A008517 -id:A008517 - cp's OEIS frontend

A111999 T(n, k) = [x^k] (-1)^nSum_{k=0..n} E2(n, n-k)(1+x)^(n-k) where E2(n, k) are the second-order Eulerian numbers. Triangle read by rows, T(n, k) for n >= 1 and 0 <= k <= n.

-1, 3, 2, -15, -20, -6, 105, 210, 130, 24, -945, -2520, -2380, -924, -120, 10395, 34650, 44100, 26432, 7308, 720, -135135, -540540, -866250, -705320, -303660, -64224, -5040, 2027025, 9459450, 18288270, 18858840, 11098780, 3678840, 623376, 40320, -34459425, -183783600, -416215800

Offset: 1

Wolfdieter Lang, Sep 12 2005

Previous name was: A triangle that converts certain binomials into triangle A008276 (diagonals of signed Stirling1 triangle A008275).
Stirling1(n,n-m) = A008275(n,n-m) = Sum_{k=0..m-1}a(m,k)*binomial(n,2*m-k).
The unsigned column sequences start with A001147, A000906 = 2*A000457, 2*|A112000|, 4*|A112001|.
The general results on the convolution of the refined partition polynomials of A133932, with u_1 = 1 and u_n = -t otherwise, can be applied here to obtain results of convolutions of these unsigned polynomials. - Tom Copeland, Sep 20 2016

			Triangle starts:
  [1]      -1;
  [2]       3,       2;
  [3]     -15,     -20,       -6;
  [4]     105,     210,      130,       24;
  [5]    -945,   -2520,    -2380,     -924,     -120;
  [6]   10395,   34650,    44100,    26432,     7308,     720;
  [7] -135135, -540540,  -866250,  -705320,  -303660,  -64224,  -5040;
  [8] 2027025, 9459450, 18288270, 18858840, 11098780, 3678840, 623376, 40320.

Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 152. Table C_{m, nu}.

Peter Bala, Diagonals of triangles with generating function exp(t*F(x)).
Steve Butler and Pavel Karasik, A note on nested sums, J. Int. Seq. 13 (2010), 10.4.4, page 4.
Tom Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms
Bishal Deb and Alan D. Sokal, Higher-order Stirling cycle and subset triangles: Total positivity, continued fractions and real-rootedness, arXiv:2507.18959 [math.CO], 2025. See p. 6.
EqWorld, Integral Transforms
David J. Jeffrey, G. A. Kalugin, N. Murdoch, Lagrange inversion and Lambert W, 17th Int'l Symp. Symb. Numer. Algor. Sci. Comp. (SYNASC 2015).
Wolfdieter Lang, First 10 rows.
Lajos Takács, On the number of distinct forests, SIAM J. Discrete Math., 3 (1990), 574-581. Table 3 gives an unsigned version of the triangle.

Row sums give A032188(m+1)*(-1)^m, m>=1. Unsigned row sums give A032188(m+1), m>=1.
Cf. A008517 (second-order Eulerian triangle) for a similar formula for |Stirling1(n, n-m)|.
Cf. A000457, A000906, A001147, A008275, A008276, A008306, A112001, A133932,

CoeffList := p -> op(PolynomialTools:-CoefficientList(p, x)):
E2 := (n, k) -> combinat[eulerian2](n, k):
poly := n -> (-1)^n*add(E2(n, n-k)*(1+x)^(n-k), k = 0..n):
seq(CoeffList(poly(n)), n = 1..8); # Peter Luschny, Feb 05 2021

a[m_, k_] := a[m, k] = Which[m < k + 1, 0, And[m == 1, k == 0], -1, k == -1, 0, True, -(2 m - k - 1)*(a[m - 1, k] + a[m - 1, k - 1])]; Table[a[m, k], {m, 9}, {k, 0, m - 1}] // Flatten (* Michael De Vlieger, Sep 23 2016 *)

a(m, k)=0 if m

From Tom Copeland, May 05 2010 (updated Sep 12 2011): (Start)

The integral from 0 to infinity w.r.t. w of

exp[-w(u+1)] (1+u*z*w)^(1/z) gives a power series, f(u,z), in z for reversed row polynomials in u of A111999, related to an Euler transform of diagonals of A008275.

Let g(u,x) be obtained from f(u,z) by replacing z^n with x^(n+1)/(n+1)!;

g(u,x)= x - u^2 x^2/2! + (2 u^3 + 3 u^4) x^3/3! - (6 u^4 + 20 u^5 + 15 u^6) x^4/4! + ... , an e.g.f. associated to f(u,z).

