A090582 -id:A090582 - cp's OEIS frontend

A049019 Irregular triangle read by rows: Row n gives numbers of preferential arrangements (onto functions) of n objects that are associated with the partition of n, taken in Abramowitz and Stegun order.

1, 1, 2, 1, 6, 6, 1, 8, 6, 36, 24, 1, 10, 20, 60, 90, 240, 120, 1, 12, 30, 20, 90, 360, 90, 480, 1080, 1800, 720, 1, 14, 42, 70, 126, 630, 420, 630, 840, 5040, 2520, 4200, 12600, 15120, 5040, 1, 16, 56, 112, 70, 168, 1008, 1680, 1260, 1680, 1344, 10080, 6720

Offset: 1

Alford Arnold

This is a refinement of A019538 with row sums in A000670.
From Tom Copeland, Sep 29 2008: (Start)
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.
Given the n X n lower triangular matrix M = [ binomial(j,k) u(j-k) ], the first column of the inverse matrix M^(-1) contains the (n-1) rows of A049019 as the coefficients of the multinomials formed from the u(j). M^(-1) can be computed as (1/u(0)){I - [I- M/u(0)]^n} / {I - [I- M/u(0)]} = - u(0)^(-n) {sum(j=1 to n)(-1)^j bin(n,j) u(0)^(n-j) M^(j-1)} where I is the identity matrix.
Another method for computing the coefficients and partitions up to (n-1) rows is to use (1-x^n)/ (1-x) = 1+x^2+x^3+ ... + x^(n-1) with x replaced either by [I- M/a(0)] or [1- g(x)/a(0)] with the n X n matrix M = [bin(j,k) a(j-k)] and g(x)= a(0) + a(1)x + a(2)x^2/2! + ... + a(n) x^n/n!. The first n terms (rows of the first column) of the resulting series (matrix) divided by a(0) contain the (n-1) rows of signed coefficients and associated partitions for A049019.
To obtain unsigned coefficients, change a(j) to -a(j) for j>0. A133314 contains other matrices and recursion formulas that could be used. The Faa di Bruno formula gives the coefficients as n! [e(1)+e(2)+...+e(n)]! / [1!^e(1) e(1)! 2!^e(2) e(2)!... n!^e(n) e(n)! ] for the partition of form [a(1)^e(1)...a(n)^e(n)] with [e(1)+2e(2)+...+ n e(n)] = n (see Abramowitz and Stegun pages 823 and 831) in agreement with Arnold's formula. (End)

			Irregular triangle starts (note the grouping by ';' when comparing with A019538):
[1] 1;
[2] 1;  2;
[3] 1;  6;  6;
[4] 1;  8,  6; 36;  24;
[5] 1; 10, 20; 60,  90; 240; 120;
[6] 1; 12, 30, 20;  90, 360,  90; 480, 1080; 1800; 720;
[7] 1; 14, 42, 70; 126, 630, 420, 630;  840, 5040, 2520; 4200, 12600; 15120; 5040;
.
a(17) = 240 because we can write
A048996(17)*A036038(17) = 4*60 = A036040(17)*A036043(17)! = 10*24.
As in A133314, 1/exp[u(.)*x] = u(0)^(-1) [ 1 ] + u(0)^(-2) [ -u(1) ] x + u(0)^(-3) [ -u(0)u(2) + 2 u(1)^2 ] x^2/2! + u(0)^(-4) [ -u(0)^2 u(3) + 6 u(0)u(1)u(2) - 6 u(1)^3 ] x^3/3! + u(0)^(-5) [ -u(0)^3 u(4) + 8 u(0)^2 u(1)u(3) + 6 u(0)^2 u(2)^2 - 36 u(0)u(1)^2 u(2) + 24 u(1)^4 ] x^4/4! + ... . These are essentially refined face polynomials for permutohedra: empty set + point + line segment + hexagon + 3-D- permutohedron + ... . - _Tom Copeland_, Oct 04 2008

Peter Luschny, Rows n = 1..36, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.
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.
Tom Copeland, Bijective mapping between face polytopes of permutohedra and partitions of integers, Math StackExchange question, 2016.
S. Forcey, The Hedra Zoo.
X. Gao, S. He, and Y. Zhan, Labelled tree graphs, Feynman diagrams and disk integrals , arXiv:1708.08701 [hep-th], 2017.
J. Loday, The Multiple Facets of the Associahedron
V. Pilaud, The Associahedron and its Friends, presentation for Seminaire Lotharingien de Combinatoire, April 4 - 6, 2016.
A. Postnikov, Positive Grassmannian and Polyhedral Subdivisions, arXiv:1806.05307 [math.CO], (cf. p. 17), 2018.

