A092477 -id:A092477 - cp's OEIS frontend

A000244 Powers of 3: a(n) = 3^n.

1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907, 43046721, 129140163, 387420489, 1162261467, 3486784401, 10460353203, 31381059609, 94143178827, 282429536481, 847288609443, 2541865828329, 7625597484987

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 3), L(1, 3), P(1, 3), T(1, 3). Essentially same as Pisot sequences E(3, 9), L(3, 9), P(3, 9), T(3, 9). See A008776 for definitions of Pisot sequences.
Number of (s(0), s(1), ..., s(2n+2)) such that 0 < s(i) < 6 and |s(i) - s(i-1)| = 1 for i = 1, 2, ..., 2n + 2, s(0) = 1, s(2n+2) = 3. - Herbert Kociemba, Jun 10 2004
a(1) = 1, a(n+1) is the least number such that there are a(n) even numbers between a(n) and a(n+1). Generalization for the sequence of powers of k: 1, k, k^2, k^3, k^4, ... There are a(n) multiples of k-1 between a(n) and a(n+1). - Amarnath Murthy, Nov 28 2004
a(n) = sum of (n+1)-th row in Triangle A105728. - Reinhard Zumkeller, Apr 18 2005
With p(n) being the number of integer partitions of n, p(i) being the number of parts of the i-th partition of n, d(i) being the number of different parts of the i-th partition of n, m(i, j) being the multiplicity of the j-th part of the i-th partition of n, Sum_{i = 1..p(n)} being the sum over i and Product_{j = 1..d(i)} being the product over j, one has: a(n) = Sum_{i = 1..p(n)} (p(i)!/(Product_{j = 1..d(i)} m(i, j)!))*2^(p(i) - 1). - Thomas Wieder, May 18 2005
For any k > 1 in the sequence, k is the first prime power appearing in the prime decomposition of repunit R_k, i.e., of A002275(k). - Lekraj Beedassy, Apr 24 2006
a(n-1) is the number of compositions of compositions. In general, (k+1)^(n-1) is the number of k-levels nested compositions (e.g., 4^(n-1) is the number of compositions of compositions of compositions, etc.). Each of the n - 1 spaces between elements can be a break for one of the k levels, or not a break at all. - Franklin T. Adams-Watters, Dec 06 2006
Let S be a binary relation on the power set P(A) of a set A having n = |A| elements such that for every element x, y of P(A), xSy if x is a subset of y. Then a(n) = |S|. - Ross La Haye, Dec 22 2006
From Manfred Boergens, Mar 28 2023: (Start)
With regard to the comment by Ross La Haye:
Cf. A001047 if either nonempty subsets are considered or x is a proper subset of y.
Cf. a(n+1) in A028243 if nonempty subsets are considered and x is a proper subset of y. (End)
If X_1, X_2, ..., X_n is a partition of the set {1, 2, ..., 2*n} into blocks of size 2 then, for n >= 1, a(n) is equal to the number of functions f : {1, 2, ..., 2*n} -> {1, 2} such that for fixed y_1, y_2, ..., y_n in {1, 2} we have f(X_i) <> {y_i}, (i = 1, 2, ..., n). - Milan Janjic, May 24 2007
This is a general comment on all sequences of the form a(n) = [(2^k)-1]^n for all positive integers k. Example 1.1.16 of Stanley's "Enumerative Combinatorics" offers a slightly different version. a(n) in the number of functions f:[n] into P([k]) - {}. a(n) is also the number of functions f:[k] into P([n]) such that the generalized intersection of f(i) for all i in [k] is the empty set. Where [n] = {1, 2, ..., n}, P([n]) is the power set of [n] and {} is the empty set. - Geoffrey Critzer, Feb 28 2009
a(n) = A064614(A000079(n)) and A064614(m)A000079(n). - Reinhard Zumkeller, Feb 08 2010
3^(n+1) = (1, 2, 2, 2, ...) dot (1, 1, 3, 9, ..., 3^n); e.g., 3^3 = 27 = (1, 2, 2, 2) dot (1, 1, 3, 9) = (1 + 2 + 6 + 18). - Gary W. Adamson, May 17 2010
a(n) is the number of generalized compositions of n when there are 3*2^i different types of i, (i = 1, 2, ...). - Milan Janjic, Sep 24 2010
For n >= 1, a(n-1) is the number of generalized compositions of n when there are 2^(i-1) different types of i, (i = 1, 2, ...). - Milan Janjic, Sep 24 2010
The sequence in question ("Powers of 3") also describes the number of moves of the k-th disk solving the [RED ; BLUE ; BLUE] or [RED ; RED ; BLUE] pre-colored Magnetic Tower of Hanoi puzzle (cf. A183111 - A183125).
a(n) is the number of Stern polynomials of degree n. See A057526. - T. D. Noe, Mar 01 2011
Positions of records in the number of odd prime factors, A087436. - Juri-Stepan Gerasimov, Mar 17 2011
Sum of coefficients of the expansion of (1+x+x^2)^n. - Adi Dani, Jun 21 2011
a(n) is the number of compositions of n elements among {0, 1, 2}; e.g., a(2) = 9 since there are the 9 compositions 0 + 0, 0 + 1, 1 + 0, 0 + 2, 1 + 1, 2 + 0, 1 + 2, 2 + 1, and 2 + 2. [From Adi Dani, Jun 21 2011; modified by editors.]
Except the first two terms, these are odd numbers n such that no x with 2 <= x <= n - 2 satisfy x^(n-1) == 1 (mod n). - Arkadiusz Wesolowski, Jul 03 2011
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 3-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
Explanation from David Applegate, Feb 20 2017: (Start)
Since the preceding comment appears in a large number of sequences, it might be worth adding a proof.
The number of compositions of n into exactly k parts is binomial(n-1,k-1).
For a p-colored composition of n such that no adjacent parts have the same color, there are exactly p choices for the color of the first part, and p-1 choices for the color of each additional part (any color other than the color of the previous one). So, for a partition into k parts, there are p (p-1)^(k-1) valid colorings.
Thus the number of p-colored compositions of n into exactly k parts such that no adjacent parts have the same color is binomial(n-1,k-1) p (p-1)^(k-1).
The total number of p-colored compositions of n such that no adjacent parts have the same color is then
Sum_{k=1..n} binomial(n-1,k-1) * p * (p-1)^(k-1) = p^n.
To see this, note that the binomial expansion of ((p - 1) + 1)^(n - 1) = Sum_{k = 0..n - 1} binomial(n - 1, k) (p - 1)^k 1^(n - 1 - k) = Sum_{k = 1..n} binomial(n - 1, k - 1) (p - 1)^(k - 1).
(End)
Also, first and least element of the matrix [1, sqrt(2); sqrt(2), 2]^(n+1). - M. F. Hasler, Nov 25 2011
One-half of the row sums of the triangular version of A035002. - J. M. Bergot, Jun 10 2013
