x,k)) = x^n. - Natalia L. Skirrow, Mar 23 2023
T(n,k) is the number of ordered partitions of [n] into k blocks. - Alois P. Heinz, Feb 21 2025

			The triangle T(n,k) begins:
  n\k 0 1    2     3      4       5        6        7        8        9      10 ...
  0:  1
  1:  0 1
  2:  0 1    2
  3:  0 1    6     6
  4:  0 1   14    36     24
  5:  0 1   30   150    240     120
  6:  0 1   62   540   1560    1800      720
  7:  0 1  126  1806   8400   16800    15120     5040
  8:  0 1  254  5796  40824  126000   191520   141120    40320
  9:  0 1  510 18150 186480  834120  1905120  2328480  1451520   362880
  10: 0 1 1022 55980 818520 5103000 16435440 29635200 30240000 16329600 3628800
  ... reformatted and extended. - _Wolfdieter Lang_, Mar 31 2017
From _Peter Bala_, Feb 04 2018: (Start)
T(4,2) = 14 alignments of length 2 of 4 strings of length 1. Examples include
  (i) A -    (ii) A -    (iii) A -
      B -         B -          - B
      C -         - C          - C
      - D         - D          - D
There are C(4,1) = 4 alignments of type (i) with a single gap character - in column 1, C(4,2) = 6 alignments of type (ii) with two gap characters in column 1 and C(4,3) = 4 alignments of type (iii) with three gap characters in column 1, giving a total of 4 + 6 + 4 = 14 alignments. (End)

Vincenzo Librandi, Rows n = 0..100, flattened
Peter Bala, Deformations of the Hadamard product of power series
F. Brenti and V. Welker, f-vectors of barycentric subdivisions, arXiv:math/0606356 [math.CO], Math. Z., 259(4), 849-865, 2008.
M. Dukes and C. D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.
Germain Kreweras, Une dualité élémentaire souvent utile dans les problèmes combinatoires, Mathématiques et Sciences Humaines 3 (1963): 31-41.
Jerry Metzger and Thomas Richards, A Prisoner Problem Variation, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.7.
Massimo Nocentini, An algebraic and combinatorial study of some infinite sequences of numbers supported by symbolic and logic computation, PhD Thesis, University of Florence, 2019. See Ex. 36.
Mircea Dan Rus, Yet another note on notation, arXiv:2501.08762 [math.HO], 2025. See p. 6.
J. B. Slowinski, The Number of Multiple Alignments, Molecular Phylogenetics and Evolution 10:2 (1998), 264-266. doi:10.1006/mpev.1998.0522
M. Z. Spivey, On Solutions to a General Combinatorial Recurrence, J. Int. Seq. 14 (2011) # 11.9.7.
Wikipedia, Barycentric subdivision
Wikipedia, Simplicial complex
Wikipedia, Simplex
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Columns k=0..10 are A000007, A000012, A000918, A001117, A000919, A001118, A000920, A135456, A133068, A133360, A133132,
Case m=1 of the polynomials defined in A278073.
Cf. A000142 (diagonal), A000670 (row sums), A000012 (alternating row sums), A210029 (central terms).
Cf. A001286, A037960, A037961, A037962, A037963.
Cf. A008292, A028246 (o.g.f. and e.g.f. of sums of powers).
Cf. A019538, A122193, A299041.
A version for partitions is A116608, or by maximum A008284.
A version for compositions is A235998, or by maximum A048004.
Classes of patterns:
- A000142 = strict
- A005649 = anti-run, complement A069321
- A019536 = necklace
- A032011 = distinct multiplicities
- A060223 = Lyndon
- A226316 = (1,2,3)-avoiding, weakly A052709, complement A335515
- A296975 = aperiodic
- A345194 = alternating, up/down A350354, complement A350252
- A349058 = weakly alternating
- A351200 = distinct runs
- A351292 = distinct run-lengths
Cf. A007318, A097805, A335456, A335457, A344605, A349050.