Then g^(-1)(u,x)=(1+u)*x - log(1+u*x) is the comp. inverse of g(u,x) in x, and, consequently, A133932 is a refinement of A111999.

With h(u,x)= 1/(dg^(-1)/dx)= (1+u*x)/(1+(1+u)*u*x),

g(u,x)=exp[x*h(u,t)d/dt] t, evaluated at t=0. Also, dg(u,x)/dx = h(u,g(u,x)). (End)

From Tom Copeland, May 06 2010: (Start)

For m,k>0, a(m,k) = Sum(j=2 to 2m-k+1): (-1)^(2m-k+1+j) C(2m-k+1,j) St1d(j,m),

where C(n,j) is the binomial coefficient and St1d(j,m) is the (j-m)-th element of the m-th subdiagonal of A008275 for (j-m)>0 and is 0 otherwise,

e.g., St1d(1,1) = 0, St1d(2,1) = -1, St1d(3,1) = -3, St1d(4,1) = -6. (End)

From Tom Copeland, Sep 03 2011 (updated Sep 12 2011): (Start)

The integral from 0 to infinity w.r.t. w of

exp[-w*(u+1)/u] (1+u*z*w)^(1/(u^2*z)) gives a power series, F(u,z), in z for the row polynomials in u of A111999.

Let G(u,x) be obtained from F(u,z) by replacing z^n with x^(n+1)/(n+1)!;

G(u,x) = x - x^2/2! + (3 + 2 u) x^3/3! - (15 + 20 u + 6 u^2) x^4/4! + ... , an e.g.f. for A111999 associated to F(u,z).

G^(-1)(u,x) = ((1+u)*u*x - log(1+u*x))/u^2 is the comp. inverse of G(u,x) in x.

With H(u,x) = 1/(dG^(-1)/dx) = (1+u*x)/(1+(1+u)*x),

G(u,x) = exp[x*H(u,t)d/dt] t, evaluated at t=0. Also, dG(u,x)/dx = H(u,G(u,x)). (End)

From Tom Copeland, Sep 16 2011: (Start)

f(u,z) and F(u,z) are expressible in terms of the incomplete gamma function Γ(v,p)(see Laplace Transforms for Power-law Functions at EqWorld):

With K(p,s) = p^(-s-1) exp(p) Γ(s+1,p),

f(u,z) = K(p,s)/(u*z) with p=(u+1)/(u*z) and s=1/z , and

F(u,z) = K(p,s)/(u*z) with p=(u+1)/(u^2*z) and s=1/(u^2*z). (End)

Diagonals of A008306 are reversed rows of A111999 (see P. Bala). - Tom Copeland, May 08 2012

New name from Peter Luschny, Feb 05 2021

A001662 Coefficients of Airey's converging factor.

Original entry on oeis.org

0, 1, 1, -1, -1, 13, -47, -73, 2447, -16811, -15551, 1726511, -18994849, 10979677, 2983409137, -48421103257, 135002366063, 10125320047141, -232033147779359, 1305952009204319, 58740282660173759, -1862057132555380307, 16905219421196907793, 527257187244811805207

Offset: 0

N. J. A. Sloane

A051711 times the coefficient in expansion of W(exp(x)) about x=1, where W is the Lambert function. - Paolo Bonzini, Jun 22 2016

The polynomials with coefficients in triangle A008517, evaluated at -1.

			G.f. = x + x^2 - x^3 - x^4 + 13*x^5 - 47*x^6 - 73*x^7 + 2447*x^8 + ... - _Michael Somos_, Jun 23 2019

