A008517 -id:A008517 - cp's OEIS frontend

A008306 Triangle T(n,k) read by rows: associated Stirling numbers of first kind (n >= 2, 1 <= k <= floor(n/2)).

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

Offset: 2

N. J. A. Sloane

Also, T(n,k) is the number of derangements (permutations with no fixed points) of {1..n} with k cycles.

The sum of the n-th row is the n-th subfactorial: A000166(n). - Gary Detlefs, Jul 14 2010

			Rows 2 through 7 are:
    1;
    2;
    6,   3;
   24,  20;
  120, 130,  15;
  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 = 2..125 of table, flattened
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.
W. Carlitz, On some polynomials of Tricomi, Bollettino dell'Unione Matematica Italiana, Serie 3, Vol. 13, (1958), n. 1, p. 58-64
Tom Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms
W. Gautschi, The incomplete gamma functions since Tricomi (Cf. p. 206-207.)
P. Gniewek and B. Jeziorski, Convergence properties of the multipole expansion of the exchange contribution to the interaction energy, arXiv preprint arXiv:1601.03923 [physics.chem-ph], 2016.
S. Karlin and J. McGregor, Many server queuing processes with Poisson input and exponential service times, Pacific Journal of Mathematics, Vol. 8, No. 1, p. 87-118, March (1958). Cf. p. 117.
R. Paris, A uniform asymptotic expansion for the incomplete gamma function, Journal of Computational and Applied Mathematics, 148 (2002), p. 223-239. (See 333. From Tom Copeland, Jan 03 2016)
M. Z. Spivey, On Solutions to a General Combinatorial Recurrence, J. Int. Seq. 14 (2011) # 11.9.7.
N. Temme, A class of polynomials related to those of Laguerre
N. Temme, Traces to Tricomi in recent work on special functions and asymptotics of integrals
A. Topuzoglu, The Carlitz rank of permutations of finite fields: A survey, Journal of Symbolic Computation, Online, Dec 07, 2013.
Eric Weisstein's World of Mathematics, Permutation Cycle
Eric Weisstein's World of Mathematics, Stirling Number of the First Kind
Shawn L. Witte, Link Nomenclature, Random Grid Diagrams, and Markov Chain Methods in Knot Theory, Ph. D. Dissertation, University of California-Davis (2020).

Cf. A000166, A106828 (another version), A079510 (rearranged triangle), A235706 (specializations).
Diagonals: A000142, A000276, A000483.
Diagonals give reversed rows of A111999.

a008306 n k = a008306_tabf !! (n-2) !! (k-1)
a008306_row n = a008306_tabf !! (n-2)
a008306_tabf = map (fst . fst) $ iterate f (([1], [2]), 3) where
   f ((us, vs), x) =
     ((vs, map (* x) $ zipWith (+) ([0] ++ us) (vs ++ [0])), x + 1)
-- Reinhard Zumkeller, Aug 05 2013

A008306 := proc(n,k) local j;
add(binomial(j,n-2*k)*A008517(n-k,j),j=0..n-k) end;
seq(print(seq(A008306(n,k),k=1..iquo(n,2))),n=2..12):
# Peter Luschny, Apr 20 2011

t[0, 0] = 1; t[n_, 0] = 0; t[n_, k_] /; k > n/2 = 0; t[n_, k_] := t[n, k] = (n - 1)*(t[n - 1, k] + t[n - 2, k - 1]); A008306 = Flatten[ Table[ t[n, k], {n, 2, 12}, {k, 1, Quotient[n, 2]}]] (* Jean-François Alcover, Jan 25 2012, after David Callan *)

{ A008306(n,k) = (-1)^(n+k) * sum(i=0,k, (-1)^i * binomial(n,i) * stirling(n-i,k-i,1) ); } \\ Max Alekseyev, Sep 08 2018

