A019538 -id:A019538 - cp's OEIS frontend

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

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

A134991 Triangle of Ward numbers T(n,k) read by rows.

Original entry on oeis.org

1, 1, 3, 1, 10, 15, 1, 25, 105, 105, 1, 56, 490, 1260, 945, 1, 119, 1918, 9450, 17325, 10395, 1, 246, 6825, 56980, 190575, 270270, 135135, 1, 501, 22935, 302995, 1636635, 4099095, 4729725, 2027025, 1, 1012, 74316, 1487200, 12122110, 47507460, 94594500, 91891800, 34459425

Offset: 1

Tom Copeland, Feb 05 2008

This is the triangle of associated Stirling numbers of the second kind, A008299, read along the diagonals.
This is also a row-reversed version of A181996 (with an additional leading 1) - see the table on p. 92 in the Ward reference. A134685 is a refinement of the Ward table.
The first and second diagonals are A001147 and A000457 and appear in the diagonals of several OEIS entries. The polynomials also appear in Carlitz (p. 85), Drake et al. (p. 8) and Smiley (p. 7).
First few polynomials (with a different offset) are
  P(0,t) = 0
  P(1,t) = 1
  P(2,t) = t
  P(3,t) = t + 3*t^2
  P(4,t) = t + 10*t^2 + 15*t^3
  P(5,t) = t + 25*t^2 + 105*t^3 + 105*t^4
These are the "face" numbers of the tropical Grassmannian G(2,n),related to phylogenetic trees (with offset 0 beginning with P(2,t)). Corresponding h-vectors are A008517. - Tom Copeland, Oct 03 2011
A133314 applied to the derivative of A(x,t) implies (a.+b.)^n = 0^n, for (b_n)=P(n+1,t) and (a_0)=1, (a_1)=-t, and (a_n)=-(1+t) P(n,t) otherwise. E.g., umbrally, (a.+b.)^2 = a_2*b_0 + 2 a_1*b_1 + a_0*b_2 = 0. - Tom Copeland, Oct 08 2011
Beginning with the second column, the rows give the faces of the Whitehouse simplicial complex with the fourth-order complex being three isolated vertices and the fifth-order being the Petersen graph with 10 vertices and 15 edges (cf. Readdy). - Tom Copeland, Oct 03 2014
Stratifications of smooth projective varieties which are fine moduli spaces for stable n-pointed rational curves. Cf. pages 20 and 30 of the Kock and Vainsencher reference and references in A134685. - Tom Copeland, May 18 2017
Named after the American mathematician Morgan Ward (1901-1963). - Amiram Eldar, Jun 26 2021

			Triangle begins:
  1
  1   3
  1  10   15
  1  25  105  105
  1  56  490 1260   945
  1 119 1918 9450 17325 10395
  ...

Louis Comtet, Advanced Combinatorics, Reidel, 1974, page 222.

G. C. Greubel, Rows n = 1..65, flattened
J. Fernando Barbero G., Jesús Salas and Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences, I: General Structure, Journal of Combinatorial Theory, Series A, Vol. 125 (2014), pp. 146-165; arXiv preprint, arXiv:1307.2010 [math.CO], 2013-2014.
J. Fernando Barbero G., Jesús Salas and Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. II. Applications, arXiv preprint arXiv:1307.5624 [math.CO], 2013.
J. Fernando Barbero G., Jesús Salas and Eduardo J. S. Villaseñor, Generalized Stirling permutations and forests: Higher-order Eulerian and Ward numbers, Electronic Journal of Combinatorics, Vol. 22, No. 3 (2015), #P3.37.
Andreas Blass, Natasha Dobrinen and Dilip Raghavan, The next best thing to a p-point, The Journal of Symbolic Logic, Vol. 80, No. 3 (2015), pp. 866-900; arXiv preprint, arXiv:1308.3790 [math.LO], 2013.
David Callan, Toufik Mansour, and Mark Shattuck Some identities for derangement and Ward number sequences and related bijections, Pure Mathematics and Applications, Vol. 25 (Edition 2), pp. 132-143, 2015.
L. Carlitz, The coefficients in an asymptotic expansion and certain related numbers, Duke Math. J., Vol. 35, No. 1 (1968), pp. 83-90.
Lane Clark, Asymptotic normality of the Ward numbers, Discrete Math. 203 (1999), no. 1-3, 41-48. [From _N. J. A. Sloane_, Feb 06 2012]
Tom Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms.
Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. II. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 3.
Bishal Deb and Alan D. Sokal, Higher-order Stirling cycle and subset triangles: Total positivity, continued fractions and real-rootedness, arXiv:2507.18959 [math.CO], 2025. See p. 6.
Satyan L. Devadoss, Daoji Huang and Dominic Spadacene, Polyhedral covers of tree space, SIAM Journal on Discrete Mathematics, Vol. 28, No. 3 (2014), pp. 1508-1514; arXiv preprint, arXiv:1311.0766 [math.CO], 2013.
S. Devadoss and J. Morava, Diagonalizing the genome I: navigation in tree spaces, arXiv:1009.3224v2 [math.AG], 2012.
S. Devadoss and O. Schuh, Cartography of Tree Space, Leonardo (MIT Press Direct), Vol. 53, Issue 3, 2019.
Andreas Dieckmann and Erik Vigren, A New Result in Form of Finite Triple Sums for a Series from Ramanujan's Notebooks, Symmetry (2022) Vol. 14, No. 6, 1090.
Ming-Jian Ding and Jiang Zeng, Proof of an explicit formula for a series from Ramanujan's Notebooks via tree functions, arXiv:2307.00566 [math.CO], 2023.
Brian Drake, Ira M. Gessel and Guoce Xin, Three Proofs and a Generalization of the Goulden-Litsyn-Shevelev Conjecture on a Sequence Arising in Algebraic Geometry, J. of Integer Sequences, Vol. 10 (2007), Article 07.3.7.
Joseph Felsenstein, The number of evolutionary trees, Systematic Biology, 27 (1978), pp. 27-33, 1978.
Giovanni Gaiffi, Nested sets, set partitions and Kirkman-Cayley dissection numbers, arXiv preprint arXiv:1404.3395 [math.CO], 2014.
David M. Jackson, Achim Kempf and Alejandro H. Morales, A robust generalization of the Legendre transform for QFT, Journal of Physics A: Mathematical and Theoretical, Vol. 50, No. 22 (2017), 225201; arXiv preprint, arXiv:1612.0046 [hep-th], 2017.
Bradley Robert Jones, On tree hook length formulae, Feynman rules and B-series, Master's thesis, Math Dept., Simon Fraser University, 2014. p. 23.
Joachim Kock and Israel Vainsencher, Kontsevich's Formula for Rational Plane Curves, Recife, 1999 (version 31/01/2003).
Toufik Mansour and Mark Shattuck, A polynomial generalization of some associated sequences related to set partitions, Periodica Mathematica Hungarica, Vol. 75, No. 2 (December 2017), pp. 398-412.
MathOverflow, Face numbers for tropical Grassmannian G'_2,7 simplicial complex?, 2011.
Andrew Elvey Price and Alan D. Sokal, Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials, arXiv:2001.01468 [math.CO], 2020.
Margaret A. Readdy, The pre-WDVV ring of physics and its topology, The Ramanujan Journal, Vol. 10, No. 2 (2005), pp. 269-281; preprint, 2002.
Lucas Randazzo, Arboretum for a generalisation of Ramanujan polynomials, The Ramanujan Journal, Vol. 54 (2019), pp. 1-14; arXiv preprint, arXiv:1905.02083 [math.CO], 2019.
Lucas Randazzo, Combinatoire bijective autour d'arbres et de chemins, doctoral thesis, Université Paris-Est, 2019.
Peter Regner, Phylogenetic Trees: Selected Combinatorial Problems, Master's Thesis, 2012, Institute of Discrete Mathematics and Geometry, TU Vienna, p. 52.
Fuquan Ren, Jean Yeh, and Roberta Rui Zhou, Context-Free Grammars for Several Triangular Arrays, Axioms, Vol. 11, Issue 6, 297, 2022.
E. G. Santos, Counting non-attacking chess pieces placements: Bishops and Anassas, arXiv:2411.16492 [math.CO], 2024. See p. 8.
Leonard M. Smiley, Completion of a Rational Function Sequence of Carlitz, arXiv:math/0006106 [math.CO], 2000.
Pierre Théorêt, Fonctions génératrices pour une classe d'équations aux différences partielles, Ann. Sci. Math. Quebec 19, 91-105 (1995).
Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See p. 1.
Morgan Ward, The representations of Stirling's numbers and Stirling's polynomials as sums of factorials, Amer. J. Math., Vol. 56, No. 1 (1934), pp. 87-95.
Bao-Xuan Zhu, Coefficientwise Hankel-total positivity of row-generating polynomials for the m-Jacobi-Rogers triangle, arXiv:2202.03793 [math.CO], 2022.