Form an array with m(0,n) = m(n,0) = 2^n; m(i,j) equals the sum of the terms to the left of m(i,j) and the sum of the terms above m(i,j), which is m(i,j) = Sum_{k=0..j-1} m(i,k) + Sum_{k=0..i-1} m(k,j). The sum of the terms in antidiagonal(n+1) = 4*a(n). - J. M. Bergot, Jul 10 2013
a(n) = A007051(n+1) - A007051(n), and A007051 are the antidiagonal sums of an array defined by m(0,k) = 1 and m(n,k) = Sum_{c = 0..k - 1} m(n, c) + Sum_{r = 0..n - 1} m(r, k), which is the sum of the terms to left of m(n, k) plus those above m(n, k). m(1, k) = A000079(k); m(2, k) = A045623(k + 1); m(k + 1, k) = A084771(k). - J. M. Bergot, Jul 16 2013
Define an array to have m(0,k) = 2^k and m(n,k) = Sum_{c = 0..k - 1} m(n, c) + Sum_{r = 0..n - 1} m(r, k), which is the sum of the terms to the left of m(n, k) plus those above m(n, k). Row n = 0 of the array comprises A000079, column k = 0 comprises A011782, row n = 1 comprises A001792. Antidiagonal sums of the array are a(n): 1 = 3^0, 1 + 2 = 3^1, 2 + 3 + 4 = 3^2, 4 + 7 + 8 + 8 = 3^3. - J. M. Bergot, Aug 02 2013
The sequence with interspersed zeros and o.g.f. x/(1 - 3*x^2), A(2*k) = 0, A(2*k + 1) = 3^k = a(k), k >= 0, can be called hexagon numbers. This is because the algebraic number rho(6) = 2*cos(Pi/6) = sqrt(3) of degree 2, with minimal polynomial C(6, x) = x^2 - 3 (see A187360, n = 6), is the length ratio of the smaller diagonal and the side in the hexagon. Hence rho(6)^n = A(n-1)*1 + A(n)*rho(6), in the power basis of the quadratic number field Q(rho(6)). One needs also A(-1) = 1. See also a Dec 02 2010 comment and the P. Steinbach reference given in A049310. - Wolfdieter Lang, Oct 02 2013
Numbers k such that sigma(3k) = 3k + sigma(k). - Jahangeer Kholdi, Nov 23 2013
All powers of 3 are perfect totient numbers (A082897), since phi(3^n) = 2 * 3^(n - 1) for n > 0, and thus Sum_{i = 0..n} phi(3^i) = 3^n. - Alonso del Arte, Apr 20 2014
The least number k > 0 such that 3^k ends in n consecutive decreasing digits is a 3-term sequence given by {1, 13, 93}. The consecutive increasing digits are {3, 23, 123}. There are 100 different 3-digit endings for 3^k. There are no k-values such that 3^k ends in '012', '234', '345', '456', '567', '678', or '789'. The k-values for which 3^k ends in '123' are given by 93 mod 100. For k = 93 + 100*x, the digit immediately before the run of '123' is {9, 5, 1, 7, 3, 9, 5, 1, 3, 7, ...} for x = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...}, respectively. Thus we see the digit before '123' will never be a 0. So there are no further terms. - Derek Orr, Jul 03 2014
All elements of A^n where A = (1, 1, 1; 1, 1, 1; 1, 1, 1). - David Neil McGrath, Jul 23 2014
Counts all walks of length n (open or closed) on the vertices of a triangle containing a loop at each vertex starting from any given vertex. - David Neil McGrath, Oct 03 2014
a(n) counts walks (closed) on the graph G(1-vertex;1-loop,1-loop,1-loop). - David Neil McGrath, Dec 11 2014
2*a(n-2) counts all permutations of a solitary closed walk of length (n) from the vertex of a triangle that contains 2 loops on each of the remaining vertices. In addition, C(m,k)=2*(2^m)*B(m+k-2,m) counts permutations of walks that contain (m) loops and (k) arcs. - David Neil McGrath, Dec 11 2014
a(n) is the sum of the coefficients of the n-th layer of Pascal's pyramid (a.k.a., Pascal's tetrahedron - see A046816). - Bob Selcoe, Apr 02 2016
Numbers n such that the trinomial x^(2*n) + x^n + 1 is irreducible over GF(2). Of these only the trinomial for n=1 is primitive. - Joerg Arndt, May 16 2016
Satisfies Benford's law [Berger-Hill, 2011]. - N. J. A. Sloane, Feb 08 2017
a(n-1) is also the number of compositions of n if the parts can be runs of any length from 1 to n, and can contain any integers from 1 to n. - Gregory L. Simay, May 26 2017
Also the number of independent vertex sets and vertex covers in the n-ladder rung graph n P_2. - Eric W. Weisstein, Sep 21 2017
Also the number of (not necessarily maximal) cliques in the n-cocktail party graph. - Eric W. Weisstein, Nov 29 2017
a(n-1) is the number of 2-compositions of n; see Hopkins & Ouvry reference. - Brian Hopkins, Aug 15 2020
a(n) is the number of faces of any dimension (vertices, edges, square faces, etc.) of the n-dimensional hypercube. For example, the 0-dimensional hypercube is a point, and its only face is itself. The 1-dimensional hypercube is a line, which has two vertices and an edge. The 2-dimensional hypercube is a square, which has four vertices, four edges, and a square face. - Kevin Long, Mar 14 2023
Number of pairs (A,B) of subsets of M={1,2,...,n} with union(A,B)=M. For nonempty subsets cf. A058481. - Manfred Boergens, Mar 28 2023
From Jianing Song, Sep 27 2023: (Start)
a(n) is the number of disjunctive clauses of n variables up to equivalence. A disjunctive clause is a propositional formula of the form l_1 OR ... OR l_m, where l_1, ..., l_m are distinct elements in {x_1, ..., x_n, NOT x_1, ..., NOT x_n} for n variables x_1, ... x_n, and no x_i and NOT x_i appear at the same time. For each 1 <= i <= n, we can have neither of x_i or NOT x_i, only x_i or only NOT x_i appearing in a disjunctive clause, so the number of such clauses is 3^n. Viewing the propositional formulas of n variables as functions {0,1}^n -> {0,1}, a disjunctive clause corresponds to a function f such that the inverse image of 0 is of the form A_1 X ... X A_n, where A_i is nonempty for all 1 <= i <= n. Since each A_i has 3 choices ({0}, {1} or {0,1}), we also find that the number of disjunctive clauses of n variables is 3^n.
Equivalently, a(n) is the number of conjunctive clauses of n variables. (End)
The finite subsequence a(2), a(3), a(4), a(5) = 9, 27, 81, 243 is one of only two geometric sequences that can be formed with all interior angles (all integer, in degrees) of a simple polygon. The other sequence is a subsequence of A007283 (see comment there). - Felix Huber, Feb 15 2024

			G.f. = 1 + 3*x + 9*x^2 + 27*x^3 + 81*x^4 + 243*x^5 + 729*x^6 + 2187*x^7 + ...

