A131689 -id:A131689 - cp's OEIS frontend

A063007 T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.

1, 1, 2, 1, 6, 6, 1, 12, 30, 20, 1, 20, 90, 140, 70, 1, 30, 210, 560, 630, 252, 1, 42, 420, 1680, 3150, 2772, 924, 1, 56, 756, 4200, 11550, 16632, 12012, 3432, 1, 72, 1260, 9240, 34650, 72072, 84084, 51480, 12870, 1, 90, 1980, 18480, 90090, 252252, 420420, 411840, 218790, 48620

Offset: 0

Henry Bottomley, Jul 02 2001

T(n,k) is the number of compatible k-sets of cluster variables in Fomin and Zelevinsky's Cluster algebra of finite type B_n. Take a row of this triangle regarded as a polynomial in x and rewrite as a polynomial in y := x+1. The coefficients of the polynomial in y give a row of triangle A008459 (squares of binomial coefficients). For example, x^2+6*x+6 = y^2+4*y+1. - Paul Boddington, Mar 07 2003
T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k N=(0,1) steps. E.g. T(2,0)=1 because we have DD; T(2,1) = 6 because we have NED, NDE, EDN, END, DEN and DNE; T(2,2)=6 because we have NNEE, NENE, NEEN, EENN, ENEN and ENNE. - Emeric Deutsch, Apr 20 2004
Another version of [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...] = 1; 1, 0; 1, 2, 0; 1, 6, 6, 0; 1, 12, 30, 20, 0; ..., where DELTA is the operator defined in A084938. - Philippe Deléham Apr 15 2005
Terms in row n are the coefficients of the Legendre polynomial P(n,2x+1) with increasing powers of x.
From Peter Bala, Oct 28 2008: (Start)
Row n of this triangle is the f-vector of the simplicial complex dual to an associahedron of type B_n (a cyclohedron) [Fomin & Reading, p.60]. See A008459 for the corresponding h-vectors for associahedra of type B_n and A001263 and A033282 respectively for the h-vectors and f-vectors for associahedra of type A_n.
An alternative description of this triangle in terms of f-vectors is as follows. Let A_n be the root lattice generated as a monoid by {e_i - e_j: 0 <= i,j <= n+1}. Let P(A_n) be the polytope formed by the convex hull of this generating set. Then the rows of this array are the f-vectors of a unimodular triangulation of P(A_n) [Ardila et al.]. A008459 is the corresponding array of h-vectors for these type A_n polytopes. See A127674 (without the signs) for the array of f-vectors for type C_n polytopes and A108556 for the array of f-vectors associated with type D_n polytopes.
The S-transform on the ring of polynomials is the linear transformation of polynomials that is defined on the basis monomials x^k by S(x^k) = binomial(x,k) = x(x-1)...(x-k+1)/k!. Let P_n(x) denote the S-transform of the n-th row polynomial of this array. In the notation of [Hetyei] these are the Stirling polynomials of the type B associahedra. The first few values are P_1(x) = 2*x + 1, P_2(x) = 3*x^2 + 3*x + 1 and P_3(x) = (10*x^3 + 15*x^2 + 11*x + 3)/3. These polynomials have their zeros on the vertical line Re x = -1/2 in the complex plane, that is, the polynomials P_n(-x) satisfy a Riemann hypothesis. See A142995 for further details. The sequence of values P_n(k) for k = 0,1,2,3, ... produces the n-th row of A108625. (End)
This is the row reversed version of triangle A104684. - Wolfdieter Lang, Sep 12 2016
T(n, k) is also the number of (n-k)-dimensional faces of a convex n-dimensional Lipschitz polytope of real functions f defined on the set X = {1, 2, ..., n+1} which satisfy the condition f(n+1) = 0 (see Gordon and Petrov). - Stefano Spezia, Sep 25 2021
The rows seem to give (up to sign) the coefficients in the expansion of the integer-valued polynomial ((x+1)*(x+2)*(x+3)*...*(x+n) / n!)^2 in the basis made of the binomial(x+i,i). - F. Chapoton, Oct 09 2022
Chapoton's observation above is correct: the precise expansion is ((x+1)*(x+2)*(x+3)*...*(x+n)/ n!)^2 = Sum_{k = 0..n} (-1)^k*T(n,n-k)*binomial(x+2*n-k, 2*n-k), as can be verified using the WZ algorithm. For example, n = 3 gives ((x+1)*(x+2)*(x+3)/3!)^2 = 20*binomial(x+6,6) - 30*binomial(x+5,5) + 12*binomial(x+4,4) - binomial(x+3,3). - Peter Bala, Jun 24 2023