function T(n, k)
    if k < 0 || k > n return 0 end
    if n == 0 && k == 0 return 1 end
    k*(T(n-1, k-1) + T(n-1, k))
end
for n in 0:7
    println([T(n, k) for k in 0:n])
end
# Peter Luschny, Mar 26 2020

A131689 := (n,k) -> Stirling2(n,k)*k!: # Peter Luschny, Sep 17 2011
# Alternatively:
A131689_row := proc(n) 1/(1-t*(exp(x)-1)); expand(series(%,x,n+1)); n!*coeff(%,x,n); PolynomialTools:-CoefficientList(%,t) end:
for n from 0 to 9 do A131689_row(n) od; # Peter Luschny, Jan 23 2017

t[n_, k_] := k!*StirlingS2[n, k]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 25 2014 *)
T[n_, k_] := If[n <= 0 || k <= 0, Boole[n == 0 && k == 0], Sum[(-1)^(i + k) Binomial[k, i] i^(n + k), {i, 0, k}]]; (* Michael Somos, Jul 08 2018 *)

{T(n, k) = if( n<0, 0, sum(i=0, k, (-1)^(k + i) * binomial(k, i) * i^n))};
/* Michael Somos, Jul 08 2018 */

@cached_function
def F(n): # Fubini polynomial
    R. = PolynomialRing(ZZ)
    if n == 0: return R(1)
    return R(sum(binomial(n, k)*F(n - k)*x for k in (1..n)))
for n in (0..9): print(F(n).list()) # Peter Luschny, May 21 2021

T(n,k) = k*(T(n-1,k-1) + T(n-1,k)) with T(0,0)=1. Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A000629(n), A033999(n), A000007(n), A000670(n), A004123(n+1), A032033(n), A094417(n), A094418(n), A094419(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6 respectively. [corrected by Philippe Deléham, Feb 11 2013]
Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A000142(n), A000670(n), A122704(n) for x=-1, 0, 1, 2 respectively. - Philippe Deléham, Oct 09 2007
Sum_{k=0..n} (-1)^k*T(n,k)/(k+1) = Bernoulli numbers A027641(n)/A027642(n). - Peter Luschny, Sep 17 2011
G.f.: F(x,t) = 1 + x*t + (x+x^2)*t^2/2! + (x+6*x^2+6*x^3)*t^3/3! + ... = Sum_{n>=0} R(n,x)*t^n/n!.
The row polynomials R(n,x) satisfy the recursion R(n+1,x) = (x+x^2)*R'(n,x) + x*R(n,x) where ' indicates differentiation with respect to x. - Philippe Deléham, Feb 11 2013
T(n,k) = [t^k] (n! [x^n] (1/(1-t*(exp(x)-1)))). - Peter Luschny, Jan 23 2017
The n-th row polynomial has the form x o x o ... o x (n factors), where o denotes the black diamond multiplication operator of Dukes and White. See also Bala, Example E8. - Peter Bala, Jan 08 2018

A001117 a(n) = 3^n - 3*2^n + 3.

Original entry on oeis.org

1, 0, 0, 6, 36, 150, 540, 1806, 5796, 18150, 55980, 171006, 519156, 1569750, 4733820, 14250606, 42850116, 128746950, 386634060, 1160688606, 3483638676, 10454061750, 31368476700, 94118013006, 282379204836, 847187946150, 2541664501740, 7625194831806

Offset: 0

N. J. A. Sloane

Differences of 0. Labeled ordered partitions into 3 parts.
Number of surjections from an n-element set onto a three-element set, with n >= 3. - Mohamed Bouhamida, Dec 15 2007
Let P(A) be the power set of an n-element set A and R be a relation on P(A) such that for all x, y of P(A), xRy if either 0) x is not a subset of y and y is not a subset of x and x and y are disjoint, or 1) x is a proper subset of y or y is a proper subset of x and x and y are intersecting. Then a(n+1) = |R|. - Ross La Haye, Mar 19 2009
For n>0, the number of rows of n colors using exactly three colors.  For n=3, the six rows are ABC, ACB, BAC, BCA, CAB, and CBA. - Robert A. Russell, Sep 25 2018

H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 212.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 33.
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).
J. F. Steffensen, Interpolation, 2nd ed., Chelsea, NY, 1950, see p. 54.
A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911, p. 31.