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..200 (terms 29 onwards updated by Sean A. Irvine, April 25 2019)
J. R. Airey, The "converging factor" in asymptotic series and the calculation of Bessel, Laguerre and other functions, Phil. Mag., 24 (1937), 521-552 [ gives 22 terms ].
J. A. Airey, The "converging factor" in asymptotic series and the calculation of Bessel, Laguerre and other functions [Annotated scanned copy]
Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210; arXiv:math/0205301 [math.CO], 2002. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
J. M. Borwein and R. M. Corless, Emerging tools for experimental mathematics, Amer. Math. Monthly, 106 (No. 10, 1999), 889-909.
R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, On the Lambert W Function, Advances in Computational Mathematics, (5), 1996, pp. 329-359.
R. M. Corless, D. J. Jeffrey and D. E. Knuth, A sequence of series for the Lambert W Function (section 2.2).
D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions, arXiv:math/0501052 [math.CA], 2005.
Vaclav Kotesovec, Graph - the asymptotic ratio
Vladimir Kruchinin, The method for obtaining expressions for coefficients of reverse generating functions, arXiv:1211.3244 [math.CO], 2012.
J. C. P. Miller, A method for the determination of converging factors ..., Proc. Camb. Phil. Soc., 48 (1952), 243-254.
J. C. P. Miller, A method for the determination of converging factors ... [Annotated scanned copy]
F. D. Murnaghan, Airey's converging factor, Proc. Nat. Acad. Sci. USA, 69 (1972), 440-441.
F. D. Murnaghan and J. W. Wrench, Jr., The Converging Factor for the Exponential Integral, Report 1535, David Taylor Model Basin, U.S. Dept. of Navy, 1963 [ gives first 67 terms ].
N. J. A. Sloane, Letter to F. D. Murnaghan, Apr 17, 1974
J. W. Wrench, Jr., Letter to N. J. A. Sloane, 24 Apr 1974
P. Wynn, Converging factors for the Weber parabolic cylinder functions of complex argument I A, Proc. Konin. Ned. Akad. Weten., Series A, 66 (1963), 721-736.
P. Wynn, Converging factors for the Weber parabolic cylinder functions of complex argument I B, Proc. Konin. Ned. Akad. Weten., Series A, 66 (1963), 737-754.
P. Wynn, Converging factors for the Weber parabolic cylinder functions ... [Annotated scan of part 2 only]

Cf. A051711, A032188, A274447, A274448, A340556.

with(combinat); A001662 := proc(n) add((-1)^k*eulerian2(n-1,k),k=0..n-1) end:
seq(A001662(i),i=0..23); # Peter Luschny, Nov 13 2012

a[0] = 0; a[n_] := Sum[ (n+k-1)! * Sum[ (-1)^j/(k-j)! * Sum[ 1/i! * StirlingS1[n-i+j-1, j-i] / (n-i+j-1)!, {i, 0, j}] * 2^(n-j-1), {j, 0, k}], {k, 0, n-1}]; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Jul 26 2013, after Vladimir Kruchinin *)
a[ n_] := If[ n < 1, 0, 2^(n - 1) Sum[ (-2)^-j StirlingS1[n - i + j - 1, j - i] Binomial[n + k - 1, n + j - 1] Binomial[n + j - 1, i], {k, 0, n - 1}, {j, 0, k}, {i, 0, j}]]; (* Michael Somos, Jun 23 2019 *)
len := 12; gf := (1/2) (LambertW[Exp[x + 1]] - 1);
ser := Series[gf, { x, 0, len}]; norm := Table[n! 4^n, {n, 0, len}];
CoefficientList[ser, x] * norm (* Peter Luschny, Jun 24 2019 *)

a(n):= if n=0 then 1 else (sum((n+k-1)!*sum(((-1)^(j)/(k-j)!*sum((1/i! *stirling1(n-i+j-1, j-i))/(n-i+j-1)!, i, 0, j))*2^(n-j-1), j, 0, k), k, 0, n-1)); /* Vladimir Kruchinin, Nov 11 2012 */

@CachedFunction
def eulerian2(n, k):
    if k==0: return 1
    elif k==n: return 0
    return eulerian2(n-1, k)*(k+1)+eulerian2(n-1, k-1)*(2*n-k-1)
def A001662(n): return add((-1)^k*eulerian2(n-1,k) for k in (0..n-1))
[A001662(m) for m in (0..23)] # Peter Luschny, Nov 13 2012

