A028246 -id:A028246 - cp's OEIS frontend

A005462 Number of simplices in barycentric subdivision of n-simplex.

1, 31, 602, 10206, 166824, 2739240, 46070640, 801496080, 14495120640, 273158645760, 5368729766400, 110055327782400, 2351983118284800, 52361635508582400, 1213240925049753600, 29227769646147072000, 731310069474496512000, 18984684514588176384000

Offset: 3

N. J. A. Sloane

R. Austin, R. K. Guy, and R. Nowakowski, unpublished notes, circa 1987.
R. K. Guy, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Micah Manary, Table of n, a(n) for n = 3..138
R. Austin, R. K. Guy, and R. Nowakowski, Unpublished notes, 1987
Rajesh Kumar Mohapatra and Tzung-Pei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics (2022) Vol. 10, No. 7, 1161.

Cf. A001298, A005460, A005461, A028246.

[Factorial(n-3)*StirlingSecond(n+2,n-2): n in [3..30]]; // G. C. Greubel, Nov 22 2022

Table[(n-3)!*StirlingS2[n+2,n-2], {n,3,30}] (* G. C. Greubel, Nov 22 2022 *)

[factorial(n-3)*stirling_number2(n+2,n-2) for n in range(3,31)] # G. C. Greubel, Nov 22 2022

Essentially Stirling numbers of second kind - see A028246.

a(n) = Stirling2(n+2,n-2)*(n-3)!. - Alois P. Heinz, Aug 28 2022

A075263 Triangle of coefficients of polynomials H(n,x) formed from the first (n+1) terms of the power series expansion of ( -x/log(1-x) )^(n+1), multiplied by n!.

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

Paul D. Hanna, Sep 13 2002

Special values: H(n,1)=0, H(2n,2)=0, H(n,-x) ~= ( x/log(1+x) )^(n+1), for x>0. H'(n,1) = -1/n!, where H'(n,x) = d/dx H(n,x).
The zeros of these polynomials are all positive reals >= 1. If we order the zeros of H(n,x), {r_k, k=0..(n-1)}, by magnitude so that r_0 = 1, r_k > r_(k-1), for 0 < k < n, then r_(n-k) = r_k/(r_k - 1) when 0 < k < n, n > 1, where r_(n/2) = 2 for even n.
Also Product_{k=0..(n-1)} r_k = n!, r_(n-1) ~ C 2^n.
I believe that these numbers are the coefficients of the Eulerian polynomials An(z) written in powers of z-1. That is, the sequence is: A0(1); A1(1), A1'(1); A2(1), A2'(1), A2''(1)/2!; A3(1), A3'(1), A3''(1)/2!, A3'''(1)/3!; A4(1), A4'(1), A4''(1)/2!, A4'''(1)/3!, A4''''(1)/4! etc. My convention: A0(z)=z, A1(z)=z, A2(z)=z+z^2, A3(z)=z+4z^2+z^3, A4(z)=z+11z^2+11z^3+z^4, etc. - Louis Zulli (zullil(AT)lafayette.edu), Jan 19 2005
H(n,2) gives 1,-1,0,2,0,-16,0,272,0,-7936,0,..., see A009006. - Philippe Deléham, Aug 20 2007
Row sums are zero except for first row. - Roger L. Bagula, Sep 11 2008
From Groux Roland, May 12 2011: (Start)
Let f(x) = (exp(x)+1)^(-1) then the n-th derivative of f equals Sum_{k=0..n} T(n,k)*(f(x))^(n+1-k).
T(n+1,0) = (n+1)*T(n,0); T(n+1,n+1) = -T(n,n) and for 0 < k < n T(n+1,k) = (n+1-k) * T(n,k) - (n-k+2)*T(n,k-1).
T(n,k) = Sum_{i=0..k} (-1)^(i+k)*(n-i)!*binomial(n-i,k-i)*S(n,n-i) where S(n,k) is a Stirling number of the second kind. (End)