T. D. Noe, Table of n, a(n) for n = 0..200
John Elias, Illustration of initial terms: Interior Sierpinski triangle
K. S. Immink, Coding Schemes for Multi-Level Channels that are Intrinsically Resistant Against Unknown Gain and/or Offset Using Reference Symbols, 2013.
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.
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260.
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260. [Annotated scanned copy]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Leipzig, 1911.
A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911. [Annotated scans of pages 30-33 only]
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Cf. A000919, A001118.

Column 3 of A019538 (n>0).

with(combstruct):ZL:=[S,{S=Sequence(U,card=r),U=Set(Z,card>=1)}, labeled]: 1, seq(count(subs(r=3,ZL),size=m),m=1..25); # Zerinvary Lajos, Mar 09 2007
A001117:=-6/(z-1)/(3*z-1)/(2*z-1); # Conjectured by Simon Plouffe in his 1992 dissertation. Gives sequence except for three leading terms.

k=3; Prepend[Table[k!StirlingS2[n,k],{n,1,30}],1] (* Robert A. Russell, Sep 25 2018 *)

a(n)=3^n-3*2^n+3 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = [n=0] + 3!*S(n, 3).
E.g.f.: 1 + (exp(x)-1)^3.
For n>=3: a(n+1) = 3*a(n) + 3*(2^n - 2) = 3*a(n) + 3*A000918(n). - Geoffrey Critzer, Feb 27 2009
G.f.: (-1-11*x^2+6*x)/((x-1)*(3*x-1)*(2*x-1)). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009

Extended with formula and alternate description by Christian G. Bower, Aug 15 1998

Simpler description from Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 07 2001

A000919 a(n) = 4^n - C(4,3)3^n + C(4,2)2^n - C(4,1).

Original entry on oeis.org

0, 0, 0, 24, 240, 1560, 8400, 40824, 186480, 818520, 3498000, 14676024, 60780720, 249401880, 1016542800, 4123173624, 16664094960, 67171367640, 270232006800, 1085570781624, 4356217681200, 17466686971800, 69992221794000, 280345359228024, 1122510953731440

Offset: 1

N. J. A. Sloane

Differences of 0: 4!*S(n,4).
Number of surjections from an n-element set onto a four-element set. - David Wasserman, Jun 06 2007
Number of rows of n colors using exactly four colors.  For n=4, the 24 rows are the 24 permutations of ABCD. - Robert A. Russell, Sep 25 2018

H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 212.
K. S. Immink, Coding Schemes for Multi-Level Channels that are Intrinsically Resistant Against Unknown Gain and/or Offset Using Reference Symbols, http://www.exp-math.uni-essen.de/~immink/pdf/jsac13.pdf, 2013. [This link no longer works, but please do not delete this reference, for historical reasons. Michel Marcus has suggested that the Immink link below points to the published version of the original reference, and I agree. - N. J. A. Sloane, May 29 2023]
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 33.
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).
J. F. Steffensen, Interpolation, 2nd ed., Chelsea, NY, 1950, see p. 54.

T. D. Noe, Table of n, a(n) for n = 1..201
K. S. Immink, Coding Schemes for Multi-Level Channels that are Intrinsically Resistant Against Unknown Gain and/or Offset Using Reference Symbols, Electronics Letters 50(1):20-22, January 2014.
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260.
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260. [Annotated scanned copy]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Ethan Soloway, Megan Triplett, and Wenshi Zhao, On the Existence of Partition of the Hypercube Graph into 3 Initial Segments, arXiv:2501.05827 [math.CO], 2025. See p. 23.
A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911, p. 31.
A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911. [Annotated scans of pages 30-33 only]
Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).

Cf. A001117, A001118.