The same as A269939, with column k = 0 removed.
A reshaped version of the triangle of associated Stirling numbers of the second kind, A008299.
A181996 is the mirror image.
Columns k = 2, 3, 4 are A000247, A000478, A058844.
Diagonal k = n is A001147.
Diagonal k = n - 1 is A000457.
Row sums are A000311.
Alternating row sums are signed factorials (-1)^(n-1)*A000142(n).
Cf. A112493.

t[n_, k_] := Sum[(-1)^i*Binomial[n, i]*Sum[(-1)^j*(k-i-j)^(n-i)/(j!*(k-i-j)!), {j, 0, k-i}], {i, 0, k}]; row[n_] := Table[t[k, k-n], {k, n+1, 2*n}]; Table[row[n], {n, 1, 9}] // Flatten (* Jean-François Alcover, Apr 23 2014, after A008299 *)

E.g.f. for the polynomials is A(x,t) = (x-t)/(t+1) + T{ (t/(t+1)) * exp[(x-t)/(t+1)] }, where T(x) is the Tree function, the e.g.f. of A000169. The compositional inverse in x (about x = 0) is B(x) = x + -t * [exp(x) - x - 1]. Special case t = 1 gives e.g.f. for A000311. These results are a special case of A134685 with u(x) = B(x).
From Tom Copeland, Oct 26 2008: (Start)
Umbral-Sheffer formalism gives, for m a positive integer and u = t/(t+1),
[P(.,t)+Q(.,x)]^m = [m Q(m-1,x) - t Q(m,x)]/(t+1) + sum(n>=1) { n^(n-1)[u exp(-u)]^n/n! [n/(t+1)+Q(.,x)]^m }, when the series is convergent for a sequence of functions Q(n,x).
Check: With t=1; Q(n,x)=0^n, for n>=0; and Q(-1,x)=0, then [P(.,1)+Q(.,x)]^m = P(m,1) = A000311(m).
(End)
Let h(x,t) = 1/(dB(x)/dx) = 1/(1-t*(exp(x)-1)), an e.g.f. in x for row polynomials in t of A019538, then the n-th row polynomial in t of the table A134991, P(n,t), is given by ((h(x,t)*d/dx)^n)x evaluated at x=0, i.e., A(x,t) = exp(x*P(.,t)) = exp(x*h(u,t)*d/du) u evaluated at u=0. Also, dA(x,t)/dx = h(A(x,t),t). - Tom Copeland, Sep 05 2011
The polynomials (1+t)/t*P(n,t) are the row polynomials of A112493. Let f(x) = (1+x)/(1-x*t). Then for n >= 0, P(n+1,t) is given by t/(1+t)*(f(x)*d/dx)^n(f(x)) evaluated at x = 0. - Peter Bala, Sep 30 2011
From Tom Copeland, Oct 04 2011: (Start)
T(n,k) = (k+1)*T(n-1,k) + (n+k+1)*T(n-1,k-1) with starting indices n=0 and k=0 beginning with P(2,t) (as suggested by a formula of David Speyer on MathOverflow).
T(n,k) = k*T(n-1,k) + (n+k-1)*T(n-1,k-1) with starting indices n=1 and k=1 of table (cf. Smiley above and Riordin ref.[10] therein).
P(n,t) = (1/(1+t))^n * Sum_{k>=1} k^(n+k-1)*(u*exp(-u))^k / k! with u=(t/(t+1)) for n>1; therefore, Sum_{k>=1} (-1)^k k^(n+k-1) x^k/k! = [1+LW(x)]^(-n) P{n,-LW(x)/[1+LW(x)]}, with LW(x) the Lambert W-Fct.
T(n,k) = Sum_{i=0..k} ((-1)^i binomial(n+k,i) Sum_{j=0..k-i} (-1)^j (k-i-j)^(n+k-i)/(j!(k-i-j)!)) from relation to A008299. (End)
The e.g.f. A(x,t) = -v * ( Sum_{j=>1} D(j-1,u) (-z)^j / j! ) where u = (x-t)/(1+t), v = 1+u, z = x/((1+t) v^2) and D(j-1,u) are the polynomials of A042977. dA/dx = 1/((1+t)(v-A)) = 1/(1-t*(exp(A)-1)). - Tom Copeland, Oct 06 2011
The general results on the convolution of the refined partition polynomials of A134685, with u_1 = 1 and u_n = -t otherwise, can be applied here to obtain results of convolutions of these polynomials. - Tom Copeland, Sep 20 2016
E.g.f.: C(u,t) = (u-t)/(1+t) - W( -((t*exp((u-t)/(1+t)))/(1+t)) ), where W is the principal value of the Lambert W-function. - Cheng Peng, Sep 11 2021
The function C(u,t) in the previous formula by Peng is precisely the function A(u,t) given in the initial 2008 formula of this section and the Oct 06 2011 formula from Copeland. As noted in A000169, Euler's tree function is T(x) = -LambertW(-x), where W(x) is the principal branch of Lambert's function, and T(x) is the e.g.f. of A000169. - Tom Copeland, May 13 2022

Reference to A181996 added by N. J. A. Sloane, Apr 05 2012

Further edits by N. J. A. Sloane, Jan 24 2020

A145901 Triangle of f-vectors of the simplicial complexes dual to the permutohedra of type B_n.