			H(0,x) = 1
H(1,x) = (1 - 1*x)/1!
H(2,x) = (2 - 3*x + 1*x^2)/2!
H(3,x) = (6 - 12*x + 7*x^2 - 1*x^3)/3!
H(4,x) = (24 - 60*x + 50*x^2 - 15*x^3 + 1*x^4)/4!
H(5,x) = (120 - 360*x + 390*x^2 - 180*x^3 + 31*x^4 - 1*x^5)/5!
H(6,x) = (720 - 2520*x + 3360*x^2 - 2100*x^3 + 602*x^4 - 63*x^5 + 1*x^5)/6!
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;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Nguyen-Huu-Bong, Some Combinatorial Properties of Summation Operators, J. Comb. Theory, Ser. A 11.3 (1971): 213-221. See Table on page 214.

Cf. A028246, A075264, A084938, A123125.

Cf. Eulerian numbers (A008292).

Flat(List([0..12], n-> List([0..n], k-> Sum([0..n-k], j->
(-1)^(n-j)*Binomial(n-k,j)*(j+1)^n )))); # G. C. Greubel, Jan 27 2020

T:= func< n,k | &+[(-1)^(n-j)*Binomial(n-k,j)*(j+1)^n: j in [0..n-k]] >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 27 2020

CL := f -> PolynomialTools:-CoefficientList(f,x):
T_row := n -> `if`(n=0, [1], CL(x^(n+1)*polylog(-n, 1-x))):
for n from 0 to 6 do T_row(n) od; # Peter Luschny, Sep 28 2017

Table[CoefficientList[x^(n+1)*Sum[k^n*(1-x)^k, {k, 0, Infinity}], x], {n, 0, 10}]//Flatten (* Roger L. Bagula, Sep 11 2008 *)
p[x_, n_]:= x^(n+1)*PolyLog[-n, 1-x]; Table[CoefficientList[p[x, n], x], {n, 0, 10}]//Flatten (* Roger L. Bagula and Gary W. Adamson, Sep 15 2008 *)

T(n,k)=if(k<0 || k>n,0,n!*polcoeff((-x/log(1-x+x^2*O(x^n)))^(n+1),k))
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

T(n,k)=sum(i=0,n-k,(-1)^(n-i)*binomial(n-k,i)*(i+1)^n)
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

/* Using e.g.f. A(x,y): */
{T(n,k)=local(X=x+x*O(x^n),Y=y+y^2*O(y^(k))); n!*polcoeff(polcoeff(-log(1-(1-exp(-X*Y))/y),n,x),k,y)}
for(n=0,10,for(k=0,n-1,print1(T(n,k),", "));print(""))

/* Deléham's DELTA: T(n,k) = [x^(n-k)*y^k] P(n,0) */
{P(n,k)=if(n<0||k<0,0,if(n==0,1, P(n,k-1)+(x*(k\2+1)+y*(-(k\2+1)*((k+1)%2)))*P(n-1,k+1)))}
{T(n,k)=polcoeff(polcoeff(P(n,0),n-k,x),k,y)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

def T(n, k): return sum( (-1)^(n-j)*binomial(n-k, j)*(j+1)^n for j in (0..n-k))
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jan 27 2020

Generated by [1, 1, 2, 2, 3, 3, ...] DELTA [ -1, 0, -2, 0, -3, 0, ...], where DELTA is the operator defined in A084938.
T(n, k) = Sum_{i=0..n-k} (-1)^(n-i)*C(n-k, i)*(i+1)^n; n >= 0, 0 <= k <= n. - Paul D. Hanna, Jul 21 2005
E.g.f.: A(x, y) = -log(1-(1-exp(-x*y))/y). - Paul D. Hanna, Jul 21 2005
p(x,n) = x^(n + 1)*Sum_{k>=0} k^n*(1 - x)^k; t(n,m) = Coefficients(p(x,n)). - Roger L. Bagula, Sep 11 2008
p(x,n) = x^(n + 1)*PolyLog(-n, 1 - x); t(n,m) = coefficients(p(x,n)) for n >= 1. - Roger L. Bagula and Gary W. Adamson, Sep 15 2008

Error in one term corrected by Benoit Cloitre, Aug 20 2007

A053440 Number of k-simplices in the first derived complex of the standard triangulation of an n-simplex. Equivalently, T(n,k) is the number of ascending chains of length k+1 of nonempty subsets of the set {1, 2, ..., n+1}.