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..200
T. Banchoff, Counting the Faces of Higher-Dimensional Cubes, Beyond the Third Dimension: Geometry, computer graphics and higher dimensions, Scientific American Library, 1996.
Arno Berger and Theodore P. Hill, Benford's law strikes back: no simple explanation in sight for mathematical gem, The Mathematical Intelligencer 33.1 (2011): 85-91.
A. Bostan, Computer Algebra for Lattice Path Combinatorics, Séminaire de Combinatoire Ph. Flajolet, Mar 28 2013.
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
F. Javier de Vega, An extension of Furstenberg's theorem of the infinitude of primes, arXiv:2003.13378 [math.NT], 2020.
Nachum Dershowitz, Between Broadway and the Hudson: A Bijection of Corridor Paths, arXiv:2006.06516 [math.CO], 2020.
Joël Gay and Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018.
Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 7
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 268
Milan Janjic, Enumerative Formulae for Some Functions on Finite Sets
Tanya Khovanova, Recursive Sequences
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.
László Németh, The trinomial transform triangle, J. Int. Seqs., Vol. 21 (2018), Article 18.7.3. Also arXiv:1807.07109 [math.NT], 2018.
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
Yash Puri and Thomas Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Eric Weisstein's World of Mathematics, Clique.
Eric Weisstein's World of Mathematics, Cocktail Party Graph.
Eric Weisstein's World of Mathematics, Hanoi Graph.
Eric Weisstein's World of Mathematics, Independent Vertex Set.
Eric Weisstein's World of Mathematics, Ladder Rung Graph.
Eric Weisstein's World of Mathematics, Sierpiński Gasket Graph.
Eric Weisstein's World of Mathematics, Vertex Cover.
Doron Zeilberger, The Amazing 3^n Theorem and its even more Amazing Proof [Discovered by Xavier G. Viennot and his École Bordelaise gang], arXiv:1208.2258, 2012.
Index entries for "core" sequences
Index entries for related partition-counting sequences
Index entries for linear recurrences with constant coefficients, signature (3).
Index entries for sequences related to Benford's law

Cf. A008776 (2*a(n), and first differences).
a(n) = A092477(n, 2) for n > 0.
a(n) = A159991(n) / A009964(n).
Cf. A100772, A035002. Row sums of A125076 and A153279.
a(n) = A217764(0, n).
Cf. A046816, A006521, A014945, A275414 (multisets).
The following are parallel families: A000079 (2^n), A004094 (2^n reversed), A028909 (2^n sorted up), A028910 (2^n sorted down), A036447 (double and reverse), A057615 (double and sort up), A263451 (double and sort down); A000244 (3^n), A004167 (3^n reversed), A321540 (3^n sorted up), A321539 (3^n sorted down), A163632 (triple and reverse), A321542 (triple and sort up), A321541 (triple and sort down).
Cf. A001047, A028243, A058481.

a000244 = (3 ^)  -- Reinhard Zumkeller, Nov 14 2011
a000244_list = iterate (* 3) 1  -- Reinhard Zumkeller, Apr 04 2012

[ 3^n : n in [0..30] ]; // Wesley Ivan Hurt, Jul 04 2014

A000244 := n->3^n; [ seq(3^n, n=0..50) ];
A000244:=-1/(-1+3*z); # Simon Plouffe in his 1992 dissertation.

Table[3^n, {n, 0, 30}] (* Stefan Steinerberger, Apr 01 2006 *)
3^Range[0, 30] (* Wesley Ivan Hurt, Jul 04 2014 *)
LinearRecurrence[{3}, {1}, 20] (* Eric W. Weisstein, Sep 21 2017 *)
CoefficientList[Series[1/(1 - 3 x), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 21 2017 *)
NestList[3#&,1,30] (* Harvey P. Dale, Feb 20 2020 *)

makelist(3^n, n, 0, 30); /* Martin Ettl, Nov 05 2012 */

A000244(n) = 3^n \\ Michael B. Porter, Nov 03 2009

def A000244(n): return 3**n # Chai Wah Wu, Nov 10 2022

val powersOf3: LazyList[BigInt] = LazyList.iterate(1: BigInt)(_ * 3)
(0 to 26).map(powersOf3()) // _Alonso del Arte, May 03 2020

a(n) = 3^n.
a(0) = 1; a(n) = 3*a(n-1).
G.f.: 1/(1-3*x).
E.g.f.: exp(3*x).
a(n) = n!*Sum_{i + j + k = n, i, j, k >= 0} 1/(i!*j!*k!). - Benoit Cloitre, Nov 01 2002
a(n) = Sum_{k = 0..n} 2^k*binomial(n, k), binomial transform of A000079.
a(n) = A090888(n, 2). - Ross La Haye, Sep 21 2004
a(n) = 2^(2n) - A005061(n). - Ross La Haye, Sep 10 2005
a(n) = A112626(n, 0). - Ross La Haye, Jan 11 2006
Hankel transform of A007854. - Philippe Deléham, Nov 26 2006
a(n) = 2*StirlingS2(n+1,3) + StirlingS2(n+2,2) = 2*(StirlingS2(n+1,3) + StirlingS2(n+1,2)) + 1. - Ross La Haye, Jun 26 2008
a(n) = 2*StirlingS2(n+1, 3) + StirlingS2(n+2, 2) = 2*(StirlingS2(n+1, 3) + StirlingS2(n+1, 2)) + 1. - Ross La Haye, Jun 09 2008
Sum_{n >= 0} 1/a(n) = 3/2. - Gary W. Adamson, Aug 29 2008
If p(i) = Fibonacci(2i-2) and if A is the Hessenberg matrix of order n defined by A(i, j) = p(j-i+1), (i <= j), A(i, j) = -1, (i = j+1), and A(i, j) = 0 otherwise, then, for n >= 1, a(n-1) = det A. - Milan Janjic, May 08 2010
G.f. A(x) = M(x)/(1-M(x))^2, M(x) - o.g.f for Motzkin numbers (A001006). - Vladimir Kruchinin, Aug 18 2010
a(n) = A133494(n+1). - Arkadiusz Wesolowski, Jul 27 2011
2/3 + 3/3^2 + 2/3^3 + 3/3^4 + 2/3^5 + ... = 9/8. [Jolley, Summation of Series, Dover, 1961]
a(n) = Sum_{k=0..n} A207543(n,k)*4^(n-k). - Philippe Deléham, Feb 25 2012
a(n) = Sum_{k=0..n} A125185(n,k). - Philippe Deléham, Feb 26 2012
Sum_{n > 0} Mobius(n)/a(n) = 0.181995386702633887827... (see A238271). - Alonso del Arte, Aug 09 2012. See also the sodium 3s orbital energy in table V of J. Chem. Phys. 53 (1970) 348.
a(n) = (tan(Pi/3))^(2*n). - Bernard Schott, May 06 2022
a(n-1) = binomial(2*n-1, n) + Sum_{k >= 1} binomial(2*n, n+3*k)*(-1)^k. - Greg Dresden, Oct 14 2022
G.f.: Sum_{k >= 0} x^k/(1-2*x)^(k+1). - Kevin Long, Mar 14 2023