Cf. A000041, A000670, A019538, A036038, A036040, A036043, A048996, A133314.

def A049019(n):
    if n == 0: return [1]
    P = lambda k: Partitions(n, min_length=k, max_length=k)
    Q = (p.to_list() for k in (1..n) for p in P(k))
    return [factorial(len(p))*SetPartitions(sum(p), p).cardinality() for p in Q]
for n in (1..7): print(A049019(n)) # Peter Luschny, Aug 30 2019

a(n) = A048996(n) * A036038(n);
a(n) = A036040(n) * factorial(A036043(n)).
A lowering operator for the unsigned multinomials in the brackets in the example is [d/du(1) 1/POP] where u(1) is treated as a continuous variable and POP is an operator that pulls off the number of parts of a partition ignoring u(0), e.g., [d/du(1) 1/POP][ u(0)u(2) + 2 u(1)^2 ] = (1/2) 2*2 u(1) = 2*u(1), analogous to the prototypical delta operator (d/dz) z^n = n z^(n-1). - Tom Copeland, Oct 04 2008
From the matrix formulation with M_m,k = 1/(m-k)!; g(x) = exp[ u(.) x]; an orthonormal vector basis x_1, ..., x_n and En(x^k) = x_k for k <= n and zero otherwise, for j=0 to n-1 the j-th signed row multinomial is given by the wedge product of x_1 with the wedge product (-1)^j * j! * u(0)^(-n) * Wedge{ En[x g(x), x^2 g(x), ..., x^(j) g(x), ~, x^(j+2) g(x), ..., x^n g(x)] } where Wedge{a,b,c} = a v b v c (the usual wedge symbol is inverted here to prevent confusion with the power notation, see Mathworld) and the (j+1)-th element is omitted from the product. Tom Copeland, Oct 06 2008 [Changed an x^n to x^(n-1) and "inner product of x_1" to "wedge". - Tom Copeland, Feb 03 2010]

Partitions for 7 and 8 from Tom Copeland, Oct 02 2008

Definition edited by N. J. A. Sloane, Nov 06 2023

A248727 A046802(x,y) --> A046802(x,y+1), transform of e.g.f. for the graded number of positroids of the totally nonnegative Grassmannians G+(k,n); enumerates faces of the stellahedra.

Original entry on oeis.org

1, 2, 1, 5, 5, 1, 16, 24, 10, 1, 65, 130, 84, 19, 1, 326, 815, 720, 265, 36, 1, 1957, 5871, 6605, 3425, 803, 69, 1, 13700, 47950, 65646, 44240, 15106, 2394, 134, 1, 109601, 438404, 707840, 589106, 267134, 63896, 7094, 263, 1

Offset: 0

Tom Copeland, Oct 12 2014

This is a transform of A046802 treating it as an array of h-vectors, so y is replaced by (y+1) in the e.g.f. for A046802.
An e.g.f. for the reversed row polynomials with signs is given by exp(a.(0;t)x) = [e^{(1+t)x} [1+t(1-e^(-x))]]^(-1) = 1 - (1+2t)x + (1+5t+5t^2)x^2/2! + ... . The reciprocal is an e.g.f. for the reversed face polynomials of the simplices A074909, i.e., exp(b.(0;t)x) = e^{(1+t)x} [1+t(1-e^(-x))] = 1 + (1+2t)x +(1+3t+3t^2) x^2/2! + ... , so the relations of A133314 apply between the two sets of polynomials. In particular, umbrally [a.(0;t)+b.(0;t)]^n vanishes except for n=0 for which it's unity, implying the two sets of Appell polynomials formed from the two bases, a_n(z;t) = (a.(0;t)+z)^n and b_n(z;t) = (b.(0;t) + z)^n, are an umbral compositional inverse pair, i.e., b_n(a.(x;t);t)= x^n = a_n(b.(x;t);t). Raising operators for these Appell polynomials are related to the polynomials of A028246, whose reverse polynomials are given by A123125 * A007318. Compare: A248727 = A007318 * A123125 * A007318 and A046802 = A007318 * A123125. See A074909 for definitions and related links. - Tom Copeland, Jan 21 2015
The o.g.f. for the umbral inverses is Og(x) = x / (1 - x b.(0;t)) = x / [(1-tx)(1-(1+t)x)] = x + (1+2t) x^2 + (1+3t+3t^2) x^3 + ... . Its compositional inverse is an o.g.f for signed A033282, the reverse f-polynomials for the simplicial duals of the Stasheff polytopes, or associahedra of type A, Oginv(x) =[1+(1+2t)x-sqrt[1+2(1+2t)x+x^2]] / (2t(1+t)x) = x - (1+2t) x^2 + (1+5t+5t^2) x^3 + ... . Contrast this with the o.g.f.s related to the corresponding h-polynomials in A046802. - Tom Copeland, Jan 24 2015
Face vectors, or coefficients of the face polynomials, of the stellahedra, or stellohedra. See p. 59 of Buchstaber and Panov. - Tom Copeland, Nov 08 2016
See A008279 for a relation between the e.g.f.s enumerating the faces of permutahedra and stellahedra. - Tom Copeland, Nov 14 2016