Original entry on oeis.org

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

Offset: 0

Rob Arthan, Jan 12 2000

T(n,k) is the number of length k+1 sequences of nonempty mutually disjoint subsets of {1,2,...,n+1}. The e.g.f. for the column corresponding to k is exp(x)*(exp(x)-1)^(k+1). - Geoffrey Critzer, Dec 20 2011

			T(2,1) = 12 because there are 12 such length 2 sequences of subsets of {1,2,3}: ({1},{2}), ({1},{3}), ({2},{3}), ({1},{2,3}), ({2},{1,3}), ({3},{1,2}) with two orderings for each. - _Geoffrey Critzer_, Dec 20 2011
Triangle begins:
   1
   3      2
   7     12      6
  15     50     60     24
  31    180    390    360    120

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
F. Brenti and V. Welker, f-vectors of barycentric subdivisions Math. Z., 259(4), 849-865, 2008.
Wikipedia, Barycentric subdivision

Other versions are A028246, A142071.
Columns k=0..1 are A000225(n+1), A028243(n+2).
Cf. A000142 (main diagonal), A002050 (row sums), A019538.

a := (n, k) -> (k+1)!*Stirling2(n+2, k+2):
seq(print(seq(a(n, k), k = 0..n)), n = 0..10);

nn = 5; a = Exp[ x] - 1 ; f[list_] := Select[list, # > 0 &];Map[f, Transpose[Table[Drop[Range[0, nn]!CoefficientList[Series[a^k  Exp[x], {x, 0, nn}],x], 1], {k, 1, 5}]]] // Grid (* Geoffrey Critzer, Dec 20 2011 *)
Table[(k+1)!*StirlingS2[n+2,k+2], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 19 2017 *)

for(n=0,10, for(k=0,n, print1((k+1)!*stirling(n+2,k+2,2), ", "))) \\ G. C. Greubel, Nov 19 2017

T(0,k) = delta(0,k), T(n,k) = delta(0,k) + (k+1)(T(n-1,k-1) + (k+2)T(n-1,k)).
E.g.f.: exp(x)*(exp(x)-1)/(1-y*(exp(x)-1)). - Vladeta Jovovic, Apr 13 2003
T(n,k) = Sum_{i = 0..n} binomial(n+1,i+1)*(k+1)!*Stirling2(i+1,k+1) = (k+1)!*Stirling2(n+2,k+2) (Brenti and Welker). - Peter Bala, Jul 12 2014
T(n,k) = (k+1)!*Stirling2(n+2, k+2). - G. C. Greubel, Nov 19 2017

More terms from James Sellers, Jan 14 2000

A005463 Number of simplices in barycentric subdivision of n-simplex.

Original entry on oeis.org

1, 63, 1932, 46620, 1020600, 21538440, 451725120, 9574044480, 207048441600, 4595022432000, 105006251750400, 2475732702643200, 60284572969420800, 1516762345722624000, 39433286715863040000, 1059143615076298752000, 29378569022287220736000, 841159994641469927424000

Offset: 4

N. J. A. Sloane

nonn

R. Austin, R. K. Guy, and R. Nowakowski, unpublished notes, circa 1987.
R. K. Guy, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 4..440
R. Austin, R. K. Guy, and R. Nowakowski, Unpublished notes, 1987
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Rajesh Kumar Mohapatra and Tzung-Pei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics (2022) Vol. 10, No. 7, 1161.

Cf. A005460, A005461, A005462, A005464, A005465.

Cf. A028246, A112494.