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).

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 89, ex. 1; also p. 210.
Miklos Bona, Combinatorics of Permutations, Chapman and Hall,2004, p.12.
G. Boole, A Treatise On The Calculus of Finite Differences, Dover Publications, 1960, p. 20.
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.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, 1989, p. 155. Also eqs.(6.10) and (6.37).
Kiran S. Kedlaya and Andrew V. Sutherland, Computing L -Series of Hyperelliptic Curves in Algorithmic Number Theory Lecture Notes in Computer Science Volume 5011/2008.
T. K. Petersen, Eulerian Numbers, Birkhauser, 2015, Section 5.6.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 33.
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.
E. Whittaker and G. Robinson, The Calculus of Observations, Blackie, London, 4th ed., 1949; p. 7.

T. D. Noe, Rows n = 1..100 of triangle, flattened
N. Arkani-Hamed, Y. Bai, S. He, and G. Yan, Scattering forms and the positive geometry of kinematics, color, and the worldsheet , arXiv:1711.09102 [hep-th], 2017. [From _Tom Copeland_, Jun 24 2018]
Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141. (See Theorem 2.1 (iv), p. 131.)
J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure, arXiv:1307.2010 [math.CO], 2013.
J. F. Barbero G., J. Salas and E. J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. II. Applications, arXiv preprint arXiv:1307.5624 [math.CO], 2013.
Paul Barry, On the f-Matrices of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1805.02274 [math.CO], 2018.
Moussa Benoumhani, The Number of Topologies on a Finite Set, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.6.
Olivier Bodini, Antoine Genitrini, Cécile Mailler, and Mehdi Naima, Strict monotonic trees arising from evolutionary processes: combinatorial and probabilistic study, hal-02865198 [math.CO] / [math.PR] / [cs.DS] / [cs.DM], 2020.
K. N. Boyadzhiev, A Series transformation formula and related polynomials, Int. J. Math. Math. Sc. 23 (2005), 3849-3866.
Xavier Bultel, Jannik Dreier, Matthieu Giraud, Marie Izaute, Timothée Kheyrkhah, Pascal Lafourcade, Dounia Lakhzoum, Vincent Marlin, and Ladislav Motá, Security Analysis and Psychological Study of Authentication Methods with PIN Codes, RCIS 2018 - IEEE 12th International Conference on Research Challenges in Information Science, 2018.
J. L. Chandon, J. LeMaire, and J. Pouget, Dénombrement des quasi-ordres sur un ensemble fini, Math. Sci. Humaines, No. 62 (1978), 61-80.
Frédéric Chapoton and Vincent Pilaud, Shuffles of deformed permutahedra, multiplihedra, constrainahedra, and biassociahedra, arXiv:2201.06896 [math.CO], 2022. See p. 12, 19.
Tom Copeland, The Bernoulli polynomials and Hirzebruch's generalized Todd class
E. Delucchi, A. Pixton, and L. Sabalka, Face vectors of subdivided simplicial complexes Discrete Math. 312 (2012), no. 2, 248--257. MR2852583 (2012j:05470). See p. 250. - _N. J. A. Sloane_, Apr 04 2014
I. Dolgachev and V. Lunts, A character formula for the representation of the Weyl group in the cohomology of the associated toric variety, Journal of Algebra, 168, 741-772, (1994).
Loïc Foissy and Frédéric Patras, Lie theory of free cocommutative and commutative cofree Hopf algebras, hal-04773293, 2024. See p. 42.
S. Fomin and N. Reading, Root systems and generalized associahedra, Lecture notes for IAS/Park-City 2004; arXiv:math/0505518 [math.CO], 2005-2008.
S. Franco and A. Hasan, Graded Quivers, Generalized Dimer Models and Toric Geometry , arXiv preprint arXiv:1904.07954 [hep-th], 2019.
T. L. Friedman and P. Klingsberg, Combinatorial identities by way of Wilf's multigraph model, Int. J. Math. Math. Sci. (2006) 96327, defn. 4.3.
X. Gao, S. He, and Y. Zhan, Labelled tree graphs, Feynman diagrams and disk integrals , arXiv:1708.08701 [hep-th], 2017. [From _Tom Copeland_, Jun 24 2018]
M. Goebel, On the number of special permutation-invariant orbits and terms, in Applicable Algebra in Engin., Comm. and Comp. (AAECC 8), Volume 8, Number 6, 1997, pp. 505-509 (Lect. Notes Comp. Sci.)
M. Griffiths and I. Mezo, A generalization of Stirling Numbers of the Second Kind via a special multiset, JIS 13 (2010) #10.2.5
J. Gubeladze and J. Love, Vertex maps between simplices, cubes, and crosspolytopes, arXiv preprint arXiv:1304.3775 [math.CO], 2013.
Jens Gulin and Kalle Åström, Alternative implementations of the Auxiliary Duplicating Permutation Invariant Training, Proc Work-in-Progress Papers at 14th Int'l Conf. Indoor Positioning Indoor Nav. (IPIN-WiP 2024). See p. 6.
Li Guo, Baxter algebras, Stirling numbers and partitions, arXiv:math/0402348 [math.AC], 2004.
M. Iida, On Triangle of numbers, Josai Mathematical Monographs, Vol. 5 (2012), 61-70;
A. Hasan, Physics and Mathematics of Graded Quivers, dissertation, Graduate Center, City University of New York, 2019.
Gábor Hetyei, The Stirling polynomial of a simplicial complex Discrete and Computational Geometry 35, Number 3, March 2006, pp 437-455.
Gábor Hetyei, Face enumeration using generalized binomial coefficients. Online draft version of the Hetyei paper referenced above.
Sergey Kitaev, Andrew Niedermaier, Jeffrey Remmel, and Manda Riehl, Generalized pattern-matching conditions for Ck wreath Sn, ISRN Comb. 2013, Article ID 634823, 20 p. (2013), eq. (87)
Germain Kreweras, Une dualité élémentaire souvent utile dans les problèmes combinatoires, Mathématiques et Sciences Humaines 3 (1963): 31-41.
Germain Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30.
Germain Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30. (Annotated scanned copy)
L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, arXiv:math/0509207 [math.CO], 2005-2006.
J. Loday, The Multiple Facets of the Associahedron [From _Tom Copeland_, Sep 29 2008]
František Marko, Sums of Consecutive Powers as a Linear Combination of Products of Two Figurate Numbers, J. Int. Seq., Vol. 25 (2022), Article 22.2.8.
Richard J. Mathar, On an Alternating Double Sum of a Triple Product of Aerated Binomial Coefficients, arXiv:2306.08022 [math.GM], 2023.
MathOverflow, How many surjections are there from a set of size n? [From _Tom Copeland_, Sep 09 2015]
E. Mendelson, Races with Ties, Math. Mag. 55 (1982), 170-175.
Lucas Chaves Meyles, Pamela E. Harris, Richter Jordaan, Gordon Rojas Kirby, Sam Sehayek, and Ethan Spingarn, Unit-Interval Parking Functions and the Permutohedron, arXiv:2305.15554 [math.CO], 2023.
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
Arthur Nunge, Eulerian polynomials on segmented permutations, arXiv:1805.01797 [math.CO], 2018.
OEIS Wiki, Sorting numbers
K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela and A. I. Solomon, Hierarchical Dobinski-type relations via substitution and the moment problem, arXiv:quant-ph/0312202, 2003.
Vincent Pilaud and V. Pons, Permutrees, arXiv preprint arXiv:1606.09643 [math.CO], 2016-2017.
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]
G. Rzadkowski, Bernoulli numbers and solitons revisited, Journal of Nonlinear Mathematical Physics, 17:1, 121-126, DOI: 10.1142/S1402925110000635
Natalie Schluter, The Taylor expansion for dropout is divergent, 2018.
Natalie Schluter, On approximating dropout noise injection, arXiv:1905.11320 [cs.LG], 2019.
Mark Shattuck, Combinatorial proofs of some Stirling number formulas, Preprint (ResearchGate), 2014.
John K. Sikora, On Calculating the Coefficients of a Polynomial Generated Sequence Using the Worpitzky Number Triangles, arXiv:1806.00887 [math.NT], 2018.
R. Simion, Convex Polytopes and Enumeration, Adv. in Appl. Math. 18 (1997) pp. 149-180.
J. Stembridge, Some permutation representations of Weyl groups associated with the cohomology of toric varieties, Adv. in Math., 106, 244-301, (1994).
D. Stephen, Topology on Finite Sets, American Mathematical Monthly, 75: 739 - 741, 1968.
S. M. Tanny, On some numbers related to the Bell numbers, Canad. Math. Bull. Vol. 17 (5), 733-738, 1975.
A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911. [Annotated scans of pages 30-33 only]
D. P. Walsh, Toy stories and combinatorial identities
Eric Weisstein's World of Mathematics, Clique Covering
Eric Weisstein's World of Mathematics, Complete Bipartite Graph
Eric Weisstein's World of Mathematics, Matching
Eric Weisstein's World of Mathematics, Minimum Edge Cover
Wikipedia, Truncated octahedron
J. Worpitzky, Studien über die Bernoullischen und Eulerschen Zahlen, J. reine u. angew. Math., 94 (1883) 203-232.