			The triangle T(n, k) starts:
n\k    0     1     2     3     4    5   6  7 ...
1:     1
2:     2     1
3:     5     5     1
4:    16    24    10     1
5:    65   130    84    19     1
6:   326   815   720   265    36    1
7:  1957  5871  6605  3425   803   69   1
8: 13700 47950 65646 44240 15106 2394 134  1
... reformatted, _Wolfdieter Lang_, Mar 27 2015

P. Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
L. Berry, S. Forcey, M. Ronco, and P. Showers, Polytopes and Hopf algebras of painted trees: Fan graphs and Stellohedra, arXiv:1608.08546 [math.CO], 2018.
L. Berry, S. Forcey, M. Ronco, and P. Showers, Species substitution, graph suspension, and graded Hopf algebras of painted tree polytopes, arXiv:1608.08546 [math.CO], 2019.
V. Buchstaber and T. Panov, Toric Topology, arXiv:1210.2368v3 [math.AT], 2014.
R. Da Rosa, D. Jensen, and D. Ranganathan, Toric graph associahedra and compactifications of M_(0,n), arXiv:1411.0537 [math.AG], 2015.
S. Forcey, M. Ronco, and P. Showers, Polytopes and algebras of grafted trees: Stellohedra, arXiv:1608.08546v2 [math.CO], 2016.
Stefan Forcey, The Hedra Zoo
I. Limonchenko, Moment-angle manifolds, 2-truncated cubes and Massey operations, arXiv:1510.07778 [math.AT], 2017.
M. Lin, Graph Cohomology, 2016, (Fig. 2.5 is a stellahedron).
T. Manneville and V. Pilaud, Compatibility fans for graphical nested complexes, arXiv:1501.07152v3 [math.CO], 2015-2016.
MathOverflow, Analogue of conic sections for the permutohedra, associahedra, and noncrossing partitions, an MO question posed by T. Copeland, 2017. (See Buchstaber references therein.)
V. Pilaud, The Associahedron and its Friends, presentation for Séminaire Lotharingien de Combinatoire, April 4-6, 2016. [From _Tom Copeland_, Jun 26 2018]

Cf. A046802, A007047, A122045, A000522, A036918, A001339, A052944.
Cf. A074909, A028246, A133314, A007318, A123125, A033282.
Cf. A008279, A008292, A090582, A097808, A130850.
Cf. A019538, A119879.

(* t = A046802 *) t[, 1] = 1; t[n, n_] = 1; t[n_, 2] = 2^(n - 1) - 1; t[n_, k_] = Sum[((i - k + 1)^i*(k - i)^(n - i - 1) - (i - k + 2)^i*(k - i - 1)^(n - i - 1))*Binomial[n - 1, i], {i, 0, k - 1}]; T[n_, j_] := Sum[Binomial[k, j]*t[n + 1, k + 1], {k, j, n}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 23 2015, after Tom Copeland *)