Column 4 of A019538.

with (combstruct):ZL:=[S,{S=Sequence(U,card=r),U=Set(Z,card>=1)}, labeled]: seq(count(subs(r=4,ZL),size=m),m=1..25); # Zerinvary Lajos, Mar 09 2007
A000919:=24/(z-1)/(3*z-1)/(2*z-1)/(4*z-1); # Simon Plouffe in his 1992 dissertation

nn = 25; CoefficientList[Series[24 x^3/((1 - x) (1 - 2 x) (1 - 3 x) (1 - 4 x)), {x, 0, nn}], x] (* T. D. Noe, Jun 20 2012 *)
k=4; Table[k!StirlingS2[n,k],{n,1,30}] (* Robert A. Russell, Sep 25 2018 *)

a(n) = 4!*stirling(n, 4, 2); \\ Altug Alkan, Sep 25 2018

G.f.: 24*x^3/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)).
a(n) = 4^n - binomial(4,3)*3^n + binomial(4,2)*2^n - binomial(4,1) = 24*A000453(n). - David Wasserman, Jun 06 2007
E.g.f.: (exp(x)-1)^4. - Geoffrey Critzer, Feb 11 2009
For n >= 4: a(n+1) = 4*a(n) + 4*(3^n - 3*2^n + 3) = 4*a(n) + 4*A001117(n). - Geoffrey Critzer, Feb 27 2009
a(n) = k!*S2(n,k), where k=4 is the number of colors and S2 is the Stirling subset number. - Robert A. Russell, Sep 25 2018

A000920 Differences of 0: 6!*Stirling2(n,6).

Original entry on oeis.org

0, 0, 0, 0, 0, 720, 15120, 191520, 1905120, 16435440, 129230640, 953029440, 6711344640, 45674188560, 302899156560, 1969147121760, 12604139926560, 79694820748080, 499018753280880, 3100376804676480, 19141689213218880, 117579844328562000

Offset: 1

N. J. A. Sloane

Number of surjections from an n-element set onto a six-element set, with n >= 6. - Mohamed Bouhamida, Dec 15 2007

Number of rows of n colors using exactly six colors. For n=6, the 720 rows are the 720 permutations of ABCDEF. - Robert A. Russell, Sep 25 2018

H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 212.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 33.
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).
J. F. Steffensen, Interpolation, 2nd ed., Chelsea, NY, 1950, see p. 54.
A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911, p. 31.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260.
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260. [Annotated scanned copy]
A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Leipzig, 1911.
A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911. [Annotated scans of pages 30-33 only]
Index entries for linear recurrences with constant coefficients, signature (21,-175,735,-1624,1764,-720).

Cf. A001117, A001118, A000918, A000919, A000770.

Column 6 of A019538.

[6^n-Binomial(6,5)*5^n+Binomial(6,4)*4^n-Binomial(6,3)*3^n+Binomial(6,2)*2^n-Binomial(6,1): n in [1..30]]; // Vincenzo Librandi, May 18 2015

720/(-1+z)/(6*z-1)/(4*z-1)/(3*z-1)/(2*z-1)/(5*z-1);

CoefficientList[Series[(720*x^5)/((x-1)*(6*x-1)*(4*x-1)*(3*x-1)*(2*x-1)*(5*x-1)),{x,0,30}],x] (* Vincenzo Librandi, Apr 11 2012 *)
k=6; Table[k!StirlingS2[n,k],{n,1,30}] (* Robert A. Russell, Sep 25 2018 *)

a(n) = 6!*stirling(n, 6, 2); \\ Altug Alkan, Sep 25 2018

a(n) = Sum((-1)^i*binomial(6, i)*(6-i)^n, i = 0 .. 5).
a(n) = 6^n-C(6,5)*5^n+C(6,4)*4^n-C(6,3)*3^n+C(6,2)*2^n-C(6,1) with n>=6. - Mohamed Bouhamida, Dec 15 2007
G.f.: 720*x^6/((x-1)*(6*x-1)*(4*x-1)*(3*x-1)*(2*x-1)*(5*x-1)). [Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009; checked and corrected by R. J. Mathar, Sep 16 2009]
a(n) = 720*A000770(n). - R. J. Mathar, Apr 30 2015
E.g.f.: (exp(x) - 1)^6. - Geoffrey Critzer, May 17 2015

A056456 Number of palindromes of length n using exactly five different symbols.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 120, 120, 1800, 1800, 16800, 16800, 126000, 126000, 834120, 834120, 5103000, 5103000, 29607600, 29607600, 165528000, 165528000, 901020120, 901020120, 4809004200, 4809004200, 25292030400

Offset: 1

Marks R. Nester

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2.]

