A340556 -id:A340556 - cp's OEIS frontend

A054654 Triangle of Stirling numbers of 1st kind, S(n, n-k), n >= 0, 0 <= k <= n.

1, 1, 0, 1, -1, 0, 1, -3, 2, 0, 1, -6, 11, -6, 0, 1, -10, 35, -50, 24, 0, 1, -15, 85, -225, 274, -120, 0, 1, -21, 175, -735, 1624, -1764, 720, 0, 1, -28, 322, -1960, 6769, -13132, 13068, -5040, 0

Offset: 0

N. J. A. Sloane, Apr 18 2000

Triangle is the matrix product of the binomial coefficients with the Stirling numbers of the first kind.
Triangle T(n,k) giving coefficients in expansion of n!*C(x,n) in powers of x. E.g., 3!*C(x,3) = x^3-3*x^2+2*x.
The matrix product of binomial coefficients with the Stirling numbers of the first kind results in the Stirling numbers of the first kind again, but the triangle is shifted by (1,1).
Essentially [1,0,1,0,1,0,1,0,...] DELTA [0,-1,-1,-2,-2,-3,-3,-4,-4,...] where DELTA is the operator defined in A084938; mirror image of the Stirling-1 triangle A048994. - Philippe Deléham, Dec 30 2006
From Doudou Kisabaka, Dec 18 2009: (Start)
The sum of the entries on each row of the triangle, starting on the 3rd row, equals 0. E.g., 1+(-3)+2+0 = 0.
The entries on the triangle can be computed as follows. T(n,r) = T(n-1,r) - (n-1)*T(n-1,r-1). T(n,r) = 0 when r equals 0 or r > n. T(n,r) = 1 if n==1. (End)

			Matrix begins:
  1, 0,  0,  0,  0,   0,    0,     0,      0, ...
  0, 1, -1,  2, -6,  24, -120,   720,  -5040, ...
  0, 0,  1, -3, 11, -50,  274, -1764,  13068, ...
  0, 0,  0,  1, -6,  35, -225,  1624, -13132, ...
  0, 0,  0,  0,  1, -10,   85,  -735,   6769, ...
  0, 0,  0,  0,  0,   1,  -15,   175,  -1960, ...
  0, 0,  0,  0,  0,   0,    1,   -21,    322, ...
  0, 0,  0,  0,  0,   0,    0,     1,    -28, ...
  0, 0,  0,  0,  0,   0,    0,     0,      1, ...
  ...
Triangle begins:
  1;
  1,   0;
  1,  -1,   0;
  1,  -3,   2,    0;
  1,  -6,  11,   -6,    0;
  1, -10,  35,  -50,   24,     0;
  1, -15,  85, -225,  274,  -120,   0;
  1, -21, 175, -735, 1624, -1764, 720, 0;
  ...

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 18, table 18:6:1 at page 152.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Eric Weisstein's World of Mathematics, Pochhammer Symbol.
Eric Weisstein's World of Mathematics, Rising Factorial.
Eric Weisstein's World of Mathematics, FallingFactorial.

Essentially Stirling numbers of first kind, multiplied by factorials - see A008276.
The Stirling2 counterpart is A106800.
Cf. A054655, A039810, A039814, A126350, A126351, A126353, A340556.

a054654 n k = a054654_tabl !! n !! k
a054654_row n = a054654_tabl !! n
a054654_tabl = map reverse a048994_tabl
-- Reinhard Zumkeller, Mar 18 2014

a054654_row := proc(n) local k; seq(coeff(expand((-1)^n*pochhammer (-x,n)),x,n-k),k=0..n) end: # Peter Luschny, Nov 28 2010
seq(seq(Stirling1(n, n-k), k=0..n), n=0..8); # Peter Luschny, Feb 21 2021

row[n_] := Reverse[ CoefficientList[ (-1)^n*Pochhammer[-x, n], x] ]; Flatten[ Table[ row[n], {n, 0, 8}]] (* Jean-François Alcover, Feb 16 2012, after Maple *)
Table[StirlingS1[n,n-k],{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Jun 17 2023 *)