Let M(n,k)= sum{i=0,..,k-1, C(n,i)[(i-k)^i*(k-i+1)^(n-i)- (i-k+1)^i*(k-i)^(n-i)]} with M(n,0)=1. Then M(n,k)= A046802(n,k), and T(n,j)= sum(k=j,..,n, C(k,j)*M(n,k)) for j>0 with T(n,0)= 1 + sum(k=1,..,n, M(n,k)) for n>0 and T(0,0)=1.
E.g.f: y * exp[x*(y+1)]/[y+1-exp(x*y)].
Row sums are A007047. Row polynomials evaluated at -1 are unity. Row polynomials evaluated at -2 are A122045.
First column is A000522. Second column appears to be A036918/2 = (A001339-1)/2 = n*A000522(n)/2.
Second diagonal is A052944. (Changed from conjecture to fact on Nov 08 2016.)
The raising operator for the reverse row polynomials with row signs is R = x - (1+t) - t e^(-D) / [1 + t(1-e^(-D))] evaluated at x = 0, with D = d/dx. Also R = x - d/dD log[exp(a.(0;t)D], or R = - d/dz log[e^(-xz) exp(a.(0;t)z)] = - d/dz log[exp(a.(-x;t)z)] with the e.g.f. defined in the comments and z replaced by D. Note that t e ^(-D) / [1+t(1-e^(-D))] = t - (t+t^2) D + (t+3t^2+2t^3) D^2/2! - ... is an e.g.f. for the signed reverse row polynomials of A028246. - Tom Copeland, Jan 23 2015
Equals A007318*(padded A090582)*A007318*A097808 = A007318*(padded (A008292*A007318))*A007318*A097808 = A007318*A130850 = A007318*(mirror of A028246). Padded means in the same way that A097805 is padded A007318. - Tom Copeland, Nov 14 2016
Umbrally, the row polynomials are p_n(x) = (1 + q.(x))^n, where (q.(x))^k = q_k(x) are the row polynomials of A130850. - Tom Copeland, Nov 16 2016
From the previous umbral statement, OP(x,d/dy) y^n =  (y + q.(x))^n, where OP(x,y) = exp[y * q.(x)] = x/((1+x)*exp(-x*y) - 1), the e.g.f. of A130850, so OP(x,d/dy) y^n evaluated at y = 1 is  p_n(x), the n-th row polynomial of this entry, with offset 0. - Tom Copeland, Jun 25 2018
Consolidating some formulas in this entry and A046082, in umbral notation for concision, 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 this entry (A248727, the face polynomials of the stellahedra), by f_n(x) = A_n(1 + x;1); the Swiss-knife polynomials of 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

Title expanded by Tom Copeland, Nov 08 2016

A130850 Triangle read by rows, 0 <= k <= n, T(n,k) = Sum_{j=0..n} A(n,j)*binomial(n-j,k) where A(n,j) are the Eulerian numbers A173018.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 6, 12, 7, 1, 24, 60, 50, 15, 1, 120, 360, 390, 180, 31, 1, 720, 2520, 3360, 2100, 602, 63, 1, 5040, 20160, 31920, 25200, 10206, 1932, 127, 1, 40320, 181440, 332640, 317520, 166824, 46620, 6050, 255, 1, 362880, 1814400, 3780000, 4233600, 2739240, 1020600, 204630, 18660, 511, 1

Offset: 0

Philippe Deléham, Aug 20 2007

Old name was: Triangle T(n,k), 0<=k<=n, read by rows given by [1,1,2,2,3,3,4,4,5,5,...] DELTA [1,0,2,0,3,0,4,0,5,0,6,0,...] where DELTA is the operator defined in A084938.
Vandervelde (2018) refers to this as the Worpitzky number triangle - N. J. A. Sloane, Mar 27 2018 [Named after the German mathematician Julius Daniel Theodor Worpitzky (1835-1895). - Amiram Eldar, Jun 24 2021]
Triangle given by A123125*A007318 (as infinite lower triangular matrices), A123125 = Euler's triangle, A007318 = Pascal's triangle; A007318*A123125 gives A046802.
Taylor coefficients of Eulerian polynomials centered at 1. - Louis Zulli, Nov 28 2015
A signed refinement is A263634. - Tom Copeland, Nov 14 2016
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 A119879, by Sw_n(x) = A_n(-1;1 + x); and the row polynomials of this entry (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 begins:
1
1      1
2      3       1
6      12      7       1
24     60      50      15      1
120    360     390     180     31      1
720    2520    3360    2100    602     63      1
5040   20160   31920   25200   10206   1932    127    1
40320  181440  332640  317520  166824  46620   6050   255   1
362880 1814400 3780000 4233600 2739240 1020600 204630 18660 511 1
...

G. C. Greubel, Table of n, a(n) for n = 0..1034 [a(438) and a(901) corrected by _Georg Fischer_, Nov 10 2021]
Nguyen-Huu-Bong, Some Combinatorial Properties of Summation Operators, J. Comb. Theory, Ser. A, Vo. 11, No. 3 (1971), pp. 213-221. See Table on page 214.
John K. Sikora, On Calculating the Coefficients of a Polynomial Generated Sequence Using the Worpitzky Number Triangles, arXiv:1806.00887 [math.NT], 2018.
Sam Vandervelde, The Worpitzky Numbers Revisited, Amer. Math. Monthly, Vol. 125, No. 3 (2018), pp. 198-206.