			The triangle T(n, k) starts:
  n\k 0  1    2     3     4      5      6      7      8     9
  0:  1
  1:  1  2
  2:  1  6    6
  3:  1 12   30    20
  4:  1 20   90   140    70
  5:  1 30  210   560   630    252
  6:  1 42  420  1680  3150   2772    924
  7:  1 56  756  4200 11550  16632  12012   3432
  8:  1 72 1260  9240 34650  72072  84084  51480  12870
  9:  1 90 1980 18480 90090 252252 420420 411840 218790 48620
... reformatted by _Wolfdieter Lang_, Sep 12 2016
From _Petros Hadjicostas_, Jul 11 2020: (Start)
Its inverse (from Table II, p. 92, in Ser's book) is
   1;
  -1/2,  1/2;
   1/3, -1/2,    1/6;
  -1/4,  9/20,  -1/4,   1/20;
   1/5, -2/5,    2/7,  -1/10,  1/70;
  -1/6,  5/14, -25/84,  5/36, -1/28,  1/252;
   1/7, -9/28,  25/84, -1/6,   9/154, -1/84, 1/924;
   ... (End)

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 366.
J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, Table I, p. 92.
D. Zagier, Integral solutions of Apery-like recurrence equations, in: Groups and Symmetries: from Neolithic Scots to John McKay, CRM Proc. Lecture Notes 47, Amer. Math. Soc., Providence, RI, 2009, pp. 349-366.

T. D. Noe, Rows n = 0..100 of triangle, flattened
F. Ardila, M. Beck, S. Hosten, J. Pfeifle and K. Seashore, Root polytopes and growth series of root lattices, arXiv:0809.5123 [math.CO], 2008.
Peter Bala, Deformations of the Hadamard product of power series.
Cyril Banderier, Combinatoire analytique des chemins et des cartes, Thesis (2001), page 49.
Paul Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009), #09.7.6.
H. J. Brothers, Pascal's Prism: Supplementary Material.
David Callan, A bijection for Delannoy paths, arXiv:2202.04649 [math.CO], 2022.
F. Chapoton, Enumerative properties of generalized associahedra, Séminaire Lotharingien de Combinatoire, B51b (2004), 16 pp.
Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015-2016.
Mark Dukes and Chris D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.
Mark Dukes and Chris D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, Electronic Journal Of Combinatorics, 23(1) (2016), #P1.45.
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. Fomin and A. Zelevinsky, Cluster algebras I: Foundations, J. Amer. Math. Soc. 15(2) (2002), 497-529.
S. Fomin and A. Zelevinsky, Y-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3, 977-1018.
J. Gordon and F. Petrov, Combinatorics of the Lipschitz Polytope, Arnold Mathematical Journal (2016).
G. Hetyei, Face enumeration using generalized binomial coefficients. This is the draft version of Hetyei's paper referenced below. [Archived version]
Gabor Hetyei, The Stirling polynomial of a simplicial complex Discrete and Computational Geometry 35(3) (2006), 437-455.
Hsien-Kuei Hwang and Satoshi Kuriki, Integrated empirical measures and generalizations of classical goodness-of-fit statistics, arXiv:2404.06040 [math.ST], 2024. See p. 11.
C. Lanczos, Applied Analysis (Annotated scans of selected pages). See page 514.
T. Manneville and V. Pilaud, Compatibility fans for graphical nested complexes, arXiv preprint arXiv:1501.07152 [math.CO], 2015.
Thomas Selig, Combinatorial aspects of sandpile models on wheel and fan graphs, arXiv:2202.06487 [math.CO], 2022.
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].
J. Ser, Les Calculs Formels des Séries de Factorielles (Annotated scans of some selected pages) See Table I, page 92.
V. Strehl, Recurrences and Legendre transform, Séminaire Lotharingien de Combinatoire, B29b (1992), 22 pp.
R. A. Sulanke, Objects counted by the central Delannoy numbers., J. Integer Seq. 6 (2003), no. 1, Article 03.1.5.
D. Zagier, Integral solutions of Apery-like recurrence equations.

See A331430 for an essentially identical triangle, except with signed entries.
Columns include A000012, A002378, A033487 on the left and A000984, A002457, A002544 on the right.
Main diagonal is A006480.
Row sums are A001850. Alternating row sums are A033999.
Cf. A007318, A008459, A009766, A046899, A104684, A109983, A127674.
Cf. A033282 (f-vectors type A associahedra), A108625, A080721 (f-vectors type D associahedra).
The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692,A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)

a063007 n k = a063007_tabl !! n !! k
a063007_row n = a063007_tabl !! n
a063007_tabl = zipWith (zipWith (*)) a007318_tabl a046899_tabl
-- Reinhard Zumkeller, Nov 18 2014

/* As triangle: */ [[Binomial(n,k)*Binomial(n+k,k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 03 2015

p := (n,x) -> orthopoly[P](n,1+2*x): seq(seq(coeff(p(n,x),x,k), k=0..n), n=0..9);

Flatten[Table[Binomial[n, k]Binomial[n + k, k], {n, 0, 10}, {k, 0, n}]] (* Harvey P. Dale, Dec 24 2011 *)
Table[CoefficientList[Hypergeometric2F1[-n, n + 1, 1, -x], x], {n, 0, 9}] // Flatten
(* Peter Luschny, Mar 09 2018 *)

{T(n, k) = local(t); if( n<0, 0, t = (x + x^2)^n; for( k=1, n, t=t'); polcoeff(t, k) / n!)} /* Michael Somos, Dec 19 2002 */