PARI
```
T(n,k)=polcoeff(n!*binomial(x,n), n-k)
```

n!*binomial(x, n) = Sum_{k=0..n} T(n, k)*x^(n-k).
(In Maple notation:) Matrix product A*B of matrix A[i,j]:=binomial(j-1,i-1) with i = 1 to p+1, j = 1 to p+1, p=8 and of matrix B[i,j]:=stirling1(j,i) with i from 1 to d, j from 1 to d, d=9.
T(n, k) = (-1)^k*Sum_{j=0..k} E2(k, j)*binomial(n+j-1, 2*k), where E2(k, j) are the second-order Eulerian numbers A340556. - Peter Luschny, Feb 21 2021

Additional comments from Thomas Wieder, Dec 29 2006

Edited by N. J. A. Sloane at the suggestion of Eric W. Weisstein, Jan 20 2008

A269939 Triangle read by rows, Ward numbers T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * Stirling2(n + m, m), for n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 1, 10, 15, 0, 1, 25, 105, 105, 0, 1, 56, 490, 1260, 945, 0, 1, 119, 1918, 9450, 17325, 10395, 0, 1, 246, 6825, 56980, 190575, 270270, 135135, 0, 1, 501, 22935, 302995, 1636635, 4099095, 4729725, 2027025

Offset: 0

Peter Luschny, Mar 26 2016

We propose to call this sequence the 'Ward set numbers' and sequence A269940 the 'Ward cycle numbers'. - Peter Luschny, Nov 25 2022

			Triangle starts:
  1;
  0, 1;
  0, 1,   3;
  0, 1,  10,   15;
  0, 1,  25,  105,   105;
  0, 1,  56,  490,  1260,    945;
  0, 1, 119, 1918,  9450,  17325,  10395;
  0, 1, 246, 6825, 56980, 190575, 270270, 135135;

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.
Peter Luschny, The P-transform.
Andrew Elvey Price and Alan D. Sokal, Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials, arXiv:2001.01468 [math.CO], 2020.
Marko Riedel, Math Stackexchange, Upper Stirling numbers and Ward numbers.
Aleks Žigon Tankosič, Recurrence Relations for Some Integer Sequences Related to Ward Numbers, arXiv:2508.04754 [math.CO], 2025. See p. 2.
Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See pp. 1, 12.

Variants: A134991 (main entry for this triangle), A181996.
Row sums are A000311.
Alternating row sums are signed factorials A133942.
Cf. A269940 (Stirling1 counterpart), A268437.

# first version
A269939 := (n,k) -> add((-1)^(m+k)*binomial(n+k,n+m)*Stirling2(n+m, m), m=0..k):
seq(seq(A269939(n,k), k=0..n), n=0..8);
# Alternatively:
T := proc(n,k) option remember;
    `if`(k=0 and n=0, 1,
    `if`(k<=0 or k>n, 0,
    k*T(n-1,k)+(n+k-1)*T(n-1,k-1))) end:
for n from 0 to 6 do seq(T(n,k),k=0..n) od;
# simple, third version
T := (n,k)->  (n+k)!*coeftayl((exp(z)-z-1)^k/k!, z=0, n+k); # Marko Riedel, Apr 14 2016

Table[Sum[(-1)^(m + k) Binomial[n + k, n + m] StirlingS2[n + m, m], {m, 0, k}], {n, 0, 8}, {k, 0, n}] // Flatten (* Michael De Vlieger, Apr 15 2016 *)

T(n) = {[Vecrev(Pol(p)) | p<-Vec(serlaplace(1/((1+y)*(1 + lambertw(-y/(1+y)*exp((x-y)/(1+y) + O(x*x^n)))))))]}
{ my(A=T(8)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 14 2022