T(n,k) = Sum_{i=0..k} (-1)^i * binomial(n,i) * |stirling1(n-i,k-i)| = (-1)^(n+k) * Sum_{i=0..k} (-1)^i * binomial(n,i) * A008275(n-i,k-i). - Max Alekseyev, Sep 08 2018
E.g.f.: 1 + Sum_{1 <= 2*k <= n} T(n, k)*t^n*u^k/n! = exp(-t*u)*(1-t)^(-u).
Recurrence: T(n, k) = (n-1)*(T(n-1, k) + T(n-2, k-1)) for 1 <= k <= n/2 with boundary conditions T(0,0) = 1, T(n,0) = 0 for n >= 1, and T(n,k) = 0 for k > n/2. - David Callan, May 16 2005
E.g.f. for column k: B(A(x)) where A(x) = log(1/1-x)-x and B(x) = x^k/k!.
From Tom Copeland, Jan 05 2016: (Start)
The row polynomials of this signed array are the orthogonal NL(n,x;x-n) = n! Sum_{k=0..n} binomial(x,n-k)*(-x)^k/k!, the normalized Laguerre polynomials of order (x-n) as discussed in Gautschi (the Temme, Carlitz, and Karlin and McGregor references come from this paper) in regard to asymptotic expansions of the upper incomplete gamma function--Tricomi's Cinderella of special functions.
e^(x*t)*(1-t)^x = Sum_{n>=0} NL(n,x;x-n)*x^n/n!.
The first few are
NL(0,x) = 1
NL(1,x) = 0
NL(2,x) = -x
NL(3,x) = 2*x
NL(4,x) = -6*x + 3*x^2.
With D=d/dx, :xD:^n = x^n D^n, :Dx:^n = D^n x^n, and K(a,b,c), the Kummer confluent hypergeometric function, NL(n,x;y-n) = n!*e^x binomial(xD+y,n)*e^(-x) = n!*e^x Sum_{k=0..n} binomial(k+y,n) (-x)^k/k! = e^x x^(-y+n) D^n (x^y e^(-x)) = e^x x^(-y+n) :Dx:^n x^(y-n)*e^(-x) = e^x*x^(-y+n)*n!*L(n,:xD:,0)*x^(y-n)*e^(-x) = n! binomial(y,n)*K(-n,y-n+1,x) = n!*e^x*(-1)^n*binomial(-xD-y+n-1,n)*e^(-x). Evaluate these expressions at y=x after the derivative operations to obtain NL(n,x;x-n). (End)

More terms from Larry Reeves (larryr(AT)acm.org), Feb 16 2001

A134991 Triangle of Ward numbers T(n,k) read by rows.

Original entry on oeis.org

1, 1, 3, 1, 10, 15, 1, 25, 105, 105, 1, 56, 490, 1260, 945, 1, 119, 1918, 9450, 17325, 10395, 1, 246, 6825, 56980, 190575, 270270, 135135, 1, 501, 22935, 302995, 1636635, 4099095, 4729725, 2027025, 1, 1012, 74316, 1487200, 12122110, 47507460, 94594500, 91891800, 34459425

Offset: 1

Tom Copeland, Feb 05 2008

This is the triangle of associated Stirling numbers of the second kind, A008299, read along the diagonals.
This is also a row-reversed version of A181996 (with an additional leading 1) - see the table on p. 92 in the Ward reference. A134685 is a refinement of the Ward table.
The first and second diagonals are A001147 and A000457 and appear in the diagonals of several OEIS entries. The polynomials also appear in Carlitz (p. 85), Drake et al. (p. 8) and Smiley (p. 7).
First few polynomials (with a different offset) are
  P(0,t) = 0
  P(1,t) = 1
  P(2,t) = t
  P(3,t) = t + 3*t^2
  P(4,t) = t + 10*t^2 + 15*t^3
  P(5,t) = t + 25*t^2 + 105*t^3 + 105*t^4
These are the "face" numbers of the tropical Grassmannian G(2,n),related to phylogenetic trees (with offset 0 beginning with P(2,t)). Corresponding h-vectors are A008517. - Tom Copeland, Oct 03 2011
A133314 applied to the derivative of A(x,t) implies (a.+b.)^n = 0^n, for (b_n)=P(n+1,t) and (a_0)=1, (a_1)=-t, and (a_n)=-(1+t) P(n,t) otherwise. E.g., umbrally, (a.+b.)^2 = a_2*b_0 + 2 a_1*b_1 + a_0*b_2 = 0. - Tom Copeland, Oct 08 2011
Beginning with the second column, the rows give the faces of the Whitehouse simplicial complex with the fourth-order complex being three isolated vertices and the fifth-order being the Petersen graph with 10 vertices and 15 edges (cf. Readdy). - Tom Copeland, Oct 03 2014
Stratifications of smooth projective varieties which are fine moduli spaces for stable n-pointed rational curves. Cf. pages 20 and 30 of the Kock and Vainsencher reference and references in A134685. - Tom Copeland, May 18 2017
Named after the American mathematician Morgan Ward (1901-1963). - Amiram Eldar, Jun 26 2021