Original entry on oeis.org

1, 1, 2, 1, 8, 8, 1, 26, 72, 48, 1, 80, 464, 768, 384, 1, 242, 2640, 8160, 9600, 3840, 1, 728, 14168, 72960, 151680, 138240, 46080, 1, 2186, 73752, 595728, 1948800, 3037440, 2257920, 645120, 1, 6560, 377504, 4612608, 22305024, 52899840

Offset: 0

Peter Bala, Oct 26 2008

The Coxeter group of type B_n may be realized as the group of n X n matrices with exactly one nonzero entry in each row and column, that entry being either +1 or -1. The order of the group is 2^n*n!. The orbit of the point (1,2,...,n) (or any sufficiently generic point (x_1,...,x_n)) under the action of this group is a set of 2^n*n! distinct points whose convex hull is defined to be the permutohedron of type B_n. The rows of this table are the f-vectors of the simplicial complexes dual to these type B permutohedra. Some examples are given in the Example section below. See A060187 for the corresponding table of h-vectors of type B permutohedra.

This is the (unsigned) triangle of connection constants between the polynomial sequences (2*x + 1)^n, n >= 0, and binomial(x+k,k), k >= 0. For example, (2*x + 1)^2 = 8*binomial(x+2,2) - 8*binomial(x+1,1) + 1 and (2*x + 1)^3 = 48*binomial(x+3,3) - 72*binomial(x+2,2) + 26*binomial(x+1,1) - 1. Cf. A163626. - Peter Bala, Jun 06 2019

			The triangle begins
n\k|..0.....1.....2.....3.....4.....5
=====================================
0..|..1
1..|..1.....2
2..|..1.....8.....8
3..|..1....26....72....48
4..|..1....80...464...768...384
5..|..1...242..2640..8160..9600..3840
...
Row 2: the permutohedron of type B_2 is an octagon with 8 vertices and 8 edges. Its dual, also an octagon, has f-vector (1,8,8) - row 3 of this triangle.
Row 3: for an appropriate choice of generic point in R_3, the permutohedron of type B_3 is realized as the great rhombicuboctahedron, also known as the truncated cuboctahedron, with 48 vertices, 72 edges and 26 faces (12 squares, 8 regular hexagons and 6 regular octagons). See the Wikipedia entry and also [Fomin and Reading p.22]. Its dual polyhedron is a simplicial polyhedron, the disdyakis dodecahedron, with 26 vertices, 72 edges and 48 triangular faces and so its f-vector is (1,26,72,48) - row 4 of this triangle.
From _Peter Bala_, Jun 06 2019: (Start)
Examples of falling factorials identities for odd numbered rows: Let (x)_n = x*(x - 1)*...*(x - n + 1) with (x)_0 = 1 denote the falling factorial power.
Row 1: 2*(x)_1 + (0 - 2*x)_1 = 0.
Row 3: 48*(x)_3 + 72*(x)_2 * (2 - 2*x)_1 + 26*(x)_1 * (2 - 2*x)_2 + (2 - 2*x)_3 = 0
Row 5: 3840*(x)_5 + 9600*(x)_4 * (4 - 2*x)_(1) + 8160*(x)_3 * (4 - 2*x)_2 + 2640*(x)_2 * (4 - 2*x)_3 + 242*(x)_1 * (4 - 2*x)_4 + (4 - 2*x)_5 = 0. (End)

Sandrine Dasse-Hartaut and Pawel Hitczenko, Greek letters in random staircase tableaux arXiv:1202.3092v1 [math.CO], 2012.
Peter Bala, Deformations of the Hadamard product of power series
M. Dukes and C. D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.
S. Fomin and N. Reading, Root systems and generalized associahedra, Lecture notes for IAS/Park-City 2004; arXiv:math/0505518 [math.CO], 2005-2008.
Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.
Gábor Hetyei, The type B permutohedron and the poset of intervals as a Tchebyshev transform, University of North Carolina-Charlotte (2019).
Shi-Mei Ma, A family of two-variable derivative polynomials for tangent and secant, El J. Combinat. 20 (1) (2013) P11.
Wikipedia, Truncated cuboctahedron

Cf. A019538 (f-vectors type A permutohedra), A060187 (h-vectors type B permutohedra), A080253 (row sums), A145905, A062715, A028246.
Cf. A258377, A258378, A258379, A258380, A258381.
Cf. A039755, A154537.

with(combinat):
T:= (n,k) -> add((-1)^(k-i)*binomial(k,i)*(2*i+1)^n,i = 0..k):
for n from 0 to 9 do
seq(T(n,k),k = 0..n);
end do;

T[n_, k_] := Sum[(-1)^(k - i)*Binomial[k, i]*(2*i + 1)^n, {i, 0, k}];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 02 2019 *)