Essentially reverse of A028246.
Variants are A130850, A075263, A163626.
Cf. A000142, A001710, A005460, A005461, A005462, A005463, A005464, A263634.
Cf. A019538, A028246, A046802, A090582, A119879, A123125, A248727.

Table[(n-k)!*StirlingS2[n+1, n-k+1], {n, 0, 10}, {k, 0, n}] (* G. C. Greubel, Nov 15 2015 *)

t(n, k) = (n-k)!*stirling(n+1, n-k+1, 2);
tabl(nn) = for (n=0, 10, for (k=0, n, print1(t(n,k),", ")); print()); \\ Michel Marcus, Nov 16 2015

from sage.combinat.combinat import eulerian_number
def A130850(n, k):
    return add(eulerian_number(n, j)*binomial(n-j, k) for j in (0..n))
for n in (0..7): [A130850(n, k) for k in (0..n)] # Peter Luschny, May 21 2013

T(n,k) = (-1)^k*A075263(n,k).
T(n,k) = (n-k)!*A008278(n+1,k+1).
T(n,n-1) = 2^n - 1 for n > 0. - Derek Orr, Dec 31 2015
E.g.f.: x/(e^(-x*t)*(1+x)-1). - Tom Copeland, Nov 14 2016
Sum_{k=1..floor(n/2)} T(n,2k) = Sum_{k=0..floor(n/2)} T(n,2k+1) = A000670(n). - Jacob Sprittulla, Oct 03 2021

New name from Peter Luschny, May 21 2013

A278075 Coefficients of the signed Fubini polynomials in ascending order, F_n(x) = Sum_{k=0..n} (-1)^nStirling2(n,k)k!*(-x)^k.

Original entry on oeis.org

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

Offset: 0

Peter Luschny, Jan 09 2017

sign

Signed version of A131689.

Integral_{x=0..1} F_n(x) = B_n(1) where B_n(x) are the Bernoulli polynomials.

			Triangle of coefficients starts:
[1]
[0,  1]
[0, -1,    2]
[0,  1,   -6,    6]
[0, -1,   14,  -36,    24]
[0,  1,  -30,  150,  -240,   120]
[0, -1,   62, -540,  1560, -1800,    720]
[0,  1, -126, 1806, -8400, 16800, -15120, 5040]

Peter Luschny, Illustration of the polynomials.
Peter Luschny, The Bernoulli Manifesto.
Grzegorz Rządkowski, Bernoulli numbers and solitons - revisited, Journal of Nonlinear Mathematical Physics, (2010) 17:1, 121-126.
J. Worpitzky, Studien über die Bernoullischen und Eulerschen Zahlen, Journal für die reine und angewandte Mathematik, 94 (1883), 203-232.

Row sums are A000012, diagonal is A000142.
Cf. A131689 (unsigned), A019538 (n>0, k>0), A090582.
Let F(n, x) = Sum_{k=0..n} T(n,k)*x^k then, apart from possible differences in the sign or the offset, we have: F(n, -5) = A094418(n), F(n, -4) = A094417(n), F(n, -3) = A032033(n), F(n, -2) = A004123(n), F(n, -1) = A000670(n), F(n, 0) = A000007(n), F(n, 1) = A000012(n), F(n, 2) = A000629(n), F(n, 3) = A201339(n), F(n, 4) = A201354(n), F(n, 5) = A201365(n).
Cf. A163626, A173018.

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

F := (n,x) -> add((-1)^n*Stirling2(n,k)*k!*(-x)^k, k=0..n):
for n from 0 to 10 do PolynomialTools:-CoefficientList(F(n,x), x) od;

T[ n_, k_] := If[ n < 0 || k < 0, 0, (-1)^(n - k) k! StirlingS2[n, k]]; (* Michael Somos, Jul 08 2018 *)