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

A048291 Number of {0,1} n X n matrices with no zero rows or columns.

Original entry on oeis.org

1, 1, 7, 265, 41503, 24997921, 57366997447, 505874809287625, 17343602252913832063, 2334958727565749108488321, 1243237913592275536716800402887, 2630119877024657776969635243647463625, 22170632855360952977731028744522744983195423

Offset: 0

Joe Keane (jgk(AT)jgk.org)

Number of relations on n labeled points such that for every point x there exists y and z such that xRy and zRx.
Also the number of edge covers in the complete bipartite graph K_{n,n}. - Eric W. Weisstein, Apr 24 2017
Counts labeled digraphs (loops allowed, no multiarcs) on n nodes where each indegree and each outdegree is >= 1. The corresponding sequence for unlabeled digraphs (1, 5, 55, 1918,... for n >= 1) seems not to be in the OEIS. - R. J. Mathar, Nov 21 2023
These relations form a subsemigroup of the semigroup of all binary relations on [n].  The zero element is the universal relation (all 1's matrix).  See Schwarz link. - Geoffrey Critzer, Jan 15 2024

			a(2) = 7:  |01|  |01|  |10|  |10|  |11|  |11|  |11|
           |10|  |11|  |01|  |11|  |01|  |10|  |11|.

Brendan McKay, Posting to sci.math.research, Jun 14 1999.

T. D. Noe, Table of n, a(n) for n = 0..32
H. Cheballah, S. Giraudo, and R. Maurice, Combinatorial Hopf algebra structure on packed square matrices, arXiv preprint arXiv:1306.6605 [math.CO], 2013.
David Dolžan and Gabriel Verret, The automorphism group of the zero-divisor digraph of matrices over an antiring, arXiv:1908.04614 [math.AC], 2019.
R. J. Mathar, The number of nXm matrices with at most k 1's in each row or column, (2014).
Steven Schlicker, Roman Vasquez, and Rachel Wofford, Integer Sequences from Configurations in the Hausdorff Metric Geometry via Edge Covers of Bipartite Graphs, J. Int. Seq. (2023) Vol. 26, Art. 23.6.6.
Stefan Schwarz, The semigroup of fully indecomposable relations and Hall relations, Czechoslovak Mathematical Journal, 23 (1973), 151-163.
R. Tauraso, Edge cover time for regular graphs, JIS 11 (2008) 08.4.4.
Eric Weisstein's World of Mathematics, Complete Bipartite Graph.
Eric Weisstein's World of Mathematics, Edge Cover.