T(n,k) = Sum_{i = 0..k} (-1)^(k-i)*binomial(k,i)*(2*i+1)^n.
Recurrence relation: T(n,k) = (2*k + 1)*T(n-1,k) + 2*k*T(n-1,k-1) with T(0,0) = 1 and T(0,k) = 0 for k >= 1.
Relation with type B Stirling numbers of the second kind: T(n,k) = 2^k*k!*A039755(n,k).
Row sums A080253. The matrix product A060187 * A007318 produces the mirror image of this triangle.
E.g.f.: exp(t)/(1 + x - x* exp(2*t)) = 1 + (1 + 2*x)*t + (1 + 8*x + 8*x^2 )*t^2/2! + ... .
From Peter Bala, Oct 13 2011: (Start)
The polynomials in the first column of the array ((1+t)*P^(-1)-t*P)^(-1), P Pascal's triangle and I the identity, are the row polynomials of this table.
The polynomials in the first column of the array ((1+t)*I-t*A062715)^(-1) are, apart from the initial 1, the row polynomials of this table with an extra factor of t. Cf. A060187. (End)
From Peter Bala, Jul 18 2013: (Start)
Integrating the above e.g.f. with respect to x from x = 0 to x = 1 gives Sum_{k = 0..n} (-1)^k*T(n,k)/(k + 1) = 2^n*Bernoulli(n,1/2), the n-th cosecant number.
The corresponding Type A result is considered in A028246 as Worpitzky's algorithm.
Also for n >= 0, Sum_{k = 0..2*n} (-1)^k*T(2*n,k)/((k + 1)*(k + 2)) = 1/2*2^(2*n)*Bernoulli(2*n,1/2) and for n >= 1, Sum_{k = 0..2*n-1} (-1)^k*T(2*n - 1,k)/((k + 1)*(k + 2)) = -1/2 * 2^(2*n)* Bernoulli(2*n,1/2).
The nonzero cosecant numbers are given by A001896/A001897. (End)
From Peter Bala, Jul 22 2014: (Start)
The row polynomials R(n,x) satisfy the recurrence equation R(n+1,x) = D(R(n,x)) with R(0,x) = 1, where D is the operator 1 + 2*x + 2*x(1 + x)*d/dx.
R(n,x) = 1/(1 + x)* Sum_{k = 0..inf} (2*k + 1)^n*(x/(1 + x))^k, valid for x in the open interval (-1/2, inf). Cf. A019538.
The shifted row polynomial x*R(n,x) = (1 + x)^n*P(n,x/(1 + x)) where P(n,x) denotes the n-th row polynomial of A060187.
The row polynomials R(n,x) have only real zeros.
Symmetry: R(n,x) = (-1)^n*R(n,-1 - x). Consequently the zeros of R(n,x) lie in the open interval (-1, 0). (End)
From Peter Bala, May 28 2015: (Start)
Recurrence for row polynomials: R(n,x) = 1 + x*Sum_{k = 0..n-1} binomial(n,k)2^(n-k)*R(k,x) with R(0,x) = 1.
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) = 1/(1 - z)*( BINOMIAL(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). A(k,z) = A(-(k + 1),-z). Cf. A019538.
For cases see A258377 (k = 1), A258378(k = 2), A258379 (k = 3), A258380 (k = 4) and A258381 (k = 5). (End)
T(n,k) = A154537(n,k)*k! = A039755(n,k)*(2^k*k!), 0 <= k <= n. - Wolfdieter Lang, Apr 19 2017
From Peter Bala, Jan 12 2018: (Start)
n-th row polynomial R(n,x) =  (1 + 2*x) o (1 + 2*x) o ... o (1 + 2*x) (n factors), where o denotes the black diamond multiplication operator of Dukes and White. See example E13 in the Bala link.
R(n,x) = Sum_{k = 0..n} binomial(n,k)*2^k*F(k,x) where F(k,x) is the Fubini polynomial of order k, the k-th row polynomial of A019538. (End)

A006327 a(n) = Fibonacci(n) - 3. Number of total preorders.

Original entry on oeis.org

0, 2, 5, 10, 18, 31, 52, 86, 141, 230, 374, 607, 984, 1594, 2581, 4178, 6762, 10943, 17708, 28654, 46365, 75022, 121390, 196415, 317808, 514226, 832037, 1346266, 2178306, 3524575, 5702884, 9227462, 14930349, 24157814, 39088166, 63245983, 102334152, 165580138

Offset: 4

N. J. A. Sloane, Simon Plouffe

Minimal cost of maximum height Huffman tree of size n. - Alex Vinokur (alexvn(AT)barak-online.net), Oct 25 2004

			G.f. = 2*x^5 + 5*x^6 + 10*x^7 + 18*x^8 + 31*x^9 + 52*x^10 + 86*x^11 + ...

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 = 4..1000
G. Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30.
G. Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30. (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. Sapounakis, I. Tasoulas and P. Tsikouras, On the Dominance Partial Ordering of Dyck Paths, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.5.
A. B. Vinokur, Huffman trees and Fibonacci numbers, Kibernetika Issue 6 (1986) 9-12 (in Russian); English translation in Cybernetics 21, Issue 6 (1986), 692-696.
Alex Vinokur, Fibonacci connection between Huffman codes and Wythoff array, arXiv:cs/0410013 [cs.DM], 2004-2005.
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

A diagonal of A079502.

Cf. A000045, A001611, A000071, A157725, A001911, A157726, A006327, A157727, A157728, A157729, A167616. [Added by N. J. A. Sloane, Jun 25 2010 in response to a comment from Aviezri S. Fraenkel]

List([4..45], n-> Fibonacci(n)-3) # G. C. Greubel, Jul 13 2019

[Fibonacci(n)-3: n in [4..45]]; // G. C. Greubel, Jul 13 2019

with(combinat):a:=n->sum(fibonacci(j),j=3..n): seq(a(n),n=2..40); # Zerinvary Lajos, Oct 03 2007
A006327:=(2+z)/(z-1)/(z**2+z-1); # conjectured by Simon Plouffe in his 1992 dissertation

Fibonacci[Range[4, 45]] - 3 (* Vladimir Joseph Stephan Orlovsky, Mar 19 2010 *)

a(n)=fibonacci(n)-3 \\ Charles R Greathouse IV, Feb 03 2014

[fibonacci(n)-3 for n in (4..45)] # G. C. Greubel, Jul 13 2019

G.f.: x^5*(2 + x)/((1-x)*(1-x-x^2)).
a(n) = a(n-1) + a(n-2) + 3.
a(n+3) = Sum_{k=-n+1..n} F(abs(n)+1). - Paul Barry, Oct 24 2007
a(n) = F(4*n) mod F(n+1) = F(n) - (F(n+4)^2 - F(n)^2)/F(2*n+4). - Gary Detlefs, Apr 02 2012

Offset corrected by Gary Detlefs, Apr 02 2012

A090582 T(n, k) = Sum_{j=0..n-k} (-1)^jbinomial(n - k + 1, j)(n - k + 1 - j)^n. Triangle read by rows, T(n, k) for 1 <= k <= n.