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

T(n, k) = (-1)^(n-k) * Stirling2(n, k) * k!.
E.g.f.: 1/(1-x*(1-exp(-t))) = Sum_{n>=0} F_n(x) t^n/n!.
T(n, k) = k*(T(n-1, k-1) - T(n-1, k)) for 0 <= k <= n, T(0, 0) = 1, otherwise 0.
Bernoulli numbers are given by B(n) = Sum_{k = 0..n} T(n, k) / (k+1) with B(1) = 1/2. - Michael Somos, Jul 08 2018
Let F_n(x) be the row polynomials of this sequence and W_n(x) the row polynomials of A163626. Then F_n(1 - x) = W_n(x) and Integral_{x=0..1} U(n, x) = Bernoulli(n, 1) for U in {W, F}. - Peter Luschny, Aug 10 2021
T(n, k) = [z^k] Sum_{k=0..n} Eulerian(n, k)*z^(k+1)*(z-1)^(n-k-1) for n >= 1, where Eulerian(n, k) = A173018(n, k). - Peter Luschny, Aug 15 2022

A084416 Triangle read by rows: T(n,k) = Sum_{i=k..n} i!*Stirling2(n,i), n >= 1, 1 <= k <= n.

Original entry on oeis.org

1, 3, 2, 13, 12, 6, 75, 74, 60, 24, 541, 540, 510, 360, 120, 4683, 4682, 4620, 4080, 2520, 720, 47293, 47292, 47166, 45360, 36960, 20160, 5040, 545835, 545834, 545580, 539784, 498960, 372960, 181440, 40320, 7087261, 7087260, 7086750, 7068600, 6882120, 6048000, 4142880, 1814400, 362880

Offset: 1

N. J. A. Sloane, Jun 24 2003

Interpolates between A000670 and factorials.
From Thomas Scheuerle, Apr 25 2022: (Start)
Number of preferential arrangements of n labeled elements when at least k ranks are required.
This sequence starts for k and n with offset 1. If it would start with k = 0, we would observe in column k = 0 an exact copy of column k = 1 with a preceding one at n = 0, k = 0. The difference between 0 ranks and one rank (all in the same rank) is only for n = 0 where k = 0 allows zero-filled ranks as an valid arrangement, too. (End)

			Triangle begins with T(n,k):
   k=   1,   2,   3,   4,   5
  n=1   1
  n=2   3,   2
  n=3  13,  12,   6
  n=4  75,  74,  60,  24
  n=5 541, 540, 510, 360, 120
...
From _Thomas Scheuerle_, Apr 25 2022: (Start)
If we would add n = 0, k = 0 to the data of this sequence:
   k=   0,   1,   2,
  n=0   1
  n=1   1,   1
  n=2   3,   3,   2
...
T(n, 3) with 3 preceding zeros is: 0,0,0,6,60,510,4620,...
This sequence has the e.g.f.: (e^x-1)^3/(2-e^x).
.
13 arrangements for n = 3 and k = 1 (one rank required):
1,2,3  1,2|3  2,3|1  1,3|2  1|2,3  2|1,3  3|1,2  1|2|3  1|3|2  2|1|3  2|3|1  3|1|2  3|2|1
12 arrangements for n = 3 and k = 2 (two ranks required):
1,2|3  2,3|1  1,3|2  1|2,3  2|1,3  3|1,2  1|2|3  1|3|2  2|1|3  2|3|1  3|1|2  3|2|1
6 arrangements for n = 3 and k = 3 (three ranks required):
1|2|3  1|3|2  2|1|3  2|3|1  3|1|2  3|2|1
. (End)

Mirror image of array in A084417.
Cf. A005649, A069321 (row sums).
A000670(n) (column k = 1), A052875(n) (column k = 2), A102232(n) (column k = 3).
Cf. A000142, A008277.

T := (n,k)->sum(i!*Stirling2(n,i),i=k..n): seq(seq(T(n,k),k=1..n),n=1..10);

row(n) = vector(n, k, sum(i=k, n, i!*stirling(n, i, 2))); \\ Michel Marcus, Apr 20 2022

E.g.f. for m-th column: (exp(x)-1)^m/(2-exp(x)). - Vladeta Jovovic, Sep 14 2003
T(n, k) = Sum_{m = k..n} A090582(n + 1, m + 1).
From Thomas Scheuerle, Apr 25 2022: (Start)
Sum_{k = 0..n} T(n, k) = A005649(n). Column k = 0 is not part of data.
Sum_{k = 1..n} T(n, k) = A069321(n).
T(n, 0) = A000670(n). Column k = 0 is not part of data.
T(n, 1) = A000670(n), for n > 0.
T(n, 2) = A052875(n).
T(n, 3) = A102232(n).
T(n, n) = n! = A000142. (End)

More terms from Emeric Deutsch, May 11 2004

More terms from Michel Marcus, Apr 20 2022

A073593 Number of cards needed to be drawn (with replacement) from a deck of n cards to have a 50% or greater chance of seeing each card at least once.