Cf. A054976, A104602, A283624, A287065, A173403.
Cf. A055601, A055599, A104601, A086193 (traceless, no loops), A086206, A322661 (adj. matr. undirected edges).
Diagonal of A183109.

seq(add((-1)^(n+k)*binomial(n, k)*(2^k-1)^n, k=0..n), n=0..15); # Robert FERREOL, Mar 10 2017

Flatten[{1,Table[Sum[Binomial[n,k]*(-1)^k*(2^(n-k)-1)^n,{k,0,n}],{n,1,15}]}] (* Vaclav Kotesovec, Jul 02 2014 *)

a(n)=sum(k=0,n,binomial(n,k)*(-1)^k*(2^(n-k)-1)^n)

import math
f = math.factorial
def A048291(n): return sum([(f(n)/f(s)/f(n - s))*(-1)**s*(2**(n - s) - 1)**n for s in range(0, n+1)]) # Indranil Ghosh, Mar 14 2017

a(n) = Sum_{s=0..n} binomial(n, s)*(-1)^s*2^((n-s)*n)*(1-2^(-n+s))^n.
From Vladeta Jovovic, Feb 23 2008: (Start)
E.g.f.: Sum_{n>=0} (2^n-1)^n*exp((1-2^n)*x)*x^n/n!.
a(n) = Sum_{i=0..n} Sum_{j=0..n} (-1)^(i+j)*binomial(n,i)*binomial(n,j)*2^(i*j). (End)
a(n) ~ 2^(n^2). - Vaclav Kotesovec, Jul 02 2014
a(n) = Sum_{s=0..n-1} binomial(n,s)*(-1)^s*A092477(n,n-s), n > 0. - R. J. Mathar, Nov 18 2023

A055601 Number of n X n binary matrices with no zero rows.

Original entry on oeis.org

1, 1, 9, 343, 50625, 28629151, 62523502209, 532875860165503, 17878103347812890625, 2375680873491867011912191, 1255325460068093790930770843649, 2644211984585174742731315532085090303, 22235498641774645581443610453175918212890625

Offset: 0

Vladeta Jovovic, Jun 01 2000

More generally, Sum_{n>=0} m^n * q^(n^2) * exp(b*q^n*x) * x^n / n! = Sum_{n>=0} (m*q^n + b)^n * x^n / n! for all q, m, b. - Paul D. Hanna, Jan 02 2008

			A(x) = 1 + x + 3^2*x^2/2! + 7^3*x^3/3! + 15^4*x^4/4! +... + (2^n-1)^n*x^n/n! +...
A(x) = exp(-x) + 2*exp(-2x) + 2^4*exp(-4x)*x^2/2! + 2^9*exp(-8x)*x^3/3! +...+ 2^(n^2)*exp(-2^n*x)*x^n/n! +...
This is a special case of the more general statement: Sum_{n>=0} m^n * F(q^n*x)^b * log( F(q^n*x) )^n / n! = Sum_{n>=0} x^n * [y^n] F(y)^(m*q^n + b) where F(x) = exp(x), q=2, m=1, b=-1. - _Paul D. Hanna_, Jan 02 2008

Kenny Lau, Table of n, a(n) for n = 0..57

Cf. A048291, A092477, A136516, A303560.

a:= n-> mul(Stirling2(n+1, 2), j=1..n): seq(a(n), n=0..10); # Zerinvary Lajos, Jan 01 2009

Join[{1},Table[(2^n-1)^n,{n,16}]] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011 *)

a(n)=n!*polcoeff(sum(k=0,n,2^(k^2)*exp(-2^k*x)*x^k/k!),n) \\ Paul D. Hanna, Jan 02 2008

a = lambda n:((1<Kenny Lau, Jul 05 2016

N = 58
base = 0
a = []
for i in range(N):
    a += [base**i]
    base = (base<<1)|1 #base = base*2+1
print(a)
# Kenny Lau, Jul 05 2016

a(n) = A092477(n, n) for n>0.
a(n) = (2^n - 1 )^n. - Avi Peretz (njk(AT)netvision.net.il), Apr 21 2001
a(n) = Sum_{k=0..n} (-1)^k*C(n, k)*2^((n-k)*n).
E.g.f.: A(x) = Sum_{n>=0} 2^(n^2) * exp(-2^n*x) * x^n/n!. - Paul D. Hanna, Jan 02 2008
O.g.f.: Sum_{n>=0} 2^(n^2)*x^n/(1 + 2^n*x)^(n+1). - Paul D. Hanna, Jan 20 2010
Sum_{n>=1} 1/a(n) = A303560. - Amiram Eldar, Nov 18 2020

A089072 Triangle read by rows: T(n,k) = k^n, n >= 1, 1 <= k <= n.

Original entry on oeis.org

1, 1, 4, 1, 8, 27, 1, 16, 81, 256, 1, 32, 243, 1024, 3125, 1, 64, 729, 4096, 15625, 46656, 1, 128, 2187, 16384, 78125, 279936, 823543, 1, 256, 6561, 65536, 390625, 1679616, 5764801, 16777216, 1, 512, 19683, 262144, 1953125, 10077696, 40353607, 134217728, 387420489

Offset: 1

Alford Arnold, Dec 04 2003

T(n, k) = number of mappings from an n-element set into a k-element set. - Clark Kimberling, Nov 26 2004
Let S be the semigroup of (full) transformations on [n].  Let a be in S with rank(a) = k.  Then T(n,k) = |a S|, the number of elements in the right principal ideal generated by a. - Geoffrey Critzer, Dec 30 2021
From Manfred Boergens, Jun 23 2024: (Start)
In the following two comments the restriction k<=n can be lifted, allowing all k>=1.
T(n,k) is the number of n X k binary matrices with row sums = 1.
T(n,k) is the number of coverings of [n] by tuples (A_1,...,A_k) in P([n])^k with disjoint A_j, with P(.) denoting the power set.
For nonempty A_j see A019538.
For tuples with "disjoint" dropped see A092477.
For tuples with nonempty A_j and with "disjoint" dropped see A218695. (End)

			Triangle begins:
  1;
  1,  4;
  1,  8,  27;
  1, 16,  81,  256;
  1, 32, 243, 1024,  3125;
  1, 64, 729, 4096, 15625, 46656;
  ...

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened
Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141. See Theorem 2.1(ii).