Original entry on oeis.org

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

Offset: 1

Hugo Pfoertner, Jan 11 2004

Let Q(m, n) = Sum_(k=0..n-1) (-1)^k * binomial(n, k) * (n-k)^m. Then Q(m,n) is the numerator of the probability P(m,n) = Q(m,n)/n^m of seeing each card at least once if m >= n cards are drawn with replacement from a deck of n cards, written in a two-dimensional array read by antidiagonals with Q(m,m) as first row and Q(m,1) as first column.
The sequence is given as a matrix with the first row containing the cases #draws = size_of_deck. The second row contains #draws = 1 + size_of_deck. If "mn" indicates m cards drawn from a deck with n cards then the locations in the matrix are:
  11 22 33 44 55 66 77 ...
  21 32 43 54 65 76 87 ...
  31 42 53 64 75 86 97 ...
  41 52 63 74 85 .. .. ...
read by antidiagonals ->:
  11, 22, 21, 33, 32, 31, 44, 43, 42, 41, 55, 54, 53, 52, ....
The probabilities are given by Q(m,n)/n^m:
.(m,n):.....11..22..21..33..32..31..44..43..42..41...55...54..53..52..51
.....Q:......1...2...1...6...6...1..24..36..14...1..120..240.150..30...1
...n^m:......1...4...1..27...8...1.256..81..16...1.3125.1024.243..32...1
%.Success:.100..50.100..22..75.100...9..44..88.100....4...23..62..94.100
P(n,n) = n!/n^n which can be approximated by sqrt(Pi*(2n+1/3))/e^n (Gosper's approximation to n!).
Let P[n] be the set of all n-permutations. Build a superset Q[n] of P[n] composed of n-permutations in which some (possibly all or none) ascents have been designated. An ascent in a permutation s[1]s[2]...s[n] is a pair of consecutive elements s[i],s[i+1] such that s[i] < s[i+1]. As a triangular array read by rows T(n,k) is the number of elements in Q[n] that have exactly k distinguished ascents, n >= 1, 0 <= k <= n-1. Row sums are A000670. E.g.f. is y/(1+y-exp(y*x)). For example, T(3,1)=6 because there are four 3-permutations with one ascent, with these we would also count 1->2 3, and 1 2->3 where exactly one ascent is designated by "->". (After Flajolet and Sedgewick.) - Geoffrey Critzer, Nov 13 2012
Sum_(k=1..n) Q(n, k)*binomial(x, k) = x^n such that Sum_{k=1..n} Q(n, i)*binomial(x+1,i+1) = Sum_{k=1..x} k^n. - David A. Corneth, Feb 17 2014
A141618(n,n-k+1) = a(n,k) * C(n,k-1) / k. - Tom Copeland, Oct 25 2014
See A074909 and above g.f.s below for associations among this array and the Bernoulli polynomials and their umbral compositional inverses. - Tom Copeland, Nov 14 2014
For connections to toric varieties and Eulerian polynomials (in addition to those noted below), see the Dolgachev and Lunts and the Stembridge links in A019538. - 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
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

			For m = 4, n = 2, we draw 4 times from a deck of two cards. Call the cards "a" and "b" - of the 16 possible combinations of draws (each of which is equally likely to occur), only two do not contain both a and b: a, a, a, a and b, b, b, b. So the probability of seeing both a and b is 14/16. Therefore Q(m, n) = 14.
Table starts:
  [1] 1;
  [2] 2,      1;
  [3] 6,      6,       1;
  [4] 24,     36,      14,      1;
  [5] 120,    240,     150,     30,      1;
  [6] 720,    1800,    1560,    540,     62,     1;
  [7] 5040,   15120,   16800,   8400,    1806,   126,    1;
  [8] 40320,  141120,  191520,  126000,  40824,  5796,   254,   1;
  [9] 362880, 1451520, 2328480, 1905120, 834120, 186480, 18150, 510, 1.

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
Kevin Buhr, Michael Press and DMB, Collecting a deck of cards on the street. Thread in NG sci.math.
José L. Cereceda, Figurate numbers and sums of powers of integers, arXiv:2001.03208 [math.NT], 2020. See triangle Table 1 p. 5.
Tom Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009, page 209.
S. Franco and A. Hasan, Graded Quivers, Generalized Dimer Models and Toric Geometry, arXiv preprint arXiv:1904.07954 [hep-th], 2019.
A. Hasan, Physics and Mathematics of Graded Quivers, dissertation, Graduate Center, City University of New York, 2019.
Hugo Pfoertner, "Hit all" probabilities: Program and results.

Cf. A073593 first m >= n giving at least 50% probability, A085813 ditto for 95%, A055775 n^n/n!, A090583 Gosper's approximation to n!.
Reflected version of A019538.
Cf. A233734 (central terms).
Cf. A074909, A008292, A028246.
Cf. A046802, A119879, A123125, A130850, and A248727.

a090582 n k = a090582_tabl !! (n-1) !! (k-1)
a090582_row n = a090582_tabl !! (n-1)
a090582_tabl = map reverse a019538_tabl
-- Reinhard Zumkeller, Dec 15 2013

T := (n, k) -> add((-1)^j*binomial(n - k + 1, j)*(n - k + 1 - j)^n, j = 0..n-k):
# Or:
T := (n, k) -> (n - k + 1)!*Stirling2(n, n - k + 1):
for n from 1 to 9 do seq( T(n, k), k = 1..n) od; # Peter Luschny, May 21 2021

In[1]:= Table[Table[k! StirlingS2[n, k], {k, n, 1, -1}], {n, 1, 6}] (* Victor Adamchik, Oct 05 2005 *)
nn=6; a=y/(1+y-Exp[y x]); Range[0,nn]! CoefficientList[Series[a, {x,0,nn}], {x,y}]//Grid (* Geoffrey Critzer, Nov 10 2012 *)

a(n)={m=ceil((-1+sqrt(1+8*n))/2);k=m+1+binomial(m,2)-n;k*sum(i=1,k,(-1)^(i+k)*i^(m-1)*binomial(k-1,i-1))} \\ David A. Corneth, Feb 17 2014

T(n, k) = (n - k + 1)!*Stirling2(n, n - k + 1), generated by Stirling numbers of the second kind multiplied by a factorial. - Victor Adamchik, Oct 05 2005
Triangle T(n,k), 1 <= k <= n, read by rows given by [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ...] DELTA [0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 10 2006
From Tom Copeland, Oct 07 2008: (Start)
G(x,t) = 1/ (1 + (1-exp(x*t))/t) = 1 + 1*x + (2 + t)*x^2/2! + (6 + 6*t + t^2)*x^3/3! + ... gives row polynomials of A090582, the f-polynomials for the permutohedra (see A019538).
G(x,t-1) = 1 + 1*x + (1 + t)*x^2/2! + (1 + 4*t + t^2)*x^3/3! + ... gives row polynomials for A008292, the h-polynomials of permutohedra.
G[(t+1)x,-1/(t+1)] = 1 + (1 + t)*x + (1 + 3*t + 2*t^2)*x^2/2! + ... gives row polynomials of A028246. (End)
From Tom Copeland, Oct 11 2011: (Start)
With e.g.f. A(x,t) = G(x,t) - 1, the compositional inverse in x is
B(x,t) = log((t+1)-t/(1+x))/t. Let h(x,t) = 1/(dB/dx) = (1+x)*(1+(1+t)x), then the row polynomial P(n,t) is given by (1/n!)*(h(x,t)*d/dx)^n x, evaluated at x=0, A=exp(x*h(y,t)*d/dy) y, eval. at y=0, and dA/dx = h(A(x,t),t). (End)
k <= 0 or k > n yields Q(n, k) = 0; Q(1,1) = 1; Q(n, k) = k * (Q(n-1, k) + Q(n-1, k-1)). - David A. Corneth, Feb 17 2014
T = A008292*A007318. - Tom Copeland, Nov 13 2016
With all offsets 0 for this entry, let A_n(x;y) = (y + E.(x))^n, an Appell sequence in y where E.(x)^k = E_k(x) are the Eulerian polynomials of A123125 with offsets -1 so that the array becomes A008292; i.e., we ignore the first row and first column of A123125. Then the row polynomials of this entry, A090582, are given by A_n(1 + x;0). Other specializations of A_n(x;y) give A028246, A046802, A119879, A130850, and A248727. - Tom Copeland, Jan 24 2020