Let b(n) = 0, 1, -1, 1, 1, -13,.. be the sequence with all signs but one reversed: b(1)=a(1), b(n)=-a(n) for n<>1. Define the e.g.f. B(x) = 2*Sum_{n>=0} b(n)*(x/2)^n/n!. B(x) satisfies exp(B(x)) = 1 + 2*x - B(x). [Bernstein/Sloane S52]
Similarly, c(0)=1, c(n)=-a(n+1) are the alternating row sums of the second-order Eulerian numbers A340556, or c(n) = E2poly(n,-1). - Peter Luschny, Feb 13 2021
a(n) = Sum_{k=0..n-1} (n+k-1)!*Sum_{j=0..k} ((-1)^j/(k-j)!)*Sum_{i=0..j} ((1/i!)*Stirling1(n-i+j-1,j-i)/(n-i+j-1)!)*2^(n-j-1), n > 0, a(0)=1. - Vladimir Kruchinin, Nov 11 2012
From Sergei N. Gladkovskii, Nov 24 2012, Aug 22 2013: (Start)
Continued fractions:
G.f.: 2*x - x/G(0) where G(k) = 1 - 2*x*k + x*(k+1)/G(k+1).
G.f.: 2*x - 2*x/U(0) where U(k) = 1 + 1/(1 - 4*x*(k+1)/U(k+1)).
G.f.: A(x) = x/G(0) where G(k) = 1 - 2*x*(k+1) + x*(k+1)/G(k+1).
G.f.: 2*x  - x*W(0) where W(k) = 1 + x*(2*k+1)/( x*(2*k+1) + 1/(1 + x*(2*k+2)/( x*(2*k+2) + 1/W(k+1)))). (End)
a(n) = 4^n * Sum_{i=1..n} Stirling2(n,i)*A013703(i)/2^(2*i+1). - Paolo Bonzini, Jun 23 2016
E.g.f.: 1/2*(LambertW(exp(4*x+1))-1). - Vladimir Kruchinin, Feb 18 2018
a(0) = 0; a(1) = 1; a(n) = 2 * a(n-1) - Sum_{k=1..n-1} binomial(n-1,k) * a(k) * a(n-k). - Ilya Gutkovskiy, Aug 28 2020

More terms from James Sellers, Dec 07 1999

Reverted to converging factors definition by Paolo Bonzini, Jun 23 2016

A008789 a(n) = n^(n+3).

Original entry on oeis.org

0, 1, 32, 729, 16384, 390625, 10077696, 282475249, 8589934592, 282429536481, 10000000000000, 379749833583241, 15407021574586368, 665416609183179841, 30491346729331195904, 1477891880035400390625, 75557863725914323419136

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Cf. A000169, A000272, A000312, A007778, A007830, A008785, A008786, A008787, A008788, A008790, A008791.

List([0..20], n-> n^(n+3)); # G. C. Greubel, Sep 11 2019

[n^(n+3): n in [0..20]]; // Vincenzo Librandi, Jun 11 2013

printlevel := -1; a := [0]; T := x->-LambertW(-x); f := series((T(x)*(1+8*T(x)+6*(T(x))^2)/(1-T(x))^7),x,24); for m from 1 to 23 do a := [op(a),op(2*m-1,f)*m! ] od; print(a); # Len Smiley, Nov 19 2001

Table[n^(n+3),{n,0,20}](* Vladimir Joseph Stephan Orlovsky, Dec 26 2010 *)

vector(20, n, (n-1)^(n+2)) \\ G. C. Greubel, Sep 11 2019

[n^(n+3) for n in (0..20)] # G. C. Greubel, Sep 11 2019

E.g.f.(x): T*(1 +8*T +6*T^2)*(1-T)^(-7); where T=T(x) is Euler's tree function (see A000169). - Len Smiley, Nov 19 2001
See A008517 and A134991 for similar e.g.f.s and diagonals of A048993. - Tom Copeland, Oct 03 2011
E.g.f.: d^3/dx^3 {x^3/(T(x)^3*(1-T(x)))}, where T(x) = Sum_{n>=1} n^(n-1)*x^n/n! is the tree function of A000169. - Peter Bala, Aug 05 2012
a(n) = n*A008788(n). - R. J. Mathar, Oct 31 2015

A008790 a(n) = n^(n+4).

Original entry on oeis.org

0, 1, 64, 2187, 65536, 1953125, 60466176, 1977326743, 68719476736, 2541865828329, 100000000000000, 4177248169415651, 184884258895036416, 8650415919381337933, 426878854210636742656, 22168378200531005859375

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Cf. A000169, A000272, A000312, A007778, A007830, A008785, A008786, A008787, A008788, A008789, A008791.