Related to triangle of Eulerian numbers A008292.
Cf. A000312, A007778, A008788, A031971 (row sums), A062206, A083282.
Cf. A085524, A085526, A120485, A226065, A352981, A252982.
Cf. A019538, A092477, A218695.

a089072 = flip (^)
a089072_row n = map (a089072 n) [1..n]
a089072_tabl = map a089072_row [1..]  -- Reinhard Zumkeller, Mar 18 2013

[k^n: k in [1..n], n in [1..12]]; // G. C. Greubel, Nov 01 2022

Column[Table[k^n, {n, 8}, {k, n}], Center] (* Alonso del Arte, Nov 14 2011 *)

flatten([[k^n for k in range(1,n+1)] for n in range(1,12)]) # G. C. Greubel, Nov 01 2022

Sum_{k=1..n} T(n, k) = A031971(n).
T(n, n) = A000312(n).
T(2*n, n) = A062206(n).
a(n) = (n + T*(1-T)/2)^T, where T = round(sqrt(2*n),0). - Gerald Hillier, Apr 12 2015
T(n,k) = A051129(n,k). - R. J. Mathar, Dec 10 2015
T(n,k) = Sum_{i=0..k} Stirling2(n,i)*binomial(k,i)*i!. - Geoffrey Critzer, Dec 30 2021
From G. C. Greubel, Nov 01 2022: (Start)
T(n, n-1) = A007778(n-1), n >= 2.
T(n, n-2) = A008788(n-2), n >= 3.
T(2*n+1, n) = A085526(n).
T(2*n-1, n) = A085524(n).
T(2*n-1, n-1) = A085526(n-1), n >= 2.
T(3*n, n) = A083282(n).
Sum_{k=1..n} (-1)^k * T(n, k) = (-1)^n * A120485(n).
Sum_{k=1..floor(n/2)} T(n-k, k) = A226065(n).
Sum_{k=1..floor(n/2)} T(n, k) = A352981(n).
Sum_{k=1..floor(n/3)} T(n, k) = A352982(n). (End)

More terms and better definition from Herman Jamke (hermanjamke(AT)fastmail.fm), Jul 10 2004

Offset corrected by Reinhard Zumkeller, Mar 18 2013

A183109 Triangle read by rows: T(n,m) = number of n X m binary matrices with no zero rows or columns (n >= 1, 1 <= m <= n).

Original entry on oeis.org

1, 1, 7, 1, 25, 265, 1, 79, 2161, 41503, 1, 241, 16081, 693601, 24997921, 1, 727, 115465, 10924399, 831719761, 57366997447, 1, 2185, 816985, 167578321, 26666530801, 3776451407065, 505874809287625

Offset: 1

Steffen Eger, Feb 01 2011

T(n,m) = T(m,n) is also the number of complete alignments between two strings of sizes m and n, respectively; i.e. the number of complete matchings in a bipartite graph
From Manfred Boergens, Jul 25 2021: (Start)
The matrices in the definition are a superset of the matrices in the comment to A019538 by Manfred Boergens.
T(n,m) is the number of coverings of [n] by tuples (A_1,...,A_m) in P([n])^m with nonempty A_j, with P(.) denoting the power set (corrected for clarity by Manfred Boergens, May 26 2024). For the disjoint case see A019538.
For tuples with "nonempty" dropped see A092477 and A329943 (amendment by Manfred Boergens, Jun 24 2024). (End)

			Triangle begins:
  1;
  1,    7;
  1,   25,    265;
  1,   79,   2161,     41503;
  1,  241,  16081,    693601,    24997921;
  1,  727, 115465,  10924399,   831719761,   57366997447;
  1, 2185, 816985, 167578321, 26666530801, 3776451407065, 505874809287625;
  ...

Indranil Ghosh, Rows 1..50, flattened
Ch. A. Charalambides, A problem of arrangements on chessboards and generalizations, Discrete Mathematics 27.2 (1979): 179-186. (Generalizations.)
D. E. Knuth, Problem 11243, Am. Math. Montly 113 (8) (2006) page 759.
John Riordan and Paul R. Stein, Arrangements on chessboards, Journal of Combinatorial Theory, Series A 12.1 (1972): 72-80. See Table page 78.

Cf. A218695 (same sequence with restriction m<=n dropped).
Cf. A058482 (this gives the general formula, but values only for m=3).
Main diagonal gives A048291.
Column 2 is A058481.
Cf. A019538, A092477, A329943.

A183109 := proc(n,m)
    add((-1)^j*binomial(m,j)*(2^(m-j)-1)^n,j=0..m) ;
end proc:
seq(seq(A183109(n,m),m=1..n),n=1..10) ; # R. J. Mathar, Dec 03 2015

Flatten[Table[Sum[ (-1)^j*Binomial[m, j]*(2^(m - j) - 1)^n, {j, 0, m}], {n, 1, 7}, {m, 1, n}]] (* Indranil Ghosh, Mar 14 2017 *)

tabl(nn) = {for(n=1, nn, for(m = 1, n, print1(sum(j=0, m, (-1)^j*binomial(m, j)*(2^(m - j) - 1)^n),", ");); print(););};
tabl(8); \\ Indranil Ghosh, Mar 14 2017

import math
f = math.factorial
def C(n,r): return f(n)//f(r)//f(n - r)
def T(n,m):
    return sum((-1)**j*C(m,j)*(2**(m - j) - 1)**n for j in range (m+1))
i=1
for n in range(1,21):
    for m in range(1, n+1):
        print(str(i)+" "+str(T(n, m)))
        i+=1 # Indranil Ghosh, Mar 14 2017

T(n,m) = Sum_{j=0..m}(-1)^j*C(m, j)*(2^(m-j)-1)^n.
Recursion: T(m,n) = Sum_{k=1..m} T(k,n-1)*C(m,k)*2^k - T(m,n-1).
From Robert FERREOL, Mar 14 2017: (Start)
T(n,m) = Sum_{i = 0 .. n,j = 0 ..m}(-1)^(n+m+i+j)*C(n,i)*C(m,j)*2^(i*j).
Inverse formula of: 2^(n*m) = Sum_{i = 0 .. n , j = 0 ..m} C(n,i)*C(m,j)*T(i,j). (End)
A019538(j) <= a(j). - Manfred Boergens, Jul 25 2021

A218695 Square array A(h,k) = (2^h-1)A(h,k-1) + Sum_{i=1..h-1} binomial(h,h-i)2^i*A(i,k-1), with A(1,k) = A(h,1) = 1; read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 25, 25, 1, 1, 79, 265, 79, 1, 1, 241, 2161, 2161, 241, 1, 1, 727, 16081, 41503, 16081, 727, 1, 1, 2185, 115465, 693601, 693601, 115465, 2185, 1, 1, 6559, 816985, 10924399, 24997921, 10924399, 816985, 6559, 1

Offset: 1

M. F. Hasler, Nov 04 2012

This symmetric table is defined in the Kreweras papers, used also in A223911. Its upper or lower triangular part equals A183109, which might provide a simpler formula.
Number of h X k binary matrices with no zero rows or columns. - Andrew Howroyd, Mar 29 2023
A(h,k) is the number of coverings of [h] by tuples (A_1,...,A_k) in P([h])^k with nonempty A_j, with P(.) denoting the power set. For the disjoint case see A019538. For tuples with "nonempty" omitted see A092477 and A329943 (amendment by Manfred Boergens, Jun 24 2024). - Manfred Boergens, May 26 2024

			Array A(h,k) begins:
=====================================================
h\k | 1   2      3        4         5           6 ...
----+------------------------------------------------
  1 | 1   1      1        1         1           1 ...
  2 | 1   7     25       79       241         727 ...
  3 | 1  25    265     2161     16081      115465 ...
  4 | 1  79   2161    41503    693601    10924399 ...
  5 | 1 241  16081   693601  24997921   831719761 ...
  6 | 1 727 115465 10924399 831719761 57366997447 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)
G. Kreweras, Dénombrements systématiques de relations binaires externes, Math. Sci. Humaines 26 (1969) 5-15.
G. Kreweras, Dénombrement des ordres étagés, Discrete Math., 53 (1985), 147-149.