A006359 Number of distributive lattices; also number of paths with n turns when light is reflected from 6 glass plates.

Original entry on oeis.org

1, 6, 21, 91, 371, 1547, 6405, 26585, 110254, 457379, 1897214, 7869927, 32645269, 135416457, 561722840, 2330091144, 9665485440, 40093544735, 166312629795, 689883899612, 2861717685450, 11870733787751, 49241167758705, 204258021937291, 847285745315256

Offset: 0

N. J. A. Sloane

Let M denotes the 6 X 6 matrix = row by row (1,1,1,1,1,1)(1,1,1,1,1,0)(1,1,1,1,0,0)(1,1,1,0,0,0)(1,1,0,0,0,0)(1,0,0,0,0,0) and A(n) the vector (x(n),y(n),z(n),t(n),u(n),v(n)) = M^n*A where A is the vector (1,1,1,1,1,1) then a(n) = x(n). - Benoit Cloitre, Apr 02 2002

J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124.
Manfred Goebel, Rewriting Techniques and Degree Bounds for Higher Order Symmetric Polynomials, Applicable Algebra in Engineering, Communication and Computing (AAECC), Volume 9, Issue 6 (1999), 559-573.
J. Haubrich, Multinacci Rijen [Multinacci sequences], Euclides (Netherlands), Vol. 74, Issue 4, 1998, pp. 131-133.
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
J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]
Emma L. L. Gao, Sergey Kitaev, and Philip B. Zhang, Pattern-avoiding alternating words, arXiv:1505.04078 [math.CO], 2015.
G. Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30.
G. Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30. (Annotated scanned copy)
Index entries for linear recurrences with constant coefficients, signature (3,6,-4,-5,1,1).

Cf. A000217, A000330, A050446, A050447, A006356, A006357, A006358.

See also A025030, A030112, A030113, A030114, A030115, A030116.

A=seq(a.j,j=0..5):grammar1:=[Q5,{ seq(Q.i=Union(Epsilon,seq(Prod(a.j,Q.j),j=5-i..5)),i=0..5), seq(a.j=Z,j=0..5) }, unlabeled]: seq(count(grammar1,size=j),j=0..22); # Zerinvary Lajos, Mar 09 2007

LinearRecurrence[{3,6,-4,-5,1,1},{1,6,21,91,371,1547},30] (* Harvey P. Dale, Sep 03 2016 *)

k=5; M(k)=matrix(k,k,i,j,if(1-sign(i+j-k),0,1)); v(k)=vector(k,i,1); a(n)=vecmax(v(k)*M(k)^n)

{a(n)=local(p=6);polcoeff(sum(k=0,p-1,(-1)^((k+1)\2)*binomial((p+k-1)\2,k)* (-x)^k)/sum(k=0,p,(-1)^((k+1)\2)*binomial((p+k)\2,k)*x^k+x*O(x^n)),n)} \\ Paul D. Hanna, Feb 06 2006

G.f.: -(z^4 + z^3 - 3z^2 - 2z + 1) / (-1 + 3z + 6z^2 - 4z^3 - 5z^4 + z^5 + z^6). - M. Goebel (manfredg(AT)ICSI.Berkeley.EDU) Jul 26 1997
a(n) = 3*a(n-1) + 6*a(n-2) - 4*a(n-3) - 5*a(n-4) + a(n-5) + a(n-6).
a(n) is asymptotic to z(6)*w(6)^n where w(6) = (1/2)/cos(6*Pi/13) and z(6) is the root 1 < x < 2 of P(6, X) = -1 - 91*X + 2366*X^2 + 26364*X^3 - 142805*X^4 - 371293*X^5 + 371293*X^6 - Benoit Cloitre, Oct 16 2002
G.f.: A(x) = (1 + 3*x - 3*x^2 - 4*x^3 + x^4 + x^5)/(1 - 3*x - 6*x^2 + 4*x^3 + 5*x^4 - x^5 - x^6). - Paul D. Hanna, Feb 06 2006
G.f.: 1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1)))))). - Paul Barry, Mar 24 2010

Alternative description from Jacques Haubrich (jhaubrich(AT)freeler.nl)

More terms from James Sellers, Dec 24 1999

A094416 Array read by antidiagonals: generalized ordered Bell numbers Bo(r,n).

Original entry on oeis.org

1, 2, 3, 3, 10, 13, 4, 21, 74, 75, 5, 36, 219, 730, 541, 6, 55, 484, 3045, 9002, 4683, 7, 78, 905, 8676, 52923, 133210, 47293, 8, 105, 1518, 19855, 194404, 1103781, 2299754, 545835, 9, 136, 2359, 39390, 544505, 5227236, 26857659, 45375130, 7087261

Offset: 1

Ralf Stephan, May 02 2004

Also, r times the number of (r+1)-level labeled linear rooted trees with n leaves.
"AIJ" (ordered, indistinct, labeled) transform of {r,r,r,...}.
Stirling transform of r^n*n!, i.e. of e.g.f. 1/(1-r*x).
Also, Bo(r,s) is ((x*d/dx)^n)(1/(1+r-r*x)) evaluated at x=1.
r-th ordered Bell polynomial (A019538) evaluated at n.
Bo(r,n) is the n-th moment of a geometric distribution with probability parameter = 1/(r+1).  Here, geometric distribution is the number of failures prior to the first success. - Geoffrey Critzer, Jan 01 2019
Row r (starting at r=0), Bo(r+1, n), is the Akiyama-Tanigawa algorithm applied to the powers of r+1. See Python program below. - Shel Kaphan, May 03 2024

			Array begins as:
  1,  3,   13,    75,     541,     4683,      47293, ...
  2, 10,   74,   730,    9002,   133210,    2299754, ...
  3, 21,  219,  3045,   52923,  1103781,   26857659, ...
  4, 36,  484,  8676,  194404,  5227236,  163978084, ...
  5, 55,  905, 19855,  544505, 17919055,  687978905, ...
  6, 78, 1518, 39390, 1277646, 49729758, 2258233998, ...