List([0..20], n-> n^(n+4)); # G. C. Greubel, Sep 11 2019

[n^(n+4): n in [0..20]]; // Vincenzo Librandi, Jun 11 2013

a:=n->mul(n,k=-3..n):seq(a(n),n=0..20); # Zerinvary Lajos, Jan 26 2008

Table[n^(n+4),{n,0,20}](* Vladimir Joseph Stephan Orlovsky, Dec 26 2010 *)

vector(20, n, (n-1)^(n+3)) \\ G. C. Greubel, Sep 11 2019

[n^(n+4) for n in (0..20)] # G. C. Greubel, Sep 11 2019

E.g.f.: T*(1 +22*T +58*T^2 +24*T^3)*(1-T)^(-9); where T is Euler's tree function (see A000169). - Len Smiley, Nov 17 2001
See A008517 and A134991 for similar e.g.f.s and diagonals of A048993. - Tom Copeland, Oct 03 2011
E.g.f.: d^4/dx^4 {x^4/(T(x)^4*(1-T(x)))}, where T(x) = Sum_{n>=1} n^(n-1)*x^n/n! is the tree function of A000169. - Peter Bala, Aug 05 2012

A002539 Eulerian numbers of the second kind: <>.

Original entry on oeis.org

1, 22, 328, 4400, 58140, 785304, 11026296, 162186912, 2507481216, 40788301824, 697929436800, 12550904017920, 236908271543040, 4687098165573120, 97049168010017280, 2099830209402931200, 47405948832458496000, 1115089078488795648000, 27290469545695931904000, 694002594415741341696000

Offset: 0

N. J. A. Sloane , Simon Plouffe, Robert G. Wilson v, Mira Bernstein

Eulerian permutations of the multiset {1,1,2,2,...,n+3,n+3} with n ascents.Eulerian permutations have the restriction that for all m, all integers between the two copies of m are less than m. In particular, the two 1s are always next to each other.

The sequence gives the Eulerian numbers <<3,0>>, <<4,1>>, <<5,2>>, <<6,3>>, ... (and in particular the offset is 0).

			For instance, a(1) = 22 because among the 7!! = 105 permutations of {1,1,2,2,3,3,4,4} selected according to the definition of Eulerian numbers of the second kind, only 22 contain n = 1 descent, namely : 11223443, 11224433, 11233244, 11233442, 11244233, 11332244, 11334422, 11442233, 12213344, 12233144, 12233441, 12244133, 13312244, 13344122, 14412233, 22113344, 22331144, 22334411, 22441133, 33112244, 33441122, 44112233. - _Jean-François Alcover_, Mar 28 2011

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Math., 2nd edition; Addison-Wesley, 1994, pp. 270-271.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..99
L. Carlitz, Some numbers related to the Stirling numbers of the first and second kind, Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat. Fiz., Numbers 544-576 (1976): 49-55. [Annotated scanned copy. The triangle is A008517.]
I. Gessel and R. P. Stanley, Stirling polynomials, J. Combin. Theory, A 24 (1978), 24-33.
O. J. Munch, Om potensproduktsummer [Norwegian, English summary], Nordisk Matematisk Tidskrift, 7 (1959), 5-19. [Annotated scanned copy]
O. J. Munch, Om potensproduktsummer [ Norwegian, English summary ], Nordisk Matematisk Tidskrift, 7 (1959), 5-19. There are errors in the last two rows of his table.

3rd diagonal of A008517, third column of A112007.

b[1]=1; b[2]=22; b[n_] := b[n] = ((n-1)*(n-1)!*n^3 - (n+2)*(n+3)*b[n-2]*n + (n*(2*n+5)-4)*b[n-1]) / (n-1); a[n_] := b[n+1]; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Mar 23 2011, updated Oct 12 2015 *)

{a=vector(30,n,1);a[2]=22;for(n=3,#a,a[n]=(n-1)!*n^3+((n*(2*n+5)-4)*a[n-1] - n*(n+2)*(n+3)*a[n-2])/(n-1));a} \\ Uses offet 1 for technical reasons. - M. F. Hasler, Sep 19 2015

a(n)= (n+5)*a(n-1) + (n+1)*A002538(n+1), n>=1, a(0)=1.
Recurrence: (n-1)*n^2*a(n) = (n-1)*(3*n^3 + 12*n^2 + 6*n + 1)*a(n-1) - (n+1)*(3*n^4 + 15*n^3 + 13*n^2 - 15*n - 4)*a(n-2) + n*(n+1)^3*(n+2)*(n+3)*a(n-3). - Vaclav Kotesovec, May 24 2014
a(n) ~ n! * n^5 * log(n) * (log(n)*(1/2+181/(24*n)) + gamma*(1+181/(12*n)) - 2 - 65/(3*n)), where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, May 24 2014

Formulas adapted for offset 0 by Vaclav Kotesovec, May 24 2014

More terms from M. F. Hasler, Sep 19 2015

A106828 Triangle T(n,k) read by rows: associated Stirling numbers of first kind (n >= 0 and 0 <= k <= floor(n/2)).