Columns 1..3 are A000012, A058481, A058482.
Main diagonal is A048291.
Cf. A019538, A056152 (unlabeled case), A052332, A092477, A183109, A223911, A329943.

c(h,k)={(h<2 || k<2) & return(1); sum(i=1,h-1,binomial(h,h-i)*2^i*c(i,k-1))+(2^h-1)*c(h,k-1)}
/* For better performance when h and k are large, insert the following memoization code before "sum(...)": cM=='cM & cM=matrix(h,k); my(s=matsize(cM));
s[1] >= h & s[2] >= k & cM[h,k] & return(cM[h,k]);
s[1]

A(m, n) = sum(k=0, m, (-1)^(m-k) * binomial(m, k) * (2^k-1)^n ) \\ Andrew Howroyd, Mar 29 2023

From Andrew Howroyd, Mar 29 2023: (Start)
A(h, k) = Sum_{i=0..h} (-1)^(h-i) * binomial(h, i) * (2^i-1)^k.
A052332(n) = Sum_{i=1..n-1} binomial(n,i)*A(i, n-i) for n > 0. (End)

A086206 Number of n X n matrices with entries in {0,1} with no zero row and with zero main diagonal.

Original entry on oeis.org

0, 1, 27, 2401, 759375, 887503681, 3938980639167, 67675234241018881, 4558916353692287109375, 1213972926354344043087129601, 1284197945649659948122178573052927, 5412701932445852698371002894178179850241, 91054366938067173656011584805755385081787109375

Offset: 1

Vladeta Jovovic, Aug 27 2003

Equivalently a(n) is the number of labeled digraphs on [n] with no out-nodes. Cf. A362013. - Geoffrey Critzer, Apr 13 2023

Andrew Howroyd, Table of n, a(n) for n = 1..50

Cf. A055601, A086193, A092477, A362013.

a(n) = {(2^(n-1)-1)^n} \\ Andrew Howroyd, Jan 05 2020

a(n) = (2^(n-1)-1)^n = Sum_{k=0..n} (-1)^k*binomial(n, k)*2^((n-k)*(n-1)).
a(n) = A092477(n, n-1).
Sum_{n>=0} a(n)*x^n/A011266(n) = (Sum_{n>=0} (-x)^n/A011266(n))*(Sum_{n>=0} 2^(n(n-1))*x^n/A011266(n)). - Geoffrey Critzer, Apr 13 2023

Terms a(11) and beyond from Andrew Howroyd, Jan 05 2020

A329943 Square array read by antidiagonals: T(n,k) is the number of right total relations between set A with n elements and set B with k elements.

Original entry on oeis.org

1, 3, 1, 7, 9, 1, 15, 49, 27, 1, 31, 225, 343, 81, 1, 63, 961, 3375, 2401, 243, 1, 127, 3969, 29791, 50625, 16807, 729, 1, 255, 16129, 250047, 923521, 759375, 117649, 2187, 1, 511, 65025, 2048383, 15752961, 28629151, 11390625, 823543, 6561, 1

Offset: 1

Roy S. Freedman, Nov 24 2019

A relation R between set A with n elements and set B with k elements is a subset of the Cartesian product A x B.  A relation R is right total if for each b in B there exists an a in A such that (a,b) in R.  T(n,k) is the number of right total relations and T(k,n) is the number of left total relations: relation R is left total if for each a in A there exists a b in B such that (a,b) in R.
From Manfred Boergens, Jun 23 2024: (Start)
T(n,k) is the number of k X n binary matrices with no 0 rows.
T(n,k) is the number of coverings of [k] by tuples (A_1,...,A_n) in P([k])^n, with P(.) denoting the power set.
Swapping n,k gives A092477 (with k<=n).
For nonempty A_j see A218695 (n,k swapped).
For disjoint A_j see A089072 (n,k swapped).
For nonempty and disjoint A_j see A019538 (n,k swapped). (End)

			T(n,k) begins, for 1 <= n,k <= 9:
    1,     1,       1,         1,           1,             1,               1
    3,     9,      27,        81,         243,           729,            2187
    7,    49,     343,      2401,       16807,        117649,          823543
   15,   225,    3375,     50625,      759375,      11390625,       170859375
   31,   961,   29791,    923521,    28629151,     887503681,     27512614111
   63,  3969,  250047,  15752961,   992436543,   62523502209,   3938980639167
  127, 16129, 2048383, 260144641, 33038369407, 4195872914689, 532875860165503

Roy S. Freedman, Some New Results on Binary Relations, arXiv:1501.01914 [cs.DM], 2015.

Cf. A218695.
The diagonal T(n,n) is A055601.
A092477 = T(k,n) is the number of left total relations between A and B.
A053440 is the number of relations that are both right unique (see A329940) and right total.
A089072 is the number of functions from A to B: relations between A and B that are both right unique and left total.
A019538 is the number of surjections between A and B: relations that are right unique, right total, and left total.
A008279 is the number of injections: relations that are right unique, left total, and left unique.
A000142 is the number of bijections: relations that are right unique, left total, right total, and left unique.

Maple
```
rt:=(n,k)->(2^n-1)^k:
```

T[n_, k_] := (2^n - 1)^k; Table[T[n - k + 1, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Amiram Eldar, Nov 25 2019 *)

MuPAD
```
rt:=(n,k)->(2^n-1)^k:
```

T(n,k) = (2^n - 1)^k.

cp's OEIS Frontend

A000244 Powers of 3: a(n) = 3^n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

Maxima

PARI

Python

Scala

Formula

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Sage

Formula

A048291 Number of {0,1} n X n matrices with no zero rows or columns.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A055601 Number of n X n binary matrices with no zero rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Python

Formula

A089072 Triangle read by rows: T(n,k) = k^n, n >= 1, 1 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

SageMath

Formula

A218695 Square array A(h,k) = (2^h-1)A(h,k-1) + Sum_{i=1..h-1} binomial(h,h-i)2^i*A(i,k-1), with A(1,k) = A(h,1) = 1; read by antidiagonals.