G. C. Greubel, Antidiagonals n = 1..50, flattened
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
P. Blasiak, K. A. Penson and A. I. Solomon, Dobinski-type relations and the log-normal distribution, arXiv:quant-ph/0303030, 2003.
C. G. Bower, Transforms

Rows 1-10 are A000670, A004123, A032033, A094417, A094418, A094419, A238464, A238465, A238466, A238467.
Columns include A014105, A094421.
Main diagonal is A094420.
Antidiagonal sums are A094422.
Cf. A019538, A131689, A344499.

A094416:= func< n,k | (&+[Factorial(j)*n^j*StirlingSecond(k,j): j in [0..k]]) >;
[A094416(n-k+1,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Jan 12 2024

Bo[, 0]=1; Bo[r, n_]:= Bo[r, n]= r*Sum[Binomial[n,k] Bo[r,n-k], {k, n}];
Table[Bo[r-n+1, n], {r, 10}, {n, r}] // Flatten (* Jean-François Alcover, Nov 03 2018 *)

# The Akiyama-Tanigawa algorithm applied to the powers of r + 1
# generates the rows. Adds one row (r=0) and one column (n=0).
# Adapted from Peter Luschny on A371568.
def f(n, r): return (r + 1)**n
def ATtransform(r, len, f):
  A = [0] * len
  R = [0] * len
  for n in range(len):
      R[n] = f(n, r)
      for j in range(n, 0, -1):
          R[j - 1] = j * (R[j] - R[j - 1])
      A[n] = R[0]
  return A
for r in range(8): print([r], ATtransform(r, 8, f)) # Shel Kaphan, May 03 2024

def A094416(n,k): return sum(factorial(j)*n^j*stirling_number2(k,j) for j in range(k+1)) # array
flatten([[A094416(n-k+1,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Jan 12 2024

E.g.f.: 1/(1 + r*(1 - exp(x))).
Bo(r, n) = Sum_{k=0..n} k!*r^k*Stirling2(n, k) = 1/(r+1) * Sum_{k>=1} k^n * (r/(r+1))^k, for r>0, n>0.
Recurrence: Bo(r, n) = r * Sum_{k=1..n} C(n, k)*Bo(r, n-k), with Bo(r, 0) = 1.
Bo(r,0) = 1, Bo(r,n) = r*Bo(r,n-1) - (r+1)*Sum_{j=1..n-1} (-1)^j * binomial(n-1,j) * Bo(r,n-j). - Seiichi Manyama, Nov 17 2023

Offset corrected by Geoffrey Critzer, Jan 01 2019

A119879 Exponential Riordan array (sech(x),x).

Original entry on oeis.org

1, 0, 1, -1, 0, 1, 0, -3, 0, 1, 5, 0, -6, 0, 1, 0, 25, 0, -10, 0, 1, -61, 0, 75, 0, -15, 0, 1, 0, -427, 0, 175, 0, -21, 0, 1, 1385, 0, -1708, 0, 350, 0, -28, 0, 1, 0, 12465, 0, -5124, 0, 630, 0, -36, 0, 1, -50521, 0, 62325, 0, -12810, 0, 1050, 0, -45, 0, 1

Offset: 0

Paul Barry, May 26 2006

Row sums have e.g.f. exp(x)*sech(x) (signed version of A009006). Inverse of masked Pascal triangle A119467. Transforms the sequence with e.g.f. g(x) to the sequence with e.g.f. g(x)*sech(x).
Coefficients of the Swiss-Knife polynomials for the computation of Euler, tangent and Bernoulli number (triangle read by rows). Another version in A153641. - Philippe Deléham, Oct 26 2013
Relations to Green functions and raising/creation and lowering/annihilation/destruction operators are presented in Hodges and Sukumar and in Copeland's discussion of this sequence and 2020 pdf. - Tom Copeland, Jul 24 2020

			Triangle begins:
     1;
     0,    1;
    -1,    0,     1;
     0,   -3,     0,   1;
     5,    0,    -6,   0,   1;
     0,   25,     0, -10,   0,   1;
   -61,    0,    75,   0, -15,   0,   1;
     0, -427,     0, 175,   0, -21,   0,  1;
  1385,    0, -1708,   0, 350,   0, -28,  0,  1;

G. C. Greubel, Rows n=0..100 of triangle, flattened
Paul Barry, Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms, J. Int. Seq. 14 (2011) # 11.2.2, example 28.
Tom Copeland, The Elliptic Lie Triad: Ricatti and KdV Equations, Infinigens, and Elliptic Genera, 2015.
Tom Copeland, Discussion of this sequence
Tom Copeland, SKipping over Dimensions, Juggling Zeros in the Matrix, 2020.
A. Hodges and C. V. Sukumar, Bernoulli, Euler, permutations and quantum algebras, Proc. R. Soc. A Oct. 2007 vol 463 no. 463 2086 2401-2414.
Peter Luschny, Additive decompositions of classical numbers via the Swiss-Knife polynomials [_Tom Copeland_, Oct 20 2015]
Miguel Méndez and Rafael Sánchez, On the combinatorics of Riordan arrays and Sheffer polynomials: monoids, operads and monops, arXiv:1707.00336 [math.CO], 2017, Section 4.3, Example 4.
Miguel A. Méndez and Rafael Sánchez Lamoneda, Monops, Monoids and Operads: The Combinatorics of Sheffer Polynomials, The Electronic Journal of Combinatorics 25(3) (2018), #P3.25.

Row sums are A155585. - Johannes W. Meijer, Apr 20 2011
Rows reversed: A081658.
Cf. A109449, A153641, A162660. - Philippe Deléham, Oct 26 2013
Cf. A000182, A046802, A119467, A133314.
Cf. A019538, A028246, A090582, A123125, A130850, A248727.