[Factorial(n-4)*StirlingSecond(n+2,n-3): n in [4..35]]; // G. C. Greubel, Nov 22 2022

a:= n-> Stirling2(2+n,n-3)*(n-4)!:
seq(a(n), n=4..21);  # Alois P. Heinz, Apr 27 2022

Table[(n-4)!*StirlingS2[n+2, n-3], {n,4,35}] (* G. C. Greubel, Nov 22 2022 *)

[factorial(n-4)*stirling_number2(n+2,n-3) for n in range(4,36)] # G. C. Greubel, Nov 22 2022

Essentially Stirling numbers of second kind - see A028246.

a(n) = (n-4)! * Stirling2(n+2, n-3). - Alois P. Heinz, Apr 27 2022

More terms from Alois P. Heinz, Apr 27 2022

A005464 Number of simplices in barycentric subdivision of n-simplex.

Original entry on oeis.org

1, 127, 6050, 204630, 5921520, 158838240, 4115105280, 105398092800, 2706620716800, 70309810771200, 1858166876966400, 50148628078348800, 1385482985542656000, 39245951652171264000, 1140942623868343296000, 34060437199245929472000, 1044402668566817624064000, 32895725269182358302720000

Offset: 5

N. J. A. Sloane

R. Austin, R. K. Guy, and R. Nowakowski, unpublished notes, circa 1987.
R. K. Guy, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 5..440
R. Austin, R. K. Guy, and R. Nowakowski, Unpublished notes, 1987
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Rajesh Kumar Mohapatra and Tzung-Pei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics (2022) Vol. 10, No. 7, 1161.

Cf. A005460, A005461, A005462, A005463, A005465.

Cf. A028246, A144969.

[Factorial(n-5)*StirlingSecond(n+2,n-4): n in [5..35]]; // G. C. Greubel, Nov 22 2022

seq((d+2)!*(63*d^5-945*d^4+5355*d^3-13951*d^2+15806*d-5304)/2903040,d=5..30) ; # R. J. Mathar, Mar 19 2018

Table[(n-5)!*StirlingS2[n+2, n-4], {n,5,35}] (* G. C. Greubel, Nov 22 2022 *)

[factorial(n-5)*stirling_number2(n+2,n-4) for n in range(5,36)] # G. C. Greubel, Nov 22 2022

Essentially Stirling numbers of second kind - see A028246.

a(n) = (n-5)! * Stirling2(n+2, n-4). - G. C. Greubel, Nov 22 2022

A161742 Third left hand column of the RSEG2 triangle A161739.

Original entry on oeis.org

1, 4, 13, 30, -14, -504, 736, 44640, -104544, -10644480, 33246720, 5425056000, -20843695872, -5185511654400, 23457840537600, 8506857655296000, -44092609863966720, -22430879475779174400, 130748316971139072000

Offset: 2

Johannes W. Meijer & Nico Baken (n.h.g.baken(AT)tudelft.nl), Jun 18 2009

Equals third left hand column of A161739 (RSEG2 triangle).
Other left hand columns are A129825 and A161743.
A008955 is a central factorial number triangle.
A028246 is Worpitzky's triangle.
A001710 (n!/2!), A001715 (n!/3!), A001720 (n!/4!), A001725 (n!/5!), A001730 (n!/6!), A049388 (n!/7!), A049389 (n!/8!), A049398 (n!/9!), A051431 (n!/10!) appear in Maple program.

nmax:=21; for n from 0 to nmax do A008955(n,0):=1 end do: for n from 0 to nmax do A008955(n,n):=(n!)^2 end do: for n from 1 to nmax do for m from 1 to n-1 do A008955(n,m):= A008955(n-1,m-1)*n^2+A008955(n-1,m) end do: end do: for n from 1 to nmax do A028246(n,1):=1 od: for n from 1 to nmax do A028246(n,n):=(n-1)! od: for n from 3 to nmax do for m from 2 to n-1 do A028246(n,m):=m*A028246(n-1,m)+(m-1)*A028246(n-1,m-1) od: od: for n from 2 to nmax do a(n):=sum(((-1)^k/((k+1)!*(k+2)!)) *(n!)*A028246(n,k+2)* A008955(k+1,k),k=0..n-2) od: seq(a(n),n=2..nmax);

a(n) = sum(((-1)^k/((k+1)!*(k+2)!))*(n!)*A028246(n, k+2)*A008955(k+1, k), k=0..n-2)

A161743 Fourth left hand column of the RSEG2 triangle A161739.