Original entry on oeis.org

1, 0, 0, 1, 0, 2, 0, 6, 3, 0, 24, 20, 0, 120, 130, 15, 0, 720, 924, 210, 0, 5040, 7308, 2380, 105, 0, 40320, 64224, 26432, 2520, 0, 362880, 623376, 303660, 44100, 945, 0, 3628800, 6636960, 3678840, 705320, 34650, 0, 39916800, 76998240, 47324376, 11098780, 866250, 10395

Offset: 0

N. J. A. Sloane, May 22 2005

Another version of the triangle is in A008306.

A signed version of this triangle is given by the exponential Riordan array [1, log(1+t)-t]. Its row sums are (-1)^n*(1-n). Another version is [1, log(1-t)+t], whose row sums are 1-n. - Paul Barry, May 10 2008

			Rows 0 though 7 are:
  1;
  0,
  0,   1;
  0,   2,
  0,   6,   3;
  0,  24,  20,
  0, 120, 130,  15;
  0, 720, 924, 210;

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 256.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 75.

Reinhard Zumkeller, Rows n = 0..124 of table, flattened
Bishal Deb and Alan D. Sokal, Higher-order Stirling cycle and subset triangles: Total positivity, continued fractions and real-rootedness, arXiv:2507.18959 [math.CO], 2025. See pp. 5, 8.
Feng-Zhen Zhao, Some Properties of Associated Stirling Numbers, Journal of Integer Sequences, Article 08.1.7, 2008.

See A008306 for more information.

Cf. A008619 (row lengths), A000166 (row sums).

a106828 n k = a106828_tabf !! n !! k
a106828_row n = a106828_tabf !! n
a106828_tabf = map (fst . fst) $ iterate f (([1], [0]), 1) where
   f ((us, vs), x) =
     ((vs, map (* x) $ zipWith (+) ([0] ++ us) (vs ++ [0])), x + 1)
-- Reinhard Zumkeller, Aug 05 2013

A106828 := (n,k) -> add(binomial(j,n-2*k)*combinat:-eulerian2(n-k,j),j=0..n-k):
seq(print(seq(A106828(n,k),k=0..iquo(n,2))),n=0..9); # Peter Luschny, Apr 20 2011 (revised Jan 13 2016)
# Second program, after David Callan in A008306:
A106828 := proc(n, k) option remember; if k = 0 then k^n elif k = 1 then (n-1)! elif n <= 2*k-1 then 0 else (n-1)*(procname(n-1, k) + procname(n-2, k-1)) fi end: seq((seq(A106828(n, k), k = 0..iquo(n, 2))), n=0..12); # Peter Luschny, Aug 24 2021

Eulerian2[n_, k_] := Eulerian2[n, k] = If[k == 0, 1, If[k == n, 0, Eulerian2[n - 1, k] (k + 1) + Eulerian2[n - 1, k - 1] (2 n - k - 1)]];
T[n_, k_] := Sum[Binomial[j, n - 2 k] Eulerian2[n - k, j], {j, 0, n - k}];
Table[T[n, k], {n, 0, 12}, {k, 0, n/2}] (* Jean-François Alcover, Jun 13 2019 *)

E2(n, m) = sum(k=0, n-m, (-1)^(n+k)*binomial(2*n+1, k)*stirling(2*n-m-k+1, n-m-k+1, 1)); \\ A008517
T(n, k) = if ((n==0) && (k==0), 1, sum(j=0, n-k, binomial(j, n-2*k)*E2(n-k, j+1))); \\ Michel Marcus, Dec 07 2021

from math import factorial
def A106828(n, k):
    return k**n if k == 0 else factorial(n-1) if k == 1 else 0 if n <= 2*k - 1 else (n - 1)*(A106828(n-1, k) + A106828(n-2, k-1))
for n in range(14): print([A106828(n, k) for k in range(n//2 + 1)])
# Mélika Tebni, Dec 07 2021, after second Maple script.