T = lambda n,k: sum((-1)^(m+k)*binomial(n+k,n+m)*stirling_number2(n+m,m) for m in (0..k))
for n in (0..6): print([T(n,k) for k in (0..n)])

# uses[PtransMatrix from A269941]
PtransMatrix(8, lambda n: 1/(n+1), lambda n, k: (-1)^k*falling_factorial(n+k,n))

T(n,k) = (-1)^k*FF(n+k,n)*P[n,k](1/(n+1)) where P is the P-transform and FF the falling factorial function. For the definition of the P-transform see the link.
T(n,k) = A268437(n,k)*FF(n+k,n)/(2*n)!.
T(n,k) = (n+k)! [z^{n+k}] (exp(z)-z-1)^k/k!. - Marko Riedel, Apr 14 2016
From Fabián Pereyra, Jan 12 2022: (Start)
T(n,k) = k*T(n-1,k) + (n+k-1)*T(n-1,k-1) for n > 0, T(0,0) = 1, T(n,0) = 0 for n > 0. (See the second Maple program.)
E.g.f.: A(x,t) = 1/((1+t)*(1 + W(-t/(1+t)*exp((x-t)/(1+t))))), where W(x) is the Lambert W-function.
T(n,k) = Sum_{j=0..k} E2(n,j)*binomial(n-j,k-j), where E2(n,k) are the second-order Eulerian numbers A340556.
T(n,k) = Sum_{j=k..n} (-1)^(n-j)*A112486(n,j)*binomial(j,k). (End)

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

A106800 Triangle of Stirling numbers of 2nd kind, S(n, n-k), n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 7, 1, 0, 1, 10, 25, 15, 1, 0, 1, 15, 65, 90, 31, 1, 0, 1, 21, 140, 350, 301, 63, 1, 0, 1, 28, 266, 1050, 1701, 966, 127, 1, 0, 1, 36, 462, 2646, 6951, 7770, 3025, 255, 1, 0, 1, 45, 750, 5880, 22827, 42525, 34105, 9330, 511, 1, 0

Offset: 0

N. J. A. Sloane

Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...] DELTA [0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, May 19 2005

			From _Gheorghe Coserea_, Jan 30 2017: (Start)
Triangle starts:
  n\k  [0]  [1]   [2]    [3]    [4]    [5]    [6]   [7] [8] [9]
  [0]   1;
  [1]   1,   0;
  [2]   1,   1,    0;
  [3]   1,   3,    1,     0;
  [4]   1,   6,    7,     1,     0;
  [5]   1,  10,   25,    15,     1,     0;
  [6]   1,  15,   65,    90,    31,     1,     0;
  [7]   1,  21,  140,   350,   301,    63,     1,    0;
  [8]   1,  28,  266,  1050,  1701,   966,   127,    1,  0;
  [9]   1,  36,  462,  2646,  6951,  7770,  3025,  255,  1,  0;
  ...
(End)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 835.
F. N. David, M. G. Kendall, and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 223.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 2, table 2.14.1 at page 24.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
A. Aboud, J.-P. Bultel, A. Chouria, J.-G. Luque, and O. Mallet, Bell polynomials in combinatorial Hopf algebras, arXiv preprint arXiv:1402.2960 [math.CO], 2014.
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure, arXiv:1307.2010 [math.CO], 2013.
Eric Weisstein's World of Mathematics, Bell Polynomial.