{T(n, k) = binomial(n, k) * binomial(n+k, k)} /* Michael Somos, Sep 22 2013 */

{T(n, k) = if( k<0 || k>n, 0, (n+k)! / (k!^2 * (n-k)!))} /* Michael Somos, Sep 22 2013 */

T(n, k) = (n+k)!/(k!^2*(n-k)!) = T(n-1, k)*(n+k)/(n-k) = T(n, k-1)*(n+k)*(n-k+1)/k^2 = T(n-1, k-1)*(n+k)*(n+k-1)/k^2.
binomial(x, n)^2 = Sum_{k>=0} T(n,k) * binomial(x, n+k). - Michael Somos, May 11 2012
T(n, k) = A109983(n, k+n). - Michael Somos, Sep 22 2013
G.f.: G(t, z) = 1/sqrt(1-2*z-4*t*z+z^2). Row generating polynomials = P_n(1+2z), i.e., T(n, k) = [z^k] P_n(1+2*z), where P_n are the Legendre polynomials. - Emeric Deutsch, Apr 20 2004
Sum_{k>=0} T(n, k)*A000172(k) = Sum_{k>=0} T(n, k)^2 = A005259(n). - Philippe Deléham, Jun 08 2005
1 + z*d/dz(log(G(t,z))) = 1 + (1 + 2*t)*z + (1 + 8*t + 8*t^2)*z^2 + ... is the o.g.f. for a signed version of A127674. - Peter Bala, Sep 02 2015
If R(n,t) denotes the n-th row polynomial then x^3 * exp( Sum_{n >= 1} R(n,t)*x^n/n ) = x^3 + (1 + 2*t)*x^4 + (1 + 5*t + 5*t^2)*x^5 + (1 + 9*t + 21*t^2 + 14*t^3)*x^6 + ... is an o.g.f for A033282. - Peter Bala, Oct 19 2015
P(n,x) := 1/(1 + x)*Integral_{t = 0..x} R(n,t) dt are (modulo differences of offset) the row polynomials of A033282. - Peter Bala, Jun 23 2016
From Peter Bala, Mar 09 2018: (Start)
R(n,x) = Sum_{k = 0..n} binomial(2*k,k)*binomial(n+k,n-k)*x^k.
R(n,x) = Sum_{k = 0..n} binomial(n,k)^2*x^k*(1 + x)^(n-k).
n*R(n,x) = (1 + 2*x)*(2*n - 1)*R(n-1,x) - (n - 1)*R(n-2,x).
R(n,x) = (-1)^n*R(n,-1 - x).
R(n,x) = 1/n! * (d/dx)^n ((x^2 + x)^n). (End)
The row polynomials are R(n,x) = hypergeom([-n, n + 1], [1], -x). - Peter Luschny, Mar 09 2018
T(n,k) = C(n+1,k)*A009766(n,k). - Bob Selcoe, Jan 18 2020 (Connects this triangle with the Catalan triangle. - N. J. A. Sloane, Jan 18 2020)
If we let A(n,k) = (-1)^(n+k)*(2*k+1)*(n*(n-1)*...*(n-(k-1)))/((n+1)*...*(n+(k+1))) for n >= 0 and k = 0..n, and we consider both T(n,k) and A(n,k) as infinite lower triangular arrays, then they are inverses of one another. (Empty products are by definition 1.) See the example below. The rational numbers |A(n,k)| appear in Table II on p. 92 in Ser's (1933) book. - Petros Hadjicostas, Jul 11 2020
From Peter Bala, Nov 28 2021: (Start)
Row polynomial R(n,x) = Sum_{k >= n} binomial(k,n)^2 * x^(k-n)/(1+x)^(k+1) for x > -1/2.
R(n,x) = 1/(1 + x)^(n+1) * hypergeom([n+1, n+1], [1], x/(1 + x)).
R(n,x) = (1 + x)^n * hypergeom([-n, -n], [1], x/(1 + x)).
R(n,x) = hypergeom([(n+1)/2, -n/2], [1], -4*x*(1 + x)).
If we set R(-1,x) = 1, we can run the recurrence n*R(n,x) = (1 + 2*x)*(2*n - 1)*R(n-1,x) - (n - 1)*R(n-2,x) backwards to give R(-n,x) = R(n-1,x).
R(n,x) = [t^n] ( (1 + t)*(1 + x*(1 + t)) )^n. (End)
n*T(n,k) = (2*n-1)*T(n-1,k) + (4*n-2)*T(n-1,k-1) - (n-1)*T(n-2,k). - Fabián Pereyra, Jun 30 2022
From Peter Bala, Oct 07 2024: (Start)
n-th row polynomial R(n,x) = Sum_{k = 0..n} binomial(n, k) * x^k o (1 + x)^(n-k), where o denotes the black diamond product of power series as defined by Dukes and White (see Bala, Section 4.4, exercise 3).
Denote this triangle by T. Then T * transpose(T) = A143007, the square array of crystal ball sequences for the A_n X A_n lattices.
Let S denote the triangle ((-1)^(n+k)*T(n, k))n,k >= 0, a signed version of this triangle. Then S^(-1) * T = A007318, Pascal's triangle; it appears that T * S^(-1) = A110098.
T = A007318 * A115951. (End)

A032033 Stirling transform of A032031.