Original entry on oeis.org

1, 2, 5, 7, 10, 13, 17, 20, 23, 27, 31, 35, 38, 42, 46, 51, 55, 59, 63, 67, 72, 76, 81, 85, 90, 94, 99, 104, 108, 113, 118, 123, 128, 133, 137, 142, 147, 152, 157, 162, 167, 173, 178, 183, 188, 193, 198, 204, 209, 214, 219, 225, 230, 235, 241, 246, 251, 257, 262

Offset: 1

Robert G. Wilson v, Aug 28 2002

nonn

A version of the coupon collector's problem (A178923).

W. Feller, An Introduction to Probability Theory and Its Applications: Volume 1.
S. Ross, A First Course in Probability, Prentice-Hall, 3rd ed., Chapter 7, Example 3g.

Jens Kruse Andersen, Table of n, a(n) for n = 1..1000
P. Erdős and A. Rényi, On a classical problem of probability theory, Magyar Tudományos Akadémia Matematikai Kutató Intézetének Közleményei, 1961
Sci.math.probability newsgroup, Collecting a deck of cards on the street, Aug 2002.
Math.stackexchange, The smallest number of boxes to buy to have probability at least 1/2 of collecting all pictures, Oct 2014.

Cf. A090582, A178923 (Coupon Collector's Problem).

f[n_] := Block[{k = 1}, While[2StirlingS2[k, n]*n!/n^k < 1, k++ ]; k]; Table[ f[n], {n, 60}]

S2(n,k) = if(k<1 || k>n,0,if(n==1,1,k*S2(n-1,k)+S2(n-1,k-1))); a(n)=if(n<0,0,k=1; while( 2*S2(k,n)*n!/n^k<1,k++); k)

a(n)=v=vector(n+1); k=1; v[n]=1.0; while(v[1]<0.5, k++; for(i=1, n, v[i]=v[i]*(n+1-i)/n+v[i+1]*i/n)); k \\ Faster program. Jens Kruse Andersen, Aug 03 2014

a(n) seems to be asymptotic to n*(log(n)+c) with c=0.3(6)...and maybe c=1/e. - Benoit Cloitre, Sep 07 2002

c likely to be closer to -log(log(2)) about 0.3665. - Henry Bottomley, Jun 01 2022

A085813 Number of cards that need to be drawn (with replacement) from a deck of n cards to have a 95% or greater chance of seeing each card at least once.

Original entry on oeis.org

1, 6, 11, 16, 21, 27, 33, 38, 44, 51, 57, 63, 70, 76, 83, 90, 96, 103, 110, 117, 124, 131, 138, 145, 152, 159, 167, 174, 181, 189, 196, 203, 211, 218, 226, 233, 241, 248, 256, 264, 271, 279, 287, 294, 302, 310, 318, 326, 333, 341, 349, 357, 365, 373, 381, 389

Offset: 1

Alfred Heiligenbrunner, Jul 25 2003

nonn

The probability that there are at most k different cards in t drawings is (k/m)^t * binomial(m,k). This also includes the cases with k-1 different cards, which we want to subtract. Inclusion and exclusion leads to the formula Sum_{k=1..m} (-1)^(m-k) * (k/m)^t * binomial(m,k).

			a(2)=6 because you have to throw a coin 6 times to get both sides at least once with probability greater than or equal to 0.95. (The probability of getting only one side in a series of 6 throws is (1/2)^6 * 2 = 1/32 = 0.03125 < 0.05.)
a(6)=27 because you have to roll a die 27 times to see all 6 possible outcomes with a probability over 0.95. (If you roll a die 27 times, the probability of getting all 6 sides at least once is 0.95658638... . If you roll the die only 26 times, the probability is 0.94798274... .)

Cf. A073593 (number of drawings for a 50% probability of seeing each card, = median) and A060293 (expected value of the number of drawings until each card is drawn once).

Cf. A090582.

f[1] = 1; f[n_] := f[n] = Block[{k = f[n - 1]}, While[ 2StirlingS2[k, n]*n!/n^k < 19/10, k++ ]; k]; Table[ f[n], {n, 1, 56}]

More terms from Robert G. Wilson v, Sep 07 2003

A300305 Expected rounded number of draws until two persons simultaneously drawing cards with replacement from two decks of cards with m and n cards, respectively, both obtain complete collections, written as triangle T(m,n), 1 <= n <= m.