See A008277 and A048993, which are the main entries for this triangle of numbers.
The Stirling1 counterpart is A054654.
Row sum: A000110.
Column 0: A000012.
Column 1: A000217.
Main Diagonal: A000007.
1st minor diagonal: A000012.
2nd minor diagonal: A000225.
3rd minor diagonal: A000392.
Cf. A008278, A340556.

seq(seq(Stirling2(n, n-k), k=0..n), n=0..8); # Peter Luschny, Feb 21 2021

Table[ StirlingS2[n, m], {n, 0, 10}, {m, n, 0, -1}]//Flatten (* Robert G. Wilson v, Jan 30 2017 *)

N=11; x='x+O('x^N); t='t; concat(apply(p->Vec(p), Vec(serlaplace(exp(t*(exp(x)-1))))))  \\ Gheorghe Coserea, Jan 30 2017
{T(n, k) = my(A, B); if( n<0 || k>n, 0, A = B = exp(x + x * O(x^n)); for(i=1, n, A = x * A'); polcoeff(A / B, n-k))}; /* Michael Somos, Aug 16 2017 */

flatten([[stirling_number2(n, n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Sep 11 2021

A(x;t) = exp(t*(exp(x)-1)) = Sum_{n>=0} P_n(t) * x^n/n!, where P_n(t) = Sum_{k=0..n} T(n,k)*t^(n-k). - Gheorghe Coserea, Jan 30 2017
Also, P_n(t) * exp(t) = (t * d/dt)^n exp(t). - Michael Somos, Aug 16 2017
T(n, k) = Sum_{j=0..k} E2(k, j)*binomial(n + k - j, 2*k), where E2(k, j) are the second-order Eulerian numbers A340556. - Peter Luschny, Feb 21 2021

A342312 T(n, k) = ((2n + 1)/2)Sum_{j, k, n} (-1)^(k + j)(n + j)binomial(2n, n - j) Stirling2(n - k + j, 1 - k + j) with T(0, 0) = 1. Triangle read by row, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 0, 3, 0, 0, 10, 0, 0, -42, 21, 0, 0, 216, -288, 36, 0, 0, -1320, 3190, -1210, 55, 0, 0, 9360, -34632, 25584, -4056, 78, 0, 0, -75600, 389340, -462000, 152460, -11970, 105, 0, 0, 685440, -4621824, 7907040, -4368320, 762960, -32640, 136

Offset: 0

Peter Luschny, Mar 08 2021

The triangle can be seen as representing the numerators of a sequence of rational polynomials. Let p_{n}(x) = Sum_{k=0..n} (T(n, k)/A342313(n, k))*x^k. Then p_{n}(1) = B_{n}(1), where B_{n}(x) are the Bernoulli polynomials.

			The triangle starts:
  [0] 1
  [1] 0, 3
  [2] 0, 0,    10
  [3] 0, 0,   -42,     21
  [4] 0, 0,   216,   -288,    36
  [5] 0, 0, -1320,   3190, -1210,    55
  [6] 0, 0,  9360, -34632, 25584, -4056, 78
The first few polynomials are P(n, k) = T(n, k) / A342313(n, k):
  1;
  0, 1/2;
  0,  0,  1/6;
  0,  0, -1/10,   1/10;
  0,  0,  3/35,  -4/21,   1/14;
  0,  0, -2/21,  29/84, -11/36,    1/18;
  0,  0, 10/77, -37/55, 164/165, -26/55, 1/22;

Levin Luschny, Illustrating the polynomials.

Cf. A014105 (main diagonal), A342313 (denominators), A340556.

T := (n, k) -> `if`(n = 0 and k = 0, 1, (n+1/2)*
add((-1)^j*(n+k+j)*binomial(2*n, n-k-j)*Stirling2(n + j, j + 1) , j= 0..n-k)):
seq(print(seq(T(n, k), k=0..n)), n=0..6);

T[0, 0] := 1; T[n_, k_] := ((2n + 1)/2) Sum[(-1)^(k+j)(n+j) Binomial[2n, n-j] StirlingS2[n-k+j, 1-k+j], {j, k, n}];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten

T(n, k) = numerator([x^k] p(n, x)) for n >= 2, where p(n, x) = (1/2)*Sum_{k=0..n-1} (-1)^k*x^(n-k)*E2(n - 1, k + 1) / binomial(2*n - 1, k + 1) and E2(n,k) denotes the second-order Eulerian numbers A340556.