T := (n,k) -> binomial(n,k)*2^(n-k)*euler(n-k,1/2): # Peter Luschny, Jan 25 2009

T[n_, k_] := Binomial[n, k] 2^(n-k) EulerE[n-k, 1/2];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 20 2018, after Peter Luschny *)

{T(n,k) = binomial(n,k)*2^(n-k)*(2/(n-k+1))*(subst(bernpol(n-k+1, x), x, 1/2) - 2^(n-k+1)*subst(bernpol(n-k+1, x), x, 1/4))};
for(n=0,5, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Feb 25 2019

@CachedFunction
def A119879_poly(n, x) :
    return 1 if n == 0  else add(A119879_poly(k, 0)*binomial(n, k)*(x^(n-k)-1+n%2) for k in range(n)[::2])
def A119879_row(n) :
    R = PolynomialRing(ZZ, 'x')
    return R(A119879_poly(n,x)).coeffs()  # Peter Luschny, Jul 16 2012
# Alternatively:

# uses[riordan_array from A256893]
riordan_array(sech(x), x, 9, exp=true) # Peter Luschny, Apr 19 2015

Number triangle whose k-th column has e.g.f. sech(x)*x^k/k!.
T(n,k) = C(n,k)*2^(n-k)*E_{n-k}(1/2) where C(n,k) is the binomial coefficient and E_{m}(x) are the Euler polynomials. - Peter Luschny, Jan 25 2009
The coefficients in ascending order of x^i of the polynomials p{0}(x) = 1 and p{n}(x) = Sum_{k=0..n-1; k even} binomial(n,k)*p{k}(0)*((n mod 2) - 1 + x^(n-k)). - Peter Luschny, Jul 16 2012
E.g.f.: exp(x*z)/cosh(x). - Peter Luschny, Aug 01 2012
Sum_{k=0..n} T(n,k)*x^k = A122045(n), A155585(n), A119880(n), A119881(n) for x = 0, 1, 2, 3 respectively. - Philippe Deléham, Oct 27 2013
With all offsets 0, let A_n(x;y) = (y + E.(x))^n, an Appell sequence in y where E.(x)^k = E_k(x) are the Eulerian polynomials of A123125. Then the row polynomials of A046802 (the h-polynomials of the stellahedra) are given by h_n(x) = A_n(x;1); the row polynomials of A248727 (the face polynomials of the stellahedra), by f_n(x) = A_n(1 + x;1); the Swiss-knife polynomials of this entry, A119879, by Sw_n(x) = A_n(-1;1 + x); and the row polynomials of the Worpitsky triangle (A130850), by w_n(x) = A(1 + x;0). Other specializations of A_n(x;y) give A090582 (the f-polynomials of the permutohedra, cf. also A019538) and A028246 (another version of the Worpitsky triangle). - Tom Copeland, Jan 24 2020
Triangle equals P*((I + P^2)/2)^(-1), where P denotes Pascal's triangle A007318. - Peter Bala, Mar 07 2024

A167137 E.g.f.: P(exp(x)-1) where P(x) is the g.f. of the partition numbers (A000041).

Original entry on oeis.org

1, 1, 5, 31, 257, 2551, 30065, 407191, 6214577, 105530071, 1972879025, 40213910551, 886979957297, 21044674731991, 534313527291185, 14448883517785111, 414475305054698417, 12568507978358276311, 401658204472560090545, 13490011548122407566871, 474964861088609044357937, 17491333169997896126211031

Offset: 0

Paul D. Hanna, Nov 03 2009

nonn

CONJECTURE: Sum_{n>=0} a(n)^m * log(1+x)^n/n! is an integer series in x for all integer m>0; see A167138 and A167139 for examples.
From Peter Bala, Jul 07 2022: (Start)
Conjecture: Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is eventually periodic with the period dividing phi(k) = A000010(k). For example, modulo 16 we obtain the sequence [1, 1, 5, 15, 1, 7, 1, 7, 1, 7, ...], with an apparent period of 2 beginning at a(4).
More generally, we conjecture that the same property holds for integer sequences having an e.g.f. of the form G(exp(x) - 1), where G(x) is an integral power series. (End)

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 31*x^3/3! + 257*x^4/4! +...
A(log(1+x)) = P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Cf. A167138, A000041, A019538 (Stirling2), A218482, A305550, A306045, A320349.

Table[Sum[PartitionsP[k]*StirlingS2[n, k]*k!, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 21 2018 *)
nmax = 20; CoefficientList[Series[1/QPochhammer[E^x - 1], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 22 2021 *)

{a(n)=if(n==0,1,n!*polcoeff(exp(sum(m=1,n,sigma(m)*(exp(x+x*O(x^n))-1)^m/m) ),n))}

{Stirling2(n, k)=if(k<0||k>n, 0, sum(i=0, k, (-1)^i*binomial(k, i)/k!*(k-i)^n))}
{a(n)=sum(k=0,n,numbpart(k)*Stirling2(n, k)*k!)}

x='x+O('x^66); Vec( serlaplace( 1/eta(exp(x)-1) ) ) \\ Joerg Arndt, Sep 18 2013

a(n) = Sum_{k=0..n} A000041(k)*Stirling2(n,k)*k! where A000041 is the partition numbers.
E.g.f.: exp( Sum_{n>=1} sigma(n)*[exp(x)-1]^n/n ).
Sum_{n>=0} a(n) * log(1+x)^n/n! = g.f. of the partition numbers (A000041).
Sum_{n>=0} a(n)^2*log(1+x)^n/n! = g.f. of A167138.
From Peter Bala, Sep 18 2013: (Start)
Sum {n >= 0} (-1)^n*a(n)*(log(1 - x))^n/n! = 1 + x + 3*x^2 + 8*x^3 + 21*x^4 + ... is the o.g.f. of A218482.
a(n) is always odd. Congruences for n >= 1: a(2*n) = 2 (mod 3); a(4*n) = 2 (mod 5); a(6*n) = 0 (mod 7); a(10*n) = 7 (mod 11); a(12*n) = 5 (mod 13); a(16*n) = 0 (mod 17). (End)
From Vaclav Kotesovec, Jun 17 2018: (Start)
a(n) ~ n! * exp((1/log(2) - 1) * Pi^2 / 24 + Pi*sqrt(n/(3*log(2)))) / (4 * sqrt(3) * n * (log(2))^n).
a(n) ~ sqrt(Pi) * exp((1/log(2) - 1) * Pi^2 / 24 + Pi*sqrt(n/(3*log(2))) - n) * n^(n + 1/2) / (2^(3/2) * sqrt(3) * n * (log(2))^n). (End)

cp's OEIS Frontend

A223522 Triangle T(n,k) represents the coefficients of (x^20*d/dx)^n, where n=1,2,3,...; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297.

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

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A134991 Triangle of Ward numbers T(n,k) read by rows.

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A145901 Triangle of f-vectors of the simplicial complexes dual to the permutohedra of type B_n.

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A006327 a(n) = Fibonacci(n) - 3. Number of total preorders.

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A090582 T(n, k) = Sum_{j=0..n-k} (-1)^j*binomial(n - k + 1, j)*(n - k + 1 - j)^n. Triangle read by rows, T(n, k) for 1 <= k <= n.

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A090582 T(n, k) = Sum_{j=0..n-k} (-1)^jbinomial(n - k + 1, j)(n - k + 1 - j)^n. Triangle read by rows, T(n, k) for 1 <= k <= n.