Original entry on oeis.org

1, 3, 21, 219, 3045, 52923, 1103781, 26857659, 746870565, 23365498683, 812198635941, 31055758599099, 1295419975298085, 58538439796931643, 2848763394161128101, 148537065755389540539, 8261178848690959117605, 488177936257344615487803, 30544839926043868901604261

Offset: 0

Christian G. Bower

Also "AIJ" (ordered, indistinct, labeled) transform of 3,3,3,3...

Third row of array A094416 (generalized ordered Bell numbers).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
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, J. Phys. A.: Math. Gen 36 (2003) L273.
Christian G. Bower, Transforms (2)
Jacob Sprittulla, On Colored Factorizations, arXiv:2008.09984 [math.CO], 2020.

Cf. A032031, A094416, A094418, A094419.

b:= proc(n, m) option remember;
     `if`(n=0, 3^m*m!, m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 04 2021

a[n_] := PolyLog[-n, 3/4]/4; a[0] = 1; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Nov 14 2011 *)
t = 30; Range[0, t]! CoefficientList[Series[1/(4 - 3 Exp[x]), {x, 0, t}], x] (* Vincenzo Librandi, Mar 16 2014 *)

a(n)=ceil(polylog(-n,3/4)/4) \\ Charles R Greathouse IV, Jul 14 2014

my(N=25,x='x+O('x^N)); Vec(serlaplace(1/(4 - 3*exp(x)))) \\ Joerg Arndt, Jan 15 2024

E.g.f.: 1/(4-3*exp(x)).
a(n) = 3 * A050352(n), n > 0.
a(n) = Sum_{k=0..n} Stirling2(n, k) * (3^k) * k!.
a(n) = (1/4) * Sum_{k>=0} k^n * (3/4)^k. - Karol A. Penson, Jan 25 2002
a(n) = Sum_{k=0..n} A131689(n,k)*3^k. - Philippe Deléham, Nov 03 2008
G.f. A(x)=B(x)/x, where B(x)=x+3*x^2+21*x^3+... = Sum_{n>=1} b(n)*x^n satisfies 4*B(x)-x = 3*B(x/(1-x)), and b(n)=3*Sum_{k=1..n-1} binomial(n-1,k-1)*b(k), b(1)=1. - Vladimir Kruchinin, Jan 27 2011
a(n) = log(4/3)*Integral_{x = 0..inf} (floor(x))^n * (4/3)^(-x) dx. - Peter Bala, Feb 14 2015
a(0) = 1; a(n) = 3 * Sum_{k=1..n} binomial(n,k) * a(n-k). - Ilya Gutkovskiy, Jan 17 2020
a(0) = 1; a(n) = 3 * a(n-1) - 4 * Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 16 2023
a(n) = (3/4) * Sum_{k=0..n} 4^k * (-1)^(n-k) * k! * Stirling2(n,k) for n > 0. - Seiichi Manyama, Jun 01 2025

A094417 Generalized ordered Bell numbers Bo(4,n).

Original entry on oeis.org

1, 4, 36, 484, 8676, 194404, 5227236, 163978084, 5878837476, 237109864804, 10625889182436, 523809809059684, 28168941794178276, 1641079211868751204, 102961115527874385636, 6921180217049667005284, 496267460209336700111076, 37807710659221213027893604

Offset: 0

Ralf Stephan, May 02 2004

nonn

Fourth row of array A094416, which has more information.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.

Cf. A000670, A004123, A032033, A050353, A094418, A094419, A131689.

Cf. A346983, A354242, A365567.

m:=20; R:=LaurentSeriesRing(RationalField(), m); b:=Coefficients(R!(1/(5 - 4*Exp(x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // Bruno Berselli, Mar 17 2014

a:= proc(n) option remember;
      `if`(n=0, 1, 4* add(binomial(n, k) *a(k), k=0..n-1))
    end:
seq(a(n), n=0..20);

max = 16; f[x_] := 1/(5-4*E^x); CoefficientList[Series[f[x], {x, 0, max}], x]*Range[0, max]! (* Jean-François Alcover, Nov 14 2011, after g.f. *)

my(N=25,x='x+O('x^N)); Vec(serlaplace(1/(5 - 4*exp(x)))) \\ Joerg Arndt, Jan 15 2024

def A094416(n,k): return sum(factorial(j)*n^j*stirling_number2(k,j) for j in range(k+1)) # array
def A094417(k): return A094416(4,k)
[A094417(n) for n in range(31)] # G. C. Greubel, Jan 12 2024