Original entry on oeis.org

1, 10, 73, 425, 1561, -2856, -73520, 380160, 15376416, -117209664, -7506967104, 72162155520, 7045087741056, -80246202992640, -11448278791372800, 149576169325363200, 30017051616972275712, -440857664887810867200

Offset: 3

Johannes W. Meijer & Nico Baken (n.h.g.baken(AT)tudelft.nl), Jun 18 2009

Equals fourth left hand column of A161739 (RSEG2 triangle).
Other left hand columns are A129825 and A161742.
A008955 is a central factorial number triangle.
A028246 is Worpitzky's triangle.
A001710 (n!/2!), A001715 (n!/3!), A001720 (n!/4!), A001725 (n!/5!), A001730 (n!/6!), A049388 (n!/7!), A049389 (n!/8!), A049398 (n!/9!), A051431 (n!/10!) appear in Maple program.

nmax:=21; for n from 0 to nmax do A008955(n,0):=1 end do: for n from 0 to nmax do A008955(n,n):=(n!)^2 end do: for n from 1 to nmax do for m from 1 to n-1 do A008955(n,m):= A008955(n-1,m-1)*n^2+A008955(n-1,m) end do: end do: for n from 1 to nmax do A028246(n,1):=1 od: for n from 1 to nmax do A028246(n,n):=(n-1)! od: for n from 3 to nmax do for m from 2 to n-1 do A028246(n,m):=m*A028246(n-1,m)+(m-1)*A028246(n-1,m-1) od: od: for n from 3 to nmax do a(n) := sum(((-1)^k/((k+2)!*(k+3)!))*(n!)*A028246(n,k+3)* A008955(k+2,k), k=0..n-3) od: seq(a(n),n=3..nmax);

a(n) = sum(((-1)^k/((k+2)!*(k+3)!))*(n!)*A028246(n, k+3)*A008955(k+2, k), k = 0..n-3).

A162508 A binomial sum of powers related to the Bernoulli numbers, triangular array, read by rows.

Original entry on oeis.org

-1, -2, 2, -4, 10, -6, -8, 38, -54, 24, -16, 130, -330, 336, -120, -32, 422, -1710, 3000, -2400, 720, -64, 1330, -8106, 21840, -29400, 19440, -5040, -128, 4118, -36414, 141624, -285600, 312480, -176400, 40320

Offset: 1

Peter Luschny, Jul 05 2009

T(n,k) = sum_{v=0..k} (-1)^v*v*binomial(k,v)*(v+1)^(n-1)
for n >= 1, k >= 1; by convention T(0,0) = 1.
Gives a representation of the Bernoulli numbers B_{n} = B_{n}(1) (with B_1 = 1/2)
B_{n} = sum_{j=0..n} sum_{k=0..j} T(j,k)/(k+1)
T(n,1) = -2^(n-1) (n>=1)
T(n,n) = (-1)^n*n! (n>=1)
sum_{k=0..n} T(n,k) = -A000007(n-1) = -1,0,0,0,0,... (n>=1)
sum_{k=0..n} abs(T(n,k)) = A162509(n) = A073146(n,n-1) (n>=1)
sum_{k=0..n} T(n,k)/(k+1) = Bernoulli(n,1)-Bernoulli(n-1,1) (n>=1)
numer(sum(T(n,k)/(k+1),k=0..n)) = A051716(n) (n>=0)
denom(sum(T(n,k)/(k+1),k=0..n)) = A051717(n) (n>=0)
Contribution from Peter Luschny, Jul 08 2009: (Start)
More generally, define the polynomials (assume p[0,0](x)=1 and 0^0=1)
p[n,k](x) = sum_{v=0..k} (-1)^v*v*binomial(k,v)*(v+1+x)^(n-1)
[1], [0, -1], [0, -2-x, 2], [0, -4-4x-x^2, 10+4x, -6], ...
then T(n,k)=p[n,k](0) and (-1)^k*k!*Stirling2(n,k)=p[n,k](-1) (cf. A019538).
Assume now k >= 1 and read by rows. Then
p[n,k](1) = -1,-3,2,-9,14,-6,-27,74,-72,24,-81,350,-582,432,-120,...
(-1)^n*(-2)^(n-k)*p[n,k](-1/2))=1,3,2,9,16,6,27,98,90,24,81,544,924,576,120,..
(-1)^n*(-2)^(n-k)*p[n,k](-3/2))=1,1,2,1,8,6,1,26,54,24,1,80,348,384,120,... (End)
Variant of A199400.