# uses[bell_transform from A264428]
# Computes the full triangle 0<=k<=n.
def A106828_row(n):
    g = lambda k: factorial(k) if k>0 else 0
    s = [g(k) for k in (0..n)]
    return bell_transform(n, s)
[A106828_row(n) for n in (0..8)] # Peter Luschny, Jan 13 2016

T(n, k) = Sum_{j=0..n-k} binomial(j, n-2*k)*E2(n-k, j), where E2 are the second-order Eulerian numbers (A008517). - Peter Luschny, Jan 13 2016

Also the Bell transform of the sequence g(k) = k! if k>0 else 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 13 2016

Removed extra 0 in row 1 from Michael Somos, Jan 19 2011

A214406 Triangle of second-order Eulerian numbers of type B.

Original entry on oeis.org

1, 1, 1, 1, 8, 3, 1, 33, 71, 15, 1, 112, 718, 744, 105, 1, 353, 5270, 14542, 9129, 945, 1, 1080, 33057, 191384, 300291, 129072, 10395, 1, 3265, 190125, 2033885, 6338915, 6524739, 2071215, 135135, 1, 9824, 1038780, 18990320, 103829590, 204889344, 150895836, 37237680, 2027025

Offset: 0

Peter Bala, Jul 17 2012

The second-order Eulerian numbers A008517 count Stirling permutations by ascents. A Stirling permutation of order n is a permutation of the multiset {1,1,2,2,...,n,n} such that for each i, 1 <= i <= n, the elements lying between the two occurrences of i are larger than i.

We define a signed Stirling permutation of order n to be a vector (x_0, x_1, ..., x_(2*n)) such that x_0 = 0 and (|x_1|, ... ,|x_(2*n)|) is a Stirling permutation of order n. We say that a signed Stirling permutation (x_0, x_1, ... , x_(2*n)) has an ascent at position j, 0 <= j <= 2*n-1, if |x_j| < |x_(j+1)|. We define T(n,k), the second-order Eulerian numbers of type B, as the number of signed Stirling permutations of order n having k ascents. An example is given below.

			Row 2: [1,8,3]:
Signed Stirling permutations of order 2
= = = = = = = = = = = = = = = = = = = =
..............ascents...................ascents
(0 2 2 1 1)......1.......(0 -2 -2 1 1).....1
(0 1 2 2 1)......2.......(0 1 -2 -2 1).....2
(0 1 1 2 2)......2.......(0 1 1 -2 -2).....1
(0 2 2 -1 -1)....1.......(0 -2 -2 -1 -1)...1
(0 -1 2 2 -1)....1.......(0 -1 -2 -2 -1)...1
(0 -1 -1 2 2)....1.......(0 -1 -1 -2 -2)...0
............................................
Triangle begins
.n\k.|..0.....1......2.......3......4........5......6
= = = = = = = = = = = = = = = = = = = = = = = = = = =
..0..|..1
..1..|..1.....1
..2..|..1.....8......3
..3..|..1....33.....71......15
..4..|..1...112....718.....744....105
..5..|..1...353...5270...14542...9129......945
..6..|..1..1080..33057..191384..300291..129072..10395
...
Recurrence example: T(4,2) = 11*T(3,1) + 5*T(3,2) = 11*33 + 5*71 = 718.

Peter Bala, Second-order Eulerian numbers of type B
Peter Bala, Notes on A214406
L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros arXiv:math/0509207 [math.CO], 2005-2006.

Cf. A001813 (row sums), A008517, A039755, A185896, A034940.

T[n_, k_] /; 0 < k <= n := T[n, k] = (4n-2k-1) T[n-1, k-1] + (2k+1) T[n-1, k]; T[, 0] = 1; T[, _] = 0;
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 11 2019 *)