			Triangle begins:
  1
  1   3
  1  10   15
  1  25  105  105
  1  56  490 1260   945
  1 119 1918 9450 17325 10395
  ...

Louis Comtet, Advanced Combinatorics, Reidel, 1974, page 222.

G. C. Greubel, Rows n = 1..65, flattened
J. Fernando Barbero G., Jesús Salas and Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences, I: General Structure, Journal of Combinatorial Theory, Series A, Vol. 125 (2014), pp. 146-165; arXiv preprint, arXiv:1307.2010 [math.CO], 2013-2014.
J. Fernando Barbero G., Jesús Salas and Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. II. Applications, arXiv preprint arXiv:1307.5624 [math.CO], 2013.
J. Fernando Barbero G., Jesús Salas and Eduardo J. S. Villaseñor, Generalized Stirling permutations and forests: Higher-order Eulerian and Ward numbers, Electronic Journal of Combinatorics, Vol. 22, No. 3 (2015), #P3.37.
Andreas Blass, Natasha Dobrinen and Dilip Raghavan, The next best thing to a p-point, The Journal of Symbolic Logic, Vol. 80, No. 3 (2015), pp. 866-900; arXiv preprint, arXiv:1308.3790 [math.LO], 2013.
David Callan, Toufik Mansour, and Mark Shattuck Some identities for derangement and Ward number sequences and related bijections, Pure Mathematics and Applications, Vol. 25 (Edition 2), pp. 132-143, 2015.
L. Carlitz, The coefficients in an asymptotic expansion and certain related numbers, Duke Math. J., Vol. 35, No. 1 (1968), pp. 83-90.
Lane Clark, Asymptotic normality of the Ward numbers, Discrete Math. 203 (1999), no. 1-3, 41-48. [From _N. J. A. Sloane_, Feb 06 2012]
Tom Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms.
Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. II. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 3.
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.
Satyan L. Devadoss, Daoji Huang and Dominic Spadacene, Polyhedral covers of tree space, SIAM Journal on Discrete Mathematics, Vol. 28, No. 3 (2014), pp. 1508-1514; arXiv preprint, arXiv:1311.0766 [math.CO], 2013.
S. Devadoss and J. Morava, Diagonalizing the genome I: navigation in tree spaces, arXiv:1009.3224v2 [math.AG], 2012.
S. Devadoss and O. Schuh, Cartography of Tree Space, Leonardo (MIT Press Direct), Vol. 53, Issue 3, 2019.
Andreas Dieckmann and Erik Vigren, A New Result in Form of Finite Triple Sums for a Series from Ramanujan's Notebooks, Symmetry (2022) Vol. 14, No. 6, 1090.
Ming-Jian Ding and Jiang Zeng, Proof of an explicit formula for a series from Ramanujan's Notebooks via tree functions, arXiv:2307.00566 [math.CO], 2023.
Brian Drake, Ira M. Gessel and Guoce Xin, Three Proofs and a Generalization of the Goulden-Litsyn-Shevelev Conjecture on a Sequence Arising in Algebraic Geometry, J. of Integer Sequences, Vol. 10 (2007), Article 07.3.7.