Another representation of the polynomials for n >= 2 is p(n, x) = (1/2)*Sum_{k=0..n} x^k*Sum_{j=k..n} ((-1)^(j + k)*((n - k + 1)!*(n + k - 2)!)/((j + n - 1)!*(n - j)!))*Stirling2(n - k + j, j - k + 1).

A341102 T(n, k) = [n, k] - {n, k}, where [n, k] are the (unsigned) Stirling cycle numbers and {n, k} the Stirling set numbers. Table T(n, k) read by rows, for n >= 3 and 1 <= k <= n-2.

Original entry on oeis.org

1, 5, 4, 23, 35, 10, 119, 243, 135, 20, 719, 1701, 1323, 385, 35, 5039, 12941, 12166, 5068, 910, 56, 40319, 109329, 115099, 59514, 15498, 1890, 84, 362879, 1026065, 1163370, 689575, 226800, 40446, 3570, 120, 3628799, 10627617, 12725075, 8263750, 3170200, 722568, 93786, 6270, 165

Offset: 3

Peter Luschny, Feb 24 2021

			Triangle starts:
[ 3] [1]
[ 4] [5,      4]
[ 5] [23,     35,      10]
[ 6] [119,    243,     135,     20]
[ 7] [719,    1701,    1323,    385,    35]
[ 8] [5039,   12941,   12166,   5068,   910,    56]
[ 9] [40319,  109329,  115099,  59514,  15498,  1890,  84]
[10] [362879, 1026065, 1163370, 689575, 226800, 40446, 3570, 120]

Peter Luschny, The difference of the Stirling cycle numbers and the Stirling set numbers, Mathematics Stack Exchange, Feb. 2021.

Cf. A132393, A048993, A048742, A033312, A000292, A340556.

# Giving full rows for n >= 0:
gf := (1 - z)^(-x) - exp(x*(exp(z) - 1));
ser := series(gf, z, 20): coeffz := n -> coeff(ser,z,n):
A341102row := n -> seq(n!*coeff(coeffz(n), x, k), k=0..n):
for n from 0 to 9 do A341102row(n) od;

T(n,k) = abs(stirling(n,k,1)) - stirling(n,k,2); \\ Michel Marcus, Feb 24 2021

for n in (3..11):
    print([stirling_number1(n, k) - stirling_number2(n, k) for k in (1..n-2)])

T(n, k) = Sum_{j=0..k} (binomial(n+j-1, 2*k) - binomial(n+k-j, 2*k))*A340556(k, j).

E.g.f.: (1 - z)^(-x) - exp(x*(exp(z) - 1)) (unrestricted rows and n >= 0).

cp's OEIS Frontend

A054654 Triangle of Stirling numbers of 1st kind, S(n, n-k), n >= 0, 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

Extensions

A269939 Triangle read by rows, Ward numbers T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * Stirling2(n + m, m), for n >= 0, 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Sage

Sage

Formula

A001662 Coefficients of Airey's converging factor.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

SageMath

Formula

Extensions

A106800 Triangle of Stirling numbers of 2nd kind, S(n, n-k), n >= 0, 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Sage

Formula

A342312 T(n, k) = ((2*n + 1)/2)*Sum_{j, k, n} (-1)^(k + j)*(n + j)*binomial(2*n, n - j)* Stirling2(n - k + j, 1 - k + j) with T(0, 0) = 1. Triangle read by row, T(n, k) for 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A341102 T(n, k) = [n, k] - {n, k}, where [n, k] are the (unsigned) Stirling cycle numbers and {n, k} the Stirling set numbers. Table T(n, k) read by rows, for n >= 3 and 1 <= k <= n-2.

Views

Author

Keywords

Examples

A342312 T(n, k) = ((2n + 1)/2)Sum_{j, k, n} (-1)^(k + j)(n + j)binomial(2n, n - j) Stirling2(n - k + j, 1 - k + j) with T(0, 0) = 1. Triangle read by row, T(n, k) for 0 <= k <= n.