			For n >= 1, k >= 1:
..................... -1
................... -2, 2
................. -4, 10, -6
.............. -8, 38, -54, 24
......... -16, 130, -330, 336, -120
..... -32, 422, -1710, 3000, -2400, 720
-64, 1330, -8106, 21840, -29400, 19440, -5040

Vincenzo Librandi, Rows n = 1..50, flattened

Cf. A162509, A073146, A051716, A051717, A199400. A019538, A028246, A038719.

T := proc(n,k) local v; if n=0 and k=0 then 1 else
add((-1)^v*v*binomial(k,v)*(v+1)^(n-1),v=0..k) fi end:
# Peter Bala's e.g.f. assuming offset 0:
egf := (x, z) -> -((1-x)/exp(z) + x)^(-2):
ser := series(egf(x, z), z, 10): coz := n -> n!*coeff(ser, z, n):
row := n -> seq(coeff(coz(n), x, k), k = 0..n):
seq(print(row(n)), n = 0..9); # Peter Luschny, Jan 28 2021

t[n_, k_] := Sum[(-1)^v*v*Binomial[k, v]*(v + 1)^(n - 1), {v, 0, k}]; Table[t[n, k], {n, 1, 8}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 26 2013, after Maple *)

def A162508(n, k):
    if n==0 and k==0: return 1
    return add((-1)^v*v*binomial(k, v)*(v+1)^(n-1) for v in (0..k))
for n in (1..8): [A162508(n, k) for k in (1..n)] # Peter Luschny, Jul 21 2014

From Peter Bala, Jul 21 2014: (Start)
T(n,k) = (-1)^k*k!*( Stirling2(n+1,k+1) - Stirling2(n,k+1) ), 1 <= k <= n.
T(n,k) = (-1)^k*(k + 1)*A038719(n+1,k+1).
E.g.f.: - B(-x,z)^2, where B(x,z) = 1/((1 + x)*exp(-z) - x) = 1 + (1 + x)*z + (1 + 3*x + 2*x^2)*z^2/2! + ... is an e.g.f. for A028246 (with an offset of 0).
Recurrence: T(n,k) = (k + 1)*T(n-1,k) - k*T(n-1,k-1).
The unsigned version of the triangle equals the matrix product A007318*A019538.
Assuming this triangle is a signed version of A199400 then the n-th row polynomial R(n,x) = 1/(1 - x)*( sum {k = 1..inf} k*(k + 1)^(n-1)*(x/(x - 1))^k ), valid for x in the open interval (-inf, 1/2). (End)

More terms from Philippe Deléham, Nov 05 2011

A249163 Triangle read by rows: the positive terms of A163626.

Original entry on oeis.org

1, 1, 1, 2, 1, 12, 1, 50, 24, 1, 180, 360, 1, 602, 3360, 720, 1, 1932, 25200, 20160, 1, 6050, 166824, 332640, 40320, 1, 18660, 1020600, 4233600, 1814400, 1, 57002, 5921520, 46070640, 46569600, 3628800, 1, 173052, 33105600, 451725120, 898128000, 239500800

Offset: 0

Paul Curtz, Dec 15 2014

We have two possibilities: with or without 0's.
Without 0's:
1,
1,
1,   2,
1,  12,
1,  50,  24,
1, 180, 360,
etc.
Sum of every row: A000670(n).
First two terms of successive columns: 1, 1, 2, 12, 24, 360, ... = A211374.
With 0's:
1,   0,    0,   0,
1,   0,    0,   0,
1,   2,    0,   0,
1,  12,    0,   0,
1,  50,   24,   0,
1, 180,  360,   0,
1, 602, 3360, 720,
etc.
The columns are essentially A000012, A028243, A028246, A228909, A228911, A228913, from Stirling numbers of the second kind S(n,3), S(n,5), S(n,7), S(n,9), S(n,11), ... .