Joseph Felsenstein, The number of evolutionary trees, Systematic Biology, 27 (1978), pp. 27-33, 1978.
Giovanni Gaiffi, Nested sets, set partitions and Kirkman-Cayley dissection numbers, arXiv preprint arXiv:1404.3395 [math.CO], 2014.
David M. Jackson, Achim Kempf and Alejandro H. Morales, A robust generalization of the Legendre transform for QFT, Journal of Physics A: Mathematical and Theoretical, Vol. 50, No. 22 (2017), 225201; arXiv preprint, arXiv:1612.0046 [hep-th], 2017.
Bradley Robert Jones, On tree hook length formulae, Feynman rules and B-series, Master's thesis, Math Dept., Simon Fraser University, 2014. p. 23.
Joachim Kock and Israel Vainsencher, Kontsevich's Formula for Rational Plane Curves, Recife, 1999 (version 31/01/2003).
Toufik Mansour and Mark Shattuck, A polynomial generalization of some associated sequences related to set partitions, Periodica Mathematica Hungarica, Vol. 75, No. 2 (December 2017), pp. 398-412.
MathOverflow, Face numbers for tropical Grassmannian G'_2,7 simplicial complex?, 2011.
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.
Margaret A. Readdy, The pre-WDVV ring of physics and its topology, The Ramanujan Journal, Vol. 10, No. 2 (2005), pp. 269-281; preprint, 2002.
Lucas Randazzo, Arboretum for a generalisation of Ramanujan polynomials, The Ramanujan Journal, Vol. 54 (2019), pp. 1-14; arXiv preprint, arXiv:1905.02083 [math.CO], 2019.
Lucas Randazzo, Combinatoire bijective autour d'arbres et de chemins, doctoral thesis, Université Paris-Est, 2019.
Peter Regner, Phylogenetic Trees: Selected Combinatorial Problems, Master's Thesis, 2012, Institute of Discrete Mathematics and Geometry, TU Vienna, p. 52.
Fuquan Ren, Jean Yeh, and Roberta Rui Zhou, Context-Free Grammars for Several Triangular Arrays, Axioms, Vol. 11, Issue 6, 297, 2022.
E. G. Santos, Counting non-attacking chess pieces placements: Bishops and Anassas, arXiv:2411.16492 [math.CO], 2024. See p. 8.
Leonard M. Smiley, Completion of a Rational Function Sequence of Carlitz, arXiv:math/0006106 [math.CO], 2000.
Pierre Théorêt, Fonctions génératrices pour une classe d'équations aux différences partielles, Ann. Sci. Math. Quebec 19, 91-105 (1995).
Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See p. 1.
Morgan Ward, The representations of Stirling's numbers and Stirling's polynomials as sums of factorials, Amer. J. Math., Vol. 56, No. 1 (1934), pp. 87-95.
Bao-Xuan Zhu, Coefficientwise Hankel-total positivity of row-generating polynomials for the m-Jacobi-Rogers triangle, arXiv:2202.03793 [math.CO], 2022.