E.g.f.: 1/(5 - 4*exp(x)).
a(n) = 4 * A050353(n) for n>0.
a(n) = Sum_{k=0..n} A131689(n,k) * 4^k. - Philippe Deléham, Nov 03 2008
E.g.f.: A(x) with A_n = 4 * Sum_{k=0..n-1} C(n,k) * A_k; A_0 = 1. - Vladimir Kruchinin, Jan 27 2011
G.f.: 2/G(0), where G(k)= 1 + 1/(1 - 8*x*(k+1)/(8*x*(k+1) - 1 + 10*x*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 30 2013
a(n) = log(5/4)*int {x = 0..inf} (floor(x))^n * (5/4)^(-x) dx. - Peter Bala, Feb 14 2015
a(0) = 1; a(n) = 4 * a(n-1) - 5 * Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 16 2023
From Seiichi Manyama, Jun 01 2025: (Start)
a(n) = (-1)^(n+1)/5 * Li_{-n}(5/4), where Li_{n}(x) is the polylogarithm function.
a(n) = (1/5) * Sum_{k>=0} k^n * (4/5)^k.
a(n) = (4/5) * Sum_{k=0..n} 5^k * (-1)^(n-k) * A131689(n,k) for n > 0. (End)

A351200 Number of patterns of length n with all distinct runs.

Original entry on oeis.org

1, 1, 3, 11, 53, 305, 2051, 15731, 135697, 1300869, 13726431, 158137851, 1975599321, 26607158781, 384347911211, 5928465081703, 97262304328573, 1691274884085061, 31073791192091251, 601539400910369671, 12238270940611270161, 261071590963047040241

Offset: 0

Gus Wiseman, Feb 09 2022

nonn

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217.

			The a(1) = 1 through a(3) = 11 patterns:
  (1)  (1,1)  (1,1,1)
       (1,2)  (1,1,2)
       (2,1)  (1,2,2)
              (1,2,3)
              (1,3,2)
              (2,1,1)
              (2,1,3)
              (2,2,1)
              (2,3,1)
              (3,1,2)
              (3,2,1)
The complement for n = 3 counts the two patterns (1,2,1) and (2,1,2).

Andrew Howroyd, Table of n, a(n) for n = 0..200
Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

The version for run-lengths instead of runs is A351292.
A000670 counts patterns, ranked by A333217.
A005649 counts anti-run patterns, complement A069321.
A005811 counts runs in binary expansion.
A032011 counts patterns with distinct multiplicities.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A060223 counts Lyndon patterns, necklaces A019536, aperiodic A296975.
A131689 counts patterns by number of distinct parts.
A238130 and A238279 count compositions by number of runs.
A297770 counts distinct runs in binary expansion.
A345194 counts alternating patterns, up/down A350354.
Counting words with all distinct runs:
- A351013 = compositions, for run-lengths A329739, ranked by A351290.
- A351016 = binary words, for run-lengths A351017.
- A351018 = binary expansions, for run-lengths A032020, ranked by A175413.
- A351202 = permutations of prime factors.
- A351642 = word structures.
Row sums of A351640.
Cf. A003242, A098504, A098859, A106356, A242882, A325545, A328592, A329740, A351014, A351204, A351291.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]] /@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n],UnsameQ@@Split[#]&]],{n,0,6}]

\\ here LahI is A111596 as row polynomials.
LahI(n,y)={sum(k=1, n, y^k*(-1)^(n-k)*(n!/k!)*binomial(n-1, k-1))}
S(n)={my(p=prod(k=1, n, 1 + y*x^k + O(x*x^n))); 1 + sum(i=1, (sqrtint(8*n+1)-1)\2, polcoef(p,i,y)*LahI(i,y))}
R(q)={[subst(serlaplace(p), y, 1) | p<-Vec(q)]}
seq(n)={my(q=S(n)); concat([1], sum(k=1, n, R(q^k-1)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) ))} \\ Andrew Howroyd, Feb 12 2022

Terms a(10) and beyond from Andrew Howroyd, Feb 12 2022

A351292 Number of patterns of length n with all distinct run-lengths.

Original entry on oeis.org

1, 1, 1, 5, 5, 9, 57, 61, 109, 161, 1265, 1317, 2469, 3577, 5785, 43901, 47165, 86337, 127665, 204853, 284197, 2280089, 2398505, 4469373, 6543453, 10570993, 14601745, 22502549, 159506453, 171281529, 314077353, 462623821, 742191037, 1031307185, 1580543969, 2141246229

Offset: 0

Gus Wiseman, Feb 10 2022

nonn

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217.