Cf. A163626, A000670, A211374; also A000012, A000392, A000481, A000771, A049447, A028243, A028246, A091137, A228909, A163626, A228911, A228913 and Worpitzky numbers for the second Bernoulli numbers A164555(n)/A027642(n).

Derivative[0][y][x] = y[x]; Derivative[1][y][x] = y[x]*(1 - y[x]); Derivative[n_][y][x] := Derivative[n][y][x] = D[Derivative[n - 1][y][x], x]; row[n_] := CoefficientList[Derivative[n][y][x], y[x]] // Rest; Table[ Select[row[n], Positive] , {n, 0, 12}] // Flatten
(* or, simply: *) Table[(-1)^k*k!*StirlingS2[n+1, k+1], {n, 0, 12}, {k, 0, n}] // Flatten // Select[#, Positive]& (* Jean-François Alcover, Dec 16 2014 *)

A293617 Array of triangles read by ascending antidiagonals, T(m, n, k) = Pochhammer(m, k) * Stirling2(n + m, k + m) with m >= 0, n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 2, 1, 0, 1, 10, 3, 7, 3, 0, 1, 15, 4, 25, 12, 2, 0, 1, 21, 5, 65, 30, 6, 1, 0, 1, 28, 6, 140, 60, 12, 15, 7, 0, 1, 36, 7, 266, 105, 20, 90, 50, 12, 0, 1, 45, 8, 462, 168, 30, 350, 195, 60, 6, 0, 1, 55, 9, 750, 252, 42, 1050, 560, 180, 24, 1, 0

Offset: 0

Peter Luschny, Oct 20 2017

			Array starts:
m\j| 0   1  2     3       4       5       6       7       8       9      10
---|-----------------------------------------------------------------------
m=0| 1,  0, 0,    0,      0,      0,      0,      0,      0,      0,      0
m=1| 1,  1, 1,    1,      3,      2,      1,      7,     12,      6,      1
m=2| 1,  3, 2,    7,     12,      6,     15,     50,     60,     24,     31
m=3| 1,  6, 3,   25,     30,     12,     90,    195,    180,     60,    301
m=4| 1, 10, 4,   65,     60,     20,    350,    560,    420,    120,   1701
m=5| 1, 15, 5,  140,    105,     30,   1050,   1330,    840,    210,   6951
m=6| 1, 21, 6,  266,    168,     42,   2646,   2772,   1512,    336,  22827
m=7| 1, 28, 7,  462,    252,     56,   5880,   5250,   2520,    504,  63987
m=8| 1, 36, 8,  750,    360,     72,  11880,   9240,   3960,    720, 159027
m=9| 1, 45, 9, 1155,    495,     90,  22275,  15345,   5940,    990, 359502
   A000217, A001296,A027480,A002378,A001297,A293475,A033486,A007531,A001298
.
m\j| ...      11      12      13      14
---|-----------------------------------------
m=0| ...,      0,      0,      0,      0, ... [A000007]
m=1| ...,     15,     50,     60,     24, ... [A028246]
m=2| ...,    180,    390,    360,    120, ... [A053440]
m=3| ...,   1050,   1680,   1260,    360, ... [A294032]
m=4| ...,   4200,   5320,   3360,    840, ...
m=5| ...,  13230,  13860,   7560,   1680, ...
m=6| ...,  35280,  31500,  15120,   3024, ...
m=7| ...,  83160,  64680,  27720,   5040, ...
m=8| ..., 178200, 122760,  47520,   7920, ...
m=9| ..., 353925, 218790,  77220,  11880, ...
         A293476,A293608,A293615,A052762, ...
.
The parameter m runs over the triangles and j indexes the triangles by reading them by rows. Let T(m, n) denote the row [T(m, n, k) for 0 <= k <= n] and T(m) denote the triangle [T(m, n) for n >= 0]. Then for instance T(2) is the triangle A053440, T(3, 2) is row 2 of A294032 (which is [25, 30, 12]) and T(3, 2, 1) = 30.
.
Remark: To adapt the sequences A028246 and A053440 to our enumeration use the exponential generating functions exp(x)/(1 - y*(exp(x) - 1)) and exp(x)*(2*exp(x) - y*exp(2*x) + 2*y*exp(x) - 1 - y)/(1 - y*(exp(x) - 1))^2 instead of those indicated in their respective entries.