The same as A269939, with column k = 0 removed.
A reshaped version of the triangle of associated Stirling numbers of the second kind, A008299.
A181996 is the mirror image.
Columns k = 2, 3, 4 are A000247, A000478, A058844.
Diagonal k = n is A001147.
Diagonal k = n - 1 is A000457.
Row sums are A000311.
Alternating row sums are signed factorials (-1)^(n-1)*A000142(n).
Cf. A112493.

t[n_, k_] := Sum[(-1)^i*Binomial[n, i]*Sum[(-1)^j*(k-i-j)^(n-i)/(j!*(k-i-j)!), {j, 0, k-i}], {i, 0, k}]; row[n_] := Table[t[k, k-n], {k, n+1, 2*n}]; Table[row[n], {n, 1, 9}] // Flatten (* Jean-François Alcover, Apr 23 2014, after A008299 *)

E.g.f. for the polynomials is A(x,t) = (x-t)/(t+1) + T{ (t/(t+1)) * exp[(x-t)/(t+1)] }, where T(x) is the Tree function, the e.g.f. of A000169. The compositional inverse in x (about x = 0) is B(x) = x + -t * [exp(x) - x - 1]. Special case t = 1 gives e.g.f. for A000311. These results are a special case of A134685 with u(x) = B(x).
From Tom Copeland, Oct 26 2008: (Start)
Umbral-Sheffer formalism gives, for m a positive integer and u = t/(t+1),
[P(.,t)+Q(.,x)]^m = [m Q(m-1,x) - t Q(m,x)]/(t+1) + sum(n>=1) { n^(n-1)[u exp(-u)]^n/n! [n/(t+1)+Q(.,x)]^m }, when the series is convergent for a sequence of functions Q(n,x).
Check: With t=1; Q(n,x)=0^n, for n>=0; and Q(-1,x)=0, then [P(.,1)+Q(.,x)]^m = P(m,1) = A000311(m).
(End)
Let h(x,t) = 1/(dB(x)/dx) = 1/(1-t*(exp(x)-1)), an e.g.f. in x for row polynomials in t of A019538, then the n-th row polynomial in t of the table A134991, P(n,t), is given by ((h(x,t)*d/dx)^n)x evaluated at x=0, i.e., A(x,t) = exp(x*P(.,t)) = exp(x*h(u,t)*d/du) u evaluated at u=0. Also, dA(x,t)/dx = h(A(x,t),t). - Tom Copeland, Sep 05 2011
The polynomials (1+t)/t*P(n,t) are the row polynomials of A112493. Let f(x) = (1+x)/(1-x*t). Then for n >= 0, P(n+1,t) is given by t/(1+t)*(f(x)*d/dx)^n(f(x)) evaluated at x = 0. - Peter Bala, Sep 30 2011
From Tom Copeland, Oct 04 2011: (Start)
T(n,k) = (k+1)*T(n-1,k) + (n+k+1)*T(n-1,k-1) with starting indices n=0 and k=0 beginning with P(2,t) (as suggested by a formula of David Speyer on MathOverflow).
T(n,k) = k*T(n-1,k) + (n+k-1)*T(n-1,k-1) with starting indices n=1 and k=1 of table (cf. Smiley above and Riordin ref.[10] therein).
P(n,t) = (1/(1+t))^n * Sum_{k>=1} k^(n+k-1)*(u*exp(-u))^k / k! with u=(t/(t+1)) for n>1; therefore, Sum_{k>=1} (-1)^k k^(n+k-1) x^k/k! = [1+LW(x)]^(-n) P{n,-LW(x)/[1+LW(x)]}, with LW(x) the Lambert W-Fct.
T(n,k) = Sum_{i=0..k} ((-1)^i binomial(n+k,i) Sum_{j=0..k-i} (-1)^j (k-i-j)^(n+k-i)/(j!(k-i-j)!)) from relation to A008299. (End)
The e.g.f. A(x,t) = -v * ( Sum_{j=>1} D(j-1,u) (-z)^j / j! ) where u = (x-t)/(1+t), v = 1+u, z = x/((1+t) v^2) and D(j-1,u) are the polynomials of A042977. dA/dx = 1/((1+t)(v-A)) = 1/(1-t*(exp(A)-1)). - Tom Copeland, Oct 06 2011
The general results on the convolution of the refined partition polynomials of A134685, with u_1 = 1 and u_n = -t otherwise, can be applied here to obtain results of convolutions of these polynomials. - Tom Copeland, Sep 20 2016
E.g.f.: C(u,t) = (u-t)/(1+t) - W( -((t*exp((u-t)/(1+t)))/(1+t)) ), where W is the principal value of the Lambert W-function. - Cheng Peng, Sep 11 2021
The function C(u,t) in the previous formula by Peng is precisely the function A(u,t) given in the initial 2008 formula of this section and the Oct 06 2011 formula from Copeland. As noted in A000169, Euler's tree function is T(x) = -LambertW(-x), where W(x) is the principal branch of Lambert's function, and T(x) is the e.g.f. of A000169. - Tom Copeland, May 13 2022

Reference to A181996 added by N. J. A. Sloane, Apr 05 2012

Further edits by N. J. A. Sloane, Jan 24 2020

A201637 Triangle of second-order Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 8, 6, 0, 1, 22, 58, 24, 0, 1, 52, 328, 444, 120, 0, 1, 114, 1452, 4400, 3708, 720, 0, 1, 240, 5610, 32120, 58140, 33984, 5040, 0, 1, 494, 19950, 195800, 644020, 785304, 341136, 40320, 0, 1, 1004, 67260, 1062500, 5765500, 12440064, 11026296, 3733920, 362880, 0

Offset: 0

Peter Luschny, Nov 11 2012

This version indexes the Eulerian numbers in the same way as Graham et al.'s Concrete Mathematics. This indexing is also used by Maple. The indexing as used by Riordan, Comtet and others, is given in A008517, which is the main entry for the second-order Eulerian numbers.