			The a(1) = 1 through a(5) = 9 patterns:
  (1)  (1,1)  (1,1,1)  (1,1,1,1)  (1,1,1,1,1)
              (1,1,2)  (1,1,1,2)  (1,1,1,1,2)
              (1,2,2)  (1,2,2,2)  (1,1,1,2,2)
              (2,1,1)  (2,1,1,1)  (1,1,2,2,2)
              (2,2,1)  (2,2,2,1)  (1,2,2,2,2)
                                  (2,1,1,1,1)
                                  (2,2,1,1,1)
                                  (2,2,2,1,1)
                                  (2,2,2,2,1)
The a(6) = 57 patterns grouped by sum:
  111111  111112  111122  112221  111223  111233  112333  122333
          111211  111221  122211  111322  111332  113332  133322
          112111  122111  211122  112222  112223  122233  221333
          211111  221111  221112  211222  113222  133222  223331
                                  221113  122222  211333  333122
                                  222112  211133  222133  333221
                                  222211  221222  222331
                                  223111  222113  233311
                                  311122  222122  331222
                                  322111  222221  332221
                                          222311  333112
                                          233111  333211
                                          311222
                                          322211
                                          331112
                                          332111

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The version for runs instead of run-lengths is A351200.
A000670 counts patterns, ranked by A333217.
A005649 counts anti-run patterns, complement A069321.
A005811 counts runs in binary expansion.
A032011 counts patterns with distinct multiplicities.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A060223 counts Lyndon patterns, necklaces A019536, aperiodic A296975.
A131689 counts patterns by number of distinct parts.
A238130 and A238279 count compositions by number of runs.
A165413 counts distinct run-lengths in binary expansion, runs A297770.
A345194 counts alternating patterns, up/down A350354.
Counting words with all distinct runs:
- A351013 = compositions, for run-lengths A329739, ranked by A351290.
- A351016 = binary words, for run-lengths A351017.
- A351018 = binary expansions, for run-lengths A032020, ranked by A175413.
- A351202 = permutations of prime factors.
- A351638 = word structures.
Row sums of A350824.
Cf. A003242, A098504, A098859, A106356, A239455, A242882, A325545, A328592, A329740, A351014, A351293.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n],UnsameQ@@Length/@Split[#]&]],{n,0,6}]

P(n) = {Vec(-1 + prod(k=1, n, 1 + y*x^k + O(x*x^n)))}
R(u,k) = {k*[subst(serlaplace(p)/y, y, k-1) | p<-u]}
seq(n)={my(u=P(n), c=poldegree(u[#u])); concat([1], sum(k=1, c, R(u, k)*sum(r=k, c, binomial(r, k)*(-1)^(r-k)) ))} \\ Andrew Howroyd, Feb 11 2022

From Andrew Howroyd, Feb 12 2022: (Start)
a(n) = Sum_{k=1..n} R(n,k)*(Sum_{r=k..n} binomial(r, k)*(-1)^(r-k)), where R(n,k) = Sum_{j=1..floor((sqrt(8*n+1)-1)/2)} k*(k-1)^(j-1) * j! * A008289(n,j).
G.f.: 1 + Sum_{r>=1} Sum_{k=1..r} R(k,x) * binomial(r, k)*(-1)^(r-k), where R(k,x) = Sum_{j>=1} k*(k-1)^(j-1) * j! * [y^j](Product_{k>=1} 1 + y*x^k).
(End)

Terms a(10) and beyond from Andrew Howroyd, Feb 11 2022

A094418 Generalized ordered Bell numbers Bo(5,n).

Original entry on oeis.org

1, 5, 55, 905, 19855, 544505, 17919055, 687978905, 30187495855, 1490155456505, 81732269223055, 4931150091426905, 324557348772511855, 23141780973332248505, 1776997406800302687055, 146197529083891406394905, 12829862285488250150167855, 1196280147496701351115120505

Offset: 0

Ralf Stephan, May 02 2004

nonn

Fifth row of array A094416, which has more information.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.

Cf. A094416, A094417, A094419, A094422, A131689.

Cf. A346984, A365568, A365569, A365570.

A094416:= func< n,k | (&+[Factorial(j)*n^j*StirlingSecond(k,j): j in [0..k]]) >;
A094418:= func< k | A094416(5,k) >;
[A094418(n): n in [0..30]]; // G. C. Greubel, Jan 12 2024

t = 30; Range[0, t]! CoefficientList[Series[1/(6 - 5 Exp[x]), {x, 0, t}], x] (* Vincenzo Librandi, Mar 16 2014 *)

my(N=25,x='x+O('x^N)); Vec(serlaplace(1/(6 - 5*exp(x)))) \\ Joerg Arndt, Jan 15 2024

def A094416(n,k): return sum(factorial(j)*n^j*stirling_number2(k,j) for j in range(k+1)) # array
def A094418(k): return A094416(5,k)
[A094418(n) for n in range(31)] # G. C. Greubel, Jan 12 2024

E.g.f.: 1/(6 - 5*exp(x)).
a(n) = Sum_{k=0..n} A131689(n,k) * 5^k. - Philippe Deléham, Nov 03 2008
a(n) ~ n! / (6*(log(6/5))^(n+1)). - Vaclav Kotesovec, Mar 14 2014
a(0) = 1; a(n) = 5 * Sum_{k=1..n} binomial(n,k) * a(n-k). - Ilya Gutkovskiy, Jan 17 2020
a(0) = 1; a(n) = 5 * a(n-1) - 6 * Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 16 2023
From Seiichi Manyama, Jun 01 2025: (Start)
a(n) = (-1)^(n+1)/6 * Li_{-n}(6/5), where Li_{n}(x) is the polylogarithm function.
a(n) = (1/6) * Sum_{k>=0} k^n * (5/6)^k.
a(n) = (5/6) * Sum_{k=0..n} 6^k * (-1)^(n-k) * A131689(n,k) for n > 0. (End)

A094419 Generalized ordered Bell numbers Bo(6,n).