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,14,-14,-71,71,154,-154,-120,120).

Cf. A056451, A001118.

k=5; Table[k! StirlingS2[Ceiling[n/2],k],{n,1,30}] (* Robert A. Russell, Sep 25 2018 *)
LinearRecurrence[{1, 14, -14, -71, 71, 154, -154, -120, 120}, {0, 0, 0, 0, 0, 0, 0, 0, 120}, 30] (* Vincenzo Librandi, Sep 29 2018 *)

a(n) = 5!*stirling((n+1)\2, 5, 2); \\ Altug Alkan, Sep 25 2018

a(n) = 5! * Stirling2( [(n+1)/2], 5).
G.f.: -120*x^9/((x-1)*(2*x-1)*(2*x+1)*(2*x^2-1)*(3*x^2-1)*(5*x^2-1)). - Colin Barker, Sep 03 2012
G.f.: k!(x^(2k-1)+x^(2k))/Product_{i=1..k}(1-ix^2), where k=5 is the number of symbols. - Robert A. Russell, Sep 25 2018

A135456 Number of surjections from an n-element set onto a seven-element set.

Original entry on oeis.org

5040, 141120, 2328480, 29635200, 322494480, 3162075840, 28805736960, 248619571200, 2060056318320, 16540688324160, 129568848121440, 995210916336000, 7524340159588560, 56163512390086080, 414847224363337920

Offset: 7

Mohamed Bouhamida, Dec 15 2007

G. C. Greubel, Table of n, a(n) for n = 7..1000
Index entries for linear recurrences with constant coefficients, signature (28,-322,1960,-6769,13132,-13068,5040).

Column k=7 of A019538 and A131689.

Cf. A000918, A000919, A000920, A001117, A001118.

LinearRecurrence[{28, -322, 1960, -6769, 13132, -13068, 5040}, {5040, 141120, 2328480, 29635200, 322494480, 3162075840, 28805736960}, 25] (* G. C. Greubel, Oct 14 2016 *)

a(n) = 7^n -C(7,6)*6^n +C(7,5)*5^n -C(7,4)*4^n +C(7,3)*3^n -C(7,2)*2^n +C(7,1) with n>=7.
a(n) = A000771(n) * 7!. - Max Alekseyev, Nov 12 2009
G.f.: -5040*x^7/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)). - Colin Barker, Oct 25 2012
E.g.f.: (exp(x) - 1)^7. - Ilya Gutkovskiy, Jun 19 2018

More terms from Max Alekseyev, Nov 12 2009

A056312 Number of reversible strings with n beads using exactly five different colors.

Original entry on oeis.org

0, 0, 0, 0, 60, 900, 8400, 63000, 417120, 2551560, 14804700, 82764900, 450518460, 2404510500, 12646078200, 65771496000, 339165516120, 1737486149760, 8855359634100, 44952367981500, 227475768907860, 1148269329527100, 5785013373810000, 29100047092479000

Offset: 1

Marks R. Nester

A string and its reverse are considered to be equivalent.

			For n=5, the 60 rows are 60 permutations of ABCDE that do not include any mutual reversals.  Each of the 60 chiral pairs, such as ABCDE-EDCBA, is then counted just once.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (13,-45,-75,695,-575,-3195,5595,4706,-14918,2160,12840,-7200).

Cf. A005418, A032120, A032121, A032122.
Column 5 of A305621.
Equals (A001118 + A056456) / 2 = A001118 - A305625 = A305625 + A056456.