			... [0]  [1]    [2]     [3]     [4]     [5]     [6]    [7]  [8]
[0] [1]
[1] [1,   0]
[2] [1,   2,     0]
[3] [1,   8,     6,      0]
[4] [1,  22,    58,     24,      0]
[5] [1,  52,   328,    444,    120,      0]
[6] [1, 114,  1452,   4400,   3708,    720,      0]
[7] [1, 240,  5610,  32120,  58140,  33984,   5040,     0]
[8] [1, 494, 19950, 195800, 644020, 785304, 341136, 40320,  0]

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, table 256.

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened
Wolfdieter Lang, On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.
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.
Dengji Qi, Note: On the second order Eulerian numbers, Australasian Journal of Combinatorics, Volume 50 (2011), Pages 183-185.

Cf. A008517, A173018.
Columns 2 and 3 respectively give A004301 and A006260.
T(2n,n) gives A290306.

A201637 := (n,k) -> combinat[eulerian2](n,k):
for n from 0 to 9 do seq(A201637(n,k),k=0..n) od;
# Illustrating the connection with the Lambert W function:
alias(W = LambertW): len := 9:
w := W(-exp((x - 1)^2 * t - x)*x) + 1:
ser := series((1 - 1/x)*(1 - 1/w), t, len + 1):
egf := simplify(subs(W(-exp(-x)*x)=(-x), ser)):
poly := n -> n!*coeff(egf, t, n):
seq(seq(coeff(poly(n), x, k), k = 0..n), n = 0..len);  # Peter Luschny, Mar 15 2025

t[0, 0] = 1; t[n_, m_] = Sum[(-1)^(n+k)*Binomial[2*n+1, k]*StirlingS1[2*n-m-k, n-m-k], {k, 0, n-m-1}]; Table[t[n, m], {n, 0, 9}, {m, 0, n}] // Flatten
(* Jean-François Alcover, Jun 28 2013 *)
E2[n_, k_] /; k == 0 = 1; E2[n_, k_] /; k < 0 || k > n = 0;
E2[n_, k_] := E2[n, k] = (2*n - 1 - k)*E2[n-1, k-1] + (k + 1)*E2[n-1, k];
Table[E2[n, k], {n, 0, 8}, {k, 0, n}] // TableForm
(* Peter Luschny, Aug 14 2022 *)

for(n=0,10, for(m=0,n, print1(if(m==0 || n==0,1,sum(k=0,n-m-1, (-1)^(n+k)* binomial(2*n+1, k)*stirling(2*n-m-k, n-m-k,1))), ", "))) \\ G. C. Greubel, Oct 24 2017

@CachedFunction
def eulerian2(n, k):
    if k==0: return 1
    if k==n: return 0
    return eulerian2(n-1, k)*(k+1)+eulerian2(n-1, k-1)*(2*n-k-1)
for n in (0..9): [eulerian2(n, k) for k in(0..n)]

T(n, k) = [x^k](n! * [t^n](1 - 1/x)*(1 - 1/w)), where w = W(-exp((x - 1)^2 * t - x)*x) + 1, and W(-exp(-x)*x) is substituted after expansion by (-x). (W is the Lambert W function.) - Peter Luschny, Mar 15 2025

Terms a(52) onward added by G. C. Greubel, Oct 24 2017

A001297 Stirling numbers of the second kind S(n+3, n).