Original entry on oeis.org

1, 6, 78, 1518, 39390, 1277646, 49729758, 2258233998, 117196187550, 6842432930766, 443879517004638, 31674687990494478, 2465744921215207710, 207943837884583262286, 18885506918597311159518, 1837699347783655374914958, 190743171535070652261555870, 21035482423625416328497024206

Offset: 0

Ralf Stephan, May 02 2004

nonn

Sixth row of array A094416, which has more information.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Wikipedia, Polylogarithm.

Cf. A032033, A094416, A094417, A094418, A131689.
Cf. A346985, A354252, A365555, A365556, A365557.
Cf. A384514, A384521, A384522, A384523.

A094416:= func< n,k | (&+[Factorial(j)*n^j*StirlingSecond(k,j): j in [0..k]]) >;
A094419:= func< k | A094416(6,k) >;
[A094419(n): n in [0..30]]; // G. C. Greubel, Jan 12 2024

t = 30; Range[0, t]! CoefficientList[Series[1/(7 - 6 Exp[x]),{x, 0, t}], x] (* Vincenzo Librandi, Mar 16 2014 *)

my(N=25,x='x+O('x^N)); Vec(serlaplace(1/(7-6*exp(x)))) \\ Joerg Arndt, Jan 15 2024

a(n) = (-1)^(n+1)*polylog(-n, 7/6)/7; \\ Seiichi Manyama, Jun 01 2025

def A094416(n,k): return sum(factorial(j)*n^j*stirling_number2(k,j) for j in range(k+1)) # array
def A094419(k): return A094416(6,k)
[A094419(n) for n in range(31)] # G. C. Greubel, Jan 12 2024

E.g.f.: 1/(7 - 6*exp(x)).
a(n) = Sum_{k=0..n} A131689(n,k) * 6^k. - Philippe Deléham, Nov 03 2008
a(n) ~ n! / (7*(log(7/6))^(n+1)). - Vaclav Kotesovec, Mar 14 2014
a(0) = 1; a(n) = 6 * Sum_{k=1..n} binomial(n,k) * a(n-k). - Ilya Gutkovskiy, Jan 17 2020
a(0) = 1; a(n) = 6 * a(n-1) - 7 * Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 17 2023
From Seiichi Manyama, Jun 01 2025: (Start)
a(n) = (-1)^(n+1)/7 * Li_{-n}(7/6), where Li_{n}(x) is the polylogarithm function.
a(n) = (1/7) * Sum_{k>=0} k^n * (6/7)^k.
a(n) = (6/7) * Sum_{k=0..n} 7^k * (-1)^(n-k) * A131689(n,k) for n > 0. (End)

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)

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

A122704 a(n) = Sum_{k=0..n} 3^(n-k)*A123125(n, k).

Original entry on oeis.org

1, 1, 4, 22, 160, 1456, 15904, 202672, 2951680, 48361216, 880405504, 17630351872, 385148108800, 9114999832576, 232311251144704, 6343764407375872, 184778982658539520, 5718564661248065536, 187389427488113557504, 6481629887083387420672, 235993351028007334051840

Offset: 0

Philippe Deléham, Oct 22 2006

a(n+1) = [1,4,22,160,1456,...] is the first Eulerian transform of A000244 (powers of 3), it is also the Stirling transform of A080599(n+1) = [1,3,12,66,450,...].

			G.f. = 1 + x + 4*x^2 + 22*x^3 + 160*x^4 + 1456*x^5 + 15904*x^6 + ... - _Michael Somos_, Jun 05 2021

Alois P. Heinz, Table of n, a(n) for n = 0..409 (first 101 terms from G. C. Greubel)
Amya Luo, Pattern Avoidance in Nonnesting Permutations, Undergraduate Thesis, Dartmouth College (2024). See p. 16.
T. J. Stieltjes, Sur quelques intégrales définies et leur développement en fractions continues, Q. J. Math., London, 24, 1890, pp. 370-382.
T. J. Stieltjes, Sur quelques intégrales définies et leur développement en fractions continues, LXXVII, p.382, Stieltjes T.J. Oeuvres complètes, tome 2, Noordhoff, 1918, 617p.
OEIS Wiki, Eulerian polynomials.
Eric Weisstein's World of Mathematics, Polylogarithm.

Cf. A076726.