[60*(StirlingSecond(n, 5)+StirlingSecond(Ceiling(n/2), 5)): n in [1..30]]; // Vincenzo Librandi, Sep 30 2018

k=5; Table[(StirlingS2[i,k]+StirlingS2[Ceiling[i/2],k])k!/2,{i,30}] (* Robert A. Russell, Nov 25 2017 *) adapted
CoefficientList[Series[-60*x^4*(120*x^7 - 17*x^6 - 50*x^5 - 32*x^4 + 20*x^3 + 10*x^2 - 2*x - 1)/((x - 1)*(2*x - 1)*(2*x + 1)*(3*x - 1)*(4*x - 1)*(5*x - 1)*(2*x^2 - 1)*(3*x^2 - 1)*(5*x^2 - 1)), {x, 0, 30}], x] (* Stefano Spezia, Sep 29 2018 *)

a(n) = 60*(stirling(n, 5, 2) + stirling(ceil(n/2), 5, 2)); \\ Altug Alkan, Sep 27 2018

a(n) = A032122(n) - 5*A032121(n) + 10*A032120(n) - 10*A005418(n+1) + 5.
G.f.: -60*x^5*(120*x^7 - 17*x^6 - 50*x^5 - 32*x^4 + 20*x^3 + 10*x^2 - 2*x - 1)/((x - 1)*(2*x - 1)*(2*x + 1)*(3*x - 1)*(4*x - 1)*(5*x - 1)*(2*x^2 - 1)*(3*x^2 - 1)*(5*x^2 - 1)). [Colin Barker, Sep 03 2012]
a(n) = k! (S2(n,k) + S2(ceiling(n/2),k)) / 2, where k=5 is the number of colors and S2 is the Stirling subset number. - Robert A. Russell, Sep 25 2018

A133068 Number of surjections from an n-element set to an eight-element set.

Original entry on oeis.org

40320, 1451520, 30240000, 479001600, 6411968640, 76592355840, 843184742400, 8734434508800, 86355926616960, 823172919528960, 7621934141203200, 68937160460313600, 611692004959217280, 5342844138794426880, 46061530905262118400, 392795626402384128000

Offset: 8

Mohamed Bouhamida, Dec 16 2007

G. C. Greubel, Table of n, a(n) for n = 8..1000
Index entries for linear recurrences with constant coefficients, signature (36,-546,4536,-22449,67284,-118124,109584,-40320).

Column k=8 of A019538 and A131689.

Cf. A000918, A000919, A000920, A001117, A001118, A135456.

[&+[(-1)^(8-k)*Binomial(8, k)*k^n: k in [1..n]]: n in [8..25]]; // Vincenzo Librandi, Oct 21 2017

CoefficientList[Series[40320*x^8/((x - 1)*(2*x - 1)*(3*x - 1)*(4*x - 1)*(5*x - 1)*(6*x - 1)*(7*x - 1)*(8*x - 1)), {x, 0, 50}], x] (* G. C. Greubel, Oct 20 2017 *)
Table[Sum[(-1)^(8 - k)*Binomial[8, k]*k^n, {k, 1, 8}], {n, 8, 20}] (* G. C. Greubel, Oct 21 2017 *)

x='x+O('x^50); Vec(40320*x^8/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1))) \\ G. C. Greubel, Oct 20 2017

a(n) = Sum_{k=1..8} ((-1)^(8-k)*binomial(8,k)*k^n).
a(n) = A049434(n) * 8!. - Max Alekseyev, Nov 13 2009
G.f.: 40320*x^8/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1)). - Colin Barker, Oct 25 2012
E.g.f.: (exp(x) - 1)^8. - Ilya Gutkovskiy, Jun 19 2018

Edited by N. J. A. Sloane, Jul 12 2008 at the suggestion of R. J. Mathar

More terms from Max Alekseyev, Nov 13 2009

cp's OEIS Frontend

A000918 a(n) = 2^n - 2.

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A019538 Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= k <= n).

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A131689 Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.

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A001117 a(n) = 3^n - 3*2^n + 3.

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A000919 a(n) = 4^n - C(4,3)*3^n + C(4,2)*2^n - C(4,1).

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A000920 Differences of 0: 6!*Stirling2(n,6).

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A131689 Triangle of numbers T(n,k) = k!Stirling2(n,k) = A000142(k)A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.

A000919 a(n) = 4^n - C(4,3)3^n + C(4,2)2^n - C(4,1).