Original entry on oeis.org

0, 1, 15, 90, 350, 1050, 2646, 5880, 11880, 22275, 39325, 66066, 106470, 165620, 249900, 367200, 527136, 741285, 1023435, 1389850, 1859550, 2454606, 3200450, 4126200, 5265000, 6654375, 8336601, 10359090, 12774790, 15642600, 19027800

Offset: 0

N. J. A. Sloane

			a(2) = 1*1*1 + 1*1*2 + 1*2*2 + 2*2*2 = 15

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.
Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 227, #16.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 223.
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..1000
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].
Steve Butler and Pavel Karasik, A note on nested sums, J. Int. Seq., Vol. 13 (2010), Article 10.4.4, page 5.
Martin Griffiths, Remodified Bessel Functions via Coincidences and Near Coincidences, Journal of Integer Sequences, Vol. 14 (2011), Article 11.7.1.
C. Krishnamachaki, The operator (xD)^n, J. Indian Math. Soc., Vol. 15 (1923), pp. 3-4. [Annotated scanned copy]
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 29.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992, arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Eric Weisstein's World of Mathematics, Stirling numbers of the 2nd kind.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A001296, A001298, A008277, A008517, A048993, A062196, A094262.

Cf. A001303.

[n^2*(n+1)^2*(n+2)*(n+3)/48: n in [0..40]]; // Vincenzo Librandi, Sep 22 2017

A001297:=-(1+8*z+6*z**2)/(z-1)**7; # Simon Plouffe in his 1992 dissertation, without the initial 0

lst={};Do[f=StirlingS2[n+3, n];AppendTo[lst, f], {n, 0, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 27 2008 *)
a[ n_] := n^2 (n + 1)^2 (n + 2) (n + 3) / 48; (* Michael Somos, Sep 04 2017 *)
Table[StirlingS2[n+3,n],{n,0,30}] (* Harvey P. Dale, Dec 30 2019 *)

{a(n) = n^2 * (n+1)^2 * (n+2) * (n+3) / 48}; /* Michael Somos, Sep 04 2017 */

[stirling_number2(n+3,n) for n in range(0, 34)] # Zerinvary Lajos, May 16 2009

G.f.: x*(1 + 8*x + 6*x^2)/(1 - x)^7. - Paul Barry, Aug 05 2004
E.g.f. with offset -2: exp(x)*(1*(x^3)/3! + 11*(x^4)/4! + 25*(x^5)/5! + 15*(x^6)/6!). For the coefficients [1, 11, 25, 15] see triangle A112493. E.g.f.: 1/48*x*exp(x)*(x^5+22*x^4+152*x^3+384*x^2+312*x+48)/48. Above given e.g.f. differentiated twice.
a(n) = (binomial(n+4, n-1) - binomial(n+3, n-2))*(binomial(n+2, n-1) - binomial(n+1, n-2)). - Zerinvary Lajos, May 12 2006
a(n) = binomial(n+1, 2)*binomial(n+3, 4). - Vladimir Shevelev, Dec 18 2011
O.g.f.: D^3(x/(1-x)) = D^4(x), where D is the operator x/(1-x)*d/dx. - Peter Bala, Jul 02 2012
a(n) = A001303(-3-n) for all n in Z. - Michael Somos, Sep 04 2017
a(n) = Sum_{k=1..n} Sum_{i=1..n} i * C(k+2,k-1). - Wesley Ivan Hurt, Sep 21 2017
From Amiram Eldar, Jan 10 2022: (Start)
Sum_{n>=1} 1/a(n) = 16*Pi^2/3 - 464/9.
Sum_{n>=1} (-1)^(n+1)/a(n) = 260/9 - 4*Pi^2/3 - 64*log(2)/3. (End)
a(n) = Sum_{0<=i<=j<=k<=n} i*j*k. - Robert FERREOL, May 25 2022

Initial zero added by N. J. A. Sloane, Jan 21 2008

Name corrected by Nathaniel Johnston, Apr 30 2011

A002538 Second-order Eulerian numbers <>.