# From Peter Luschny, Jun 27 2019: (Start)
seq(subs(x=3, add(combinat:-eulerian1(n,k)*x^k, k=0..n)), n=0..20);
# Alternative:
gf := -2/(exp(2*x) - 3): series(gf, x, 21): seq(n!*coeff(%, x, n), n=0..20);
# (End)
# Or via the recurrence of the Fubini polynomials:
F := proc(n) option remember; if n = 0 then return 1 fi;
expand(add(binomial(n, k)*F(n - k)*x, k = 1..n)) end:
a := n -> 2^n*subs(x = 1/2, F(n)):
seq(a(n), n = 0..20); # Peter Luschny, May 21 2021
# next Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n, j)*2^(j-1), j=1..n))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, May 30 2021

CoefficientList[Series[(Exp[x]-2*Cosh[x])/(2*Exp[x]-3*Cosh[x]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 24 2013 *)
Table[Sum[2^(n+1)*k^n/3^(k+1), {k, 0, Infinity}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 28 2013 *)
Round@Table[(-1)^(n+1) (PolyLog[-n, Sqrt[3]] + PolyLog[-n, -Sqrt[3]])/3, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 31 2015 *)
Table[Sum[StirlingS2[n, k]*2^(n-k)*k!, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 13 2018 *)
Eulerian1[0, 0] = 1; Eulerian1[n_, k_] := Sum[(-1)^j (k-j+1)^n Binomial[ n+1, j], {j, 0, k+1}]; Table[Sum[Eulerian1[n, k]*3^k, {k, 0, n}], {n, 0, 20}] (* Jean-François Alcover, Jul 13 2019, after Peter Luschny *)
a[n_] := (-2)^(n + 1) PolyLog[-n, 3] / 3;
Table[a[n], {n, 0, 20}] (* Peter Luschny, Aug 20 2021 *)

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

{a(n) = if(n<0, 0, n!*polcoeff( 2/(3 - exp(2*x + x*O(x^n))), n))}; /* Michael Somos, Jun 05 2021 */

O.g.f.: Sum_{n>=0} n! * x^n / Product_{k=1..n} (1-2*k*x). - Paul D. Hanna, Jul 20 2011
a(n) = Sum_{k=0..n} A131689(n,k)*2^(n-k). - Philippe Deléham, Oct 09 2007
a(n) = A_{n}(3) where A_{n}(x) are the Eulerian polynomials. - Peter Luschny, Aug 03 2010
E.g.f.: (exp(x) - 2*cosh(x))/(2*exp(x) - 3*cosh(x)) =1 + x/(U(0)-x) where U(k)=  4*k+1 - x/(1 + x/(4*k+3 - x/(1 + x/U(k+1)))); (continued fraction, 4-step). - Sergei N. Gladkovskii, Nov 08 2012
G.f.: 1 + x/G(0) where G(k) = 1 - x*2*(2*k+2) + x^2*(k+1)*(k+2)*(1-2^2)/G(k+1); (continued fraction due to T. J. Stieltjes). - Sergei N. Gladkovskii, Jan 11 2013
a(n) ~ n!/3 * (2/log(3))^(n+1). - Vaclav Kotesovec, Jun 24 2013
G.f.: 1/Q(0), where Q(k) = 1 - x*(4*k+1) - 3*x^2*(k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Sep 30 2013
a(n) = Sum_{k>=0} 2^(n+1)*k^n/3^(k+1). - Vaclav Kotesovec, Nov 28 2013
a(n) = 2^n*log(3)* Integral_{x >= 0} (floor(x))^n * 3^(-x) dx. - Peter Bala, Feb 14 2015
From Karol A. Penson, Sep 04 2015: (Start)
E.g.f.: 2/(3-exp(2*x)).
Special values of the generalized hypergeometric function n_F_(n-1):
a(n) = (2^(n+1)/9) * hypergeom([2,2,..2],[1,1,..1],1/3), where the sequence in the first square bracket ("upper" parameters) has n elements all equal to 2 whereas the sequence in the second square bracket ("lower" parameters) has n-1 elements all equal to 1.
Example: a(4) = (2^5/9) * hypergeom([2,2,2,2],[1,1,1],1/3) = 16. (End)
a(n) = (-1)^(n+1)*(Li_{-n}(sqrt(3)) + Li_{-n}(-sqrt(3)))/3, where Li_n(x) is the polylogarithm. - Vladimir Reshetnikov, Oct 31 2015
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * 2^(k - 1) * a(n-k). - Ilya Gutkovskiy, Jan 16 2020
a(n) = 2^n*F_{n}(1/2), where F_{n}(x) is the Fubini polynomial. This is another way to state the above formula from Ilya Gutkovskiy. - Peter Luschny, May 21 2021
a(n+1) = -2*a(n) + 3*Sum_{k=0..n} binomial(n, k)*a(k)*a(n-k). - Michael Somos, Jun 05 2021
a(n) = (-2)^(n + 1)*PolyLog(-n, 3)/3. - Peter Luschny, Aug 20 2021

a(7) corrected (was 206672) and more terms from Peter Luschny, Aug 03 2010

More terms from Vaclav Kotesovec, Jul 13 2018

cp's OEIS Frontend

A063007 T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.

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A094417 Generalized ordered Bell numbers Bo(4,n).

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A351200 Number of patterns of length n with all distinct runs.

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A351292 Number of patterns of length n with all distinct run-lengths.

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A094418 Generalized ordered Bell numbers Bo(5,n).

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