Original entry on oeis.org

1, 3, 4, 6, 6, 7, 8, 8, 9, 10, 11, 11, 12, 12, 14, 15, 15, 15, 15, 16, 18, 18, 18, 18, 18, 19, 20, 22, 22, 22, 22, 22, 22, 23, 24, 26, 25, 25, 25, 26, 26, 26, 27, 29, 31, 29, 29, 29, 29, 29, 30, 31, 32, 33, 35, 33, 33, 33, 33, 33, 34, 34, 35, 36, 38, 40, 37, 37, 37, 37, 37, 37, 38, 38, 39, 41, 42, 45

Offset: 1

Hugo Pfoertner, Mar 07 2018

This is the two-person version of the coupon collector's problem.

			T(1,1)=1, T(2,1)=3, T(2,2)=round(11/3)=4, T(3,1)=round(11/2)=6, T(3,2)=round(57/10)=6, T(3,3)=round(1909/280)=7.
The triangle starts:
   1
   3   4
   6   6   7
   8   8   9  10
  11  11  12  12  14
  15  15  15  15  16  18
  18  18  18  18  19  20  22
  22  22  22  22  22  23  24  26
  25  25  25  26  26  26  27  29  31
  ...

IBM Research, Ponder This Challenge February 2018. Solution for 3 persons from Robert Lang.

Cf. A073593, A090582, A135736 (first column in triangle), A300306 (diagonal in triangle).

T(m,n) = round(1 - Sum_{j=0..m} Sum_{k=0..n} ( (-1)^(m-j+n-k) * binomial(m,j) * binomial(n,k) * j * k / (m*n-j*k) )) excluding term with j=m and k=n in summation.

A344397 a(n) = Stirling2(n, floor(n/2)) * floor(n/2)!.

Original entry on oeis.org

1, 0, 1, 1, 14, 30, 540, 1806, 40824, 186480, 5103000, 29607600, 953029440, 6711344640, 248619571200, 2060056318320, 86355926616960, 823172919528960, 38528927611574400, 415357755774998400, 21473732319740064000, 258323865658578720000, 14620825330739032204800

Offset: 0

Peter Luschny, May 21 2021

nonn

Cf. A131689, A210029, A019538, A090582.

a := n -> add((-1)^k*binomial((2*n-1)/4 + (-1)^n/4,k)*((2*n-1)/4 + (-1)^n/4 - k)^n, k = 0..n/2):
# Alternative, via Fubini recurrence:
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 -> coeff(F(n), x, iquo(n, 2));
seq(a(n), n = 0..22);

a[n_] := StirlingS2[n, Floor[n/2]] * Floor[n/2]!; Array[a, 23, 0] (* Amiram Eldar, May 22 2021 *)

def a(n): return stirling_number2(n, n//2) * factorial(n//2)
print([a(n) for n in range(23)])

a(n) = [x^(floor(n/2))] F(n, x), the middle coefficient of the Fubini polynomial.

a(n) = Sum_{k=0..n/2} (-1)^k*binomial((2*n - 1)/4 + (-1)^n/4, k)*((2*n - 1)/4 + (-1)^n/4 - k)^n.

cp's OEIS Frontend

A049019 Irregular triangle read by rows: Row n gives numbers of preferential arrangements (onto functions) of n objects that are associated with the partition of n, taken in Abramowitz and Stegun order.

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A248727 A046802(x,y) --> A046802(x,y+1), transform of e.g.f. for the graded number of positroids of the totally nonnegative Grassmannians G+(k,n); enumerates faces of the stellahedra.

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A130850 Triangle read by rows, 0 <= k <= n, T(n,k) = Sum_{j=0..n} A(n,j)*binomial(n-j,k) where A(n,j) are the Eulerian numbers A173018.

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A278075 Coefficients of the signed Fubini polynomials in ascending order, F_n(x) = Sum_{k=0..n} (-1)^n*Stirling2(n,k)*k!*(-x)^k.

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A084416 Triangle read by rows: T(n,k) = Sum_{i=k..n} i!*Stirling2(n,i), n >= 1, 1 <= k <= n.

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A073593 Number of cards needed to be drawn (with replacement) from a deck of n cards to have a 50% or greater chance of seeing each card at least once.

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A085813 Number of cards that need to be drawn (with replacement) from a deck of n cards to have a 95% or greater chance of seeing each card at least once.

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A278075 Coefficients of the signed Fubini polynomials in ascending order, F_n(x) = Sum_{k=0..n} (-1)^nStirling2(n,k)k!*(-x)^k.