Eric Weisstein's World of Mathematics, Nørlund Polynomial.

A000217(n) = T(n, 1, 0), A001296(n) = T(n, 2, 0), A027480(n) = T(n, 2, 1),
A002378(n) = T(n, 2, 2), A001297(n) = T(n, 3, 0), A293475(n) = T(n, 3, 1),
A033486(n) = T(n, 3, 2), A007531(n) = T(n, 3, 3), A001298(n) = T(n, 4, 0),
A293476(n) = T(n, 4, 1), A293608(n) = T(n, 4, 2), A293615(n) = T(n, 4, 3),
A052762(n) = T(n, 4, 4), A052787(n) = T(n, 5, 5), A000225(n) = T(1, n, 1),
A028243(n) = T(1, n, 2), A028244(n) = T(1, n, 3), A028245(n) = T(1, n, 4),
A032180(n) = T(1, n, 5), A228909(n) = T(1, n, 6), A228910(n) = T(1, n, 7),
A000225(n) = T(2, n, 0), A007820(n) = T(n, n, 0).
A028246(n,k) = T(1, n, k), A053440(n,k) = T(2, n, k), A294032(n,k) = T(3, n, k),
A293926(n,k) = T(n, n, k), A124320(n,k) = T(n, k, k), A156991(n,k) = T(k, n, n).
Cf. A293616.

A293617 := proc(m, n, k) option remember:
if m = 0 then 0^n elif k < 0 or k > n then 0 elif n = 0 then 1 else
(k+m)*A293617(m,n-1,k) + k*A293617(m,n-1,k-1) + A293617(m-1,n,k) fi end:
for m in [$0..4] do for n in [$0..6] do print(seq(A293617(m, n, k), k=0..n)) od od;
# Sample uses:
A027480 := n -> A293617(n, 2, 1): A293608 := n -> A293617(n, 4, 2):
# Flatten:
a := proc(n) local w; w := proc(k) local t, s; t := 1; s := 1;
while t <= k do s := s + 1; t := t + s od; [s - 1, s - t + k] end:
seq(A293617(n - k, w(k)[1], w(k)[2]), k=0..n) end: seq(a(n), n = 0..11);

T[m_, n_, k_] := Pochhammer[m, k] StirlingS2[n + m, k + m];
For[m = 0, m < 7, m++, Print[Table[T[m, n, k], {n,0,6}, {k,0,n}]]]
A293617Row[m_, n_] := Table[T[m, n, k], {k,0,n}];
(* Sample use: *)
A293926Row[n_] := A293617Row[n, n];

T(m,n,k) = (k + m)*T(m, n-1, k) + k*T(m, n-1, k-1) + T(m-1, n, k) with boundary conditions T(0, n, k) = 0^n; T(m, n, k) = 0 if k<0 or k>n; and T(m, 0, k) = 0^k.

T(m,n,k) = Pochhammer(m, k)*binomial(n + m, k + m)*NorlundPolynomial(n - k, -k - m).

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A005462 Number of simplices in barycentric subdivision of n-simplex.

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A075263 Triangle of coefficients of polynomials H(n,x) formed from the first (n+1) terms of the power series expansion of ( -x/log(1-x) )^(n+1), multiplied by n!.

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A053440 Number of k-simplices in the first derived complex of the standard triangulation of an n-simplex. Equivalently, T(n,k) is the number of ascending chains of length k+1 of nonempty subsets of the set {1, 2, ..., n+1}.

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A005463 Number of simplices in barycentric subdivision of n-simplex.

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A005464 Number of simplices in barycentric subdivision of n-simplex.

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A161742 Third left hand column of the RSEG2 triangle A161739.

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