T(n,k) = Sum_{i = 0..k} (-1)^(i-k)*binomial(2*n+1,k-i)*S(n+i,i), where S(n,k) = 1/(2^k*k!)*Sum_{j = 0..k} (-1)^(k-j)*binomial(k,j)*(2*j+1)^n = A039755(n,k).
It appears that Sum_{k = 0..n} (-1)^(k+1)*T(n,k)/((2*n-k)*binomial(2*n,k)) = (-1)^n *(2^n-2)*Bernoulli(n)/n.
Recurrence equation: T(n,k) = (4*n-2*k-1)*T(n-1,k-1) + (2*k+1)*T(n-1,k), for n,k >= 0.
The row polynomials R(n,x) may be calculated by means of the recurrence equation R(0,x) = 1 and for n >= 0, R(n,x^2) = (1 - x^2)^(2*n)*d/dx( x/(1-x^2)^(2*n-1)*R(n-1,x^2) ). Equivalently, x*R(n,x^2)/(1 - x^2)^(2*n+1)) = D^n(x), where D is the differential operator x/(1 - x^2)*d/dx.
Another recurrence is R(n+1,x) = 2*x*(1 - x)*d/dx(R(n,x)) + (1 + (4*n+1)*x)*R(n,x). It follows that the row polynomials R(n,x) have only real zeros (apply Liu and Wang, Corollary 1.2 with f(x) = R(n,x) and g(x) = R'(n,x)).
For n >= 0, the rational functions Q(n,x) := R(n,x)/(1 - x)^(2*n+1) are the o.g.f.'s for the diagonals of the type B Stirling numbers of the second kind A039755. They appear to satisfy the semi-orthogonality property Integral_{x = 0..oo} (1 - x)*Q(n,x)*Q(m,x) dx = (-1)^n*(2^(n+m) - 2)*Bernoulli(n+m)/(n+m), for n, m >= 0 but excluding the case (n,m) = (0,0). A similar result holds for the row polynomials of A185896.
Row sums are A001813.
Define functions F(n,z) := Sum_{k >= 0} (2*k+1)^(k+n)*z^k/k!, n = 0,1,2,.... Then exp(-x/2)*F(n,x/2*exp(-x)) = R(n,x)/(1 - x)^(2*n+1). - Peter Bala, Jul 26 2012

Missing 1 in data inserted by Jean-François Alcover, Nov 11 2019

A004301 Second-order Eulerian numbers <>.

Original entry on oeis.org

0, 6, 58, 328, 1452, 5610, 19950, 67260, 218848, 695038, 2170626, 6699696, 20507988, 62407890, 189123286, 571432036, 1722945672, 5187185766, 15600353130, 46882846680, 140820504700, 422822222266, 1269221639358, 3809241974028, 11431014253872, 34299887862990

Offset: 2

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

See A008517 for the definition of second-order Eulerian numbers.

			G.f. = 6*x^3 + 58*x^4 + 328*x^5 + 1452*x^6 + 5610*x^7 + 19950*x^8 + ...

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd edition. Addison-Wesley, Reading, MA, 1994, p. 270.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 2..1000
I. Gessel and R. P. Stanley, Stirling polynomials, J. Combin. Theory, A 24 (1978), 24-33.
Wikipedia, Eulerian numbers of the second kind
Index entries for linear recurrences with constant coefficients, signature (10,-40,82,-91,52,-12).

3rd column of A008517.
2nd column of A201637.
Equals the fourth right hand column of triangle A163936. - Johannes W. Meijer, Oct 16 2009

LinearRecurrence[{10, -40, 82, -91, 52, -12}, {0, 6, 58, 328, 1452, 5610}, 26] (* Jean-François Alcover, Feb 27 2019 *)

{a(n) = if( n<0, 0, (9*3^n - (12 + 8*n)*2^n + (3 + 6*n + 4*n^2))/2)}; /* Michael Somos, Oct 13 2002 */

From Michael Somos, Oct 13 2002: (Start)
G.f.: x^3(6-2x-12x^2)/((1-x)^3(1-2x)^2(1-3x)).
a(n) = A008517(n, 3) = (9*3^n - (12+8*n)*2^n + (3+6*n+4*n^2))/2. (End)
a(n) = Sum_{k=0..n-3} (-1)^(n+k)*binomial(2*n+1, k)*Stirling1(2*n-k-2, n-k-2). - Johannes W. Meijer, Oct 16 2009

Edited by Olivier Gérard, Mar 28 2011

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A112493 Triangle read by rows, T(n, k) = Sum_{j=0..n} C(n-j, n-k)*E2(n, j), where E2 are the second-order Eulerian numbers A201637, for n >= 0 and 0 <= k <= n.

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A001298 Stirling numbers of the second kind S(n+4, n).

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A111999 T(n, k) = [x^k] (-1)^n*Sum_{k=0..n} E2(n, n-k)*(1+x)^(n-k) where E2(n, k) are the second-order Eulerian numbers. Triangle read by rows, T(n, k) for n >= 1 and 0 <= k <= n.

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A001662 Coefficients of Airey's converging factor.

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A008789 a(n) = n^(n+3).

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A008790 a(n) = n^(n+4).

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A111999 T(n, k) = [x^k] (-1)^nSum_{k=0..n} E2(n, n-k)(1+x)^(n-k) where E2(n, k) are the second-order Eulerian numbers. Triangle read by rows, T(n, k) for n >= 1 and 0 <= k <= n.