Original entry on oeis.org

1, 8, 58, 444, 3708, 33984, 341136, 3733920, 44339040, 568356480, 7827719040, 115336085760, 1810992556800, 30196376985600, 532953524275200, 9927928075161600, 194677319705702400, 4008789120817152000, 86495828444928000000, 1951566265951948800000, 45958933902500720640000

Offset: 1

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

Second-order Eulerian numbers <> count permutations of the multiset {1,1,2,2,...,n,n} with k ascents and the restriction that for any m <= n, all numbers between the two copies of m are less than m.
a(n) = number of edges in the Hasse diagram for the Bruhat order on permutations of [n+1]. - David Callan, Sep 03 2005
Proof. As explained on page 1 of the Stanley link, edges in the Hasse diagram of the (strong) Bruhat order on S_n are associated with pairs (pi,(i,j)) with pi in S_n and 1 <= i < j <= n, such that pi_i < pi_j and each entry of pi lying between pi_i and pi_j in POSITION does not lie between pi_i and pi_j in VALUE.
For example, pi = (3, 5, 1, 2, 4) gives edges for the (i,j) pairs (1,2), (1,5), (3,4), (4,5) but not, e.g., for (i,j) = (3,5) because 2 lies between pi_3=1 and pi_5=4 both in position and in value.
Let us count edges for a given pair (i,j). Consider the j-i+1 entries pi_i, pi_(i+1),...,pi_j. There are (j-i+1)! possible orderings for their values and (i,j) contributes an edge <=> the values of pi_i, pi_j are adjacent in this ordering with pi_i < pi_j.
There are (j-i)! such orderings (just coalesce the items pi_i, pi_j into a single item). The net result is that (i,j) contributes an edge 1/(j-i+1) of the time. So the total number of edges in the Hasse diagram is Sum_{1 <= i < j <= n} n!/(j-i+1) = (n+1)!(H_(n+1) - 2) + n! where H_n = 1 + 1/2 + 1/3 +  ... + 1/n is the harmonic sum. QED - David Callan, Mar 07 2006
Number of reentrant corners along the lower contours of all deco polyominoes of height n+2. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column. a(n) = Sum_{k>=1} k*A121579(n+2,k). - Emeric Deutsch, Aug 16 2006
a(n) is the total sum of the cycle maxima minus the cycle minima over all permutations of [n+1]. a(2) = 8 = 2+2+1+2+1+0: (123), (132), (12)(3), (13)(2), (1)(23), (1)(2)(3). - Alois P. Heinz, Dec 22 2023
a(n-1) is the number of parking functions of order n with exactly n-1 lucky cars, where a lucky car is a car which parks in the spot it prefers. - Kimberly P. Hadaway, Jun 20 2024

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd ed. Addison-Wesley, Reading, MA, 1994, p. 270.
J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 83.
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 = 1..447 (first 100 terms from T. D. Noe)
E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
Steve Butler, Kimberly Hadaway, Victoria Lenius, Preston Martens, and Marshall Moats, Lucky cars and lucky spots in parking functions, arXiv:2412.07873 [math.CO], 2024. See p. 4.
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.]
Emma Colaric, Ryan DeMuse, Jeremy L. Martin, and Mei Yin, Interval parking functions, arXiv:2006.09321 [math.CO], 2020.
I. Gessel and R. P. Stanley, Stirling polynomials, J. Combin. Theory, A 24 (1978), 24-33. (See Table 1.)
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.
Jean-Christophe Novelli and Jean-Yves Thibon, Duplicial algebras and Lagrange inversion, arXiv preprint arXiv:1209.5959 [math.CO], 2012.
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].
J. Ser, Les Calculs Formels des Séries de Factorielles (Annotated scans of some selected pages)
R. P. Stanley, A survey of the Bruhat order of the symmetric group
Albert Tarn, Approximations to certain square roots and the series of numbers connected therewith [Annotated scanned copy]

Second diagonal of A008517 and second column of A112007.

Cf. A121579.

[n le 1 select n else (n+2)*Self(n-1) + n*Factorial(n): n in [1..30]]; // Vincenzo Librandi, Aug 11 2018

egf:= (x+2*log(1-x))/(x-1)^3:
a:= n-> n!*coeff(series(egf, x, n+1), x, n):
seq(a(n), n=1..21);  # Peter Luschny, Feb 12 2021
# Alternative:
a := n -> (n + 1)! * ((n + 2)*harmonic(n + 2) - 2*n - 3);
seq(a(n), n = 1..22);  # Peter Luschny, Apr 09 2024

Table[(-1)^(n + 1)* Sum[(-1)^(n - k) k (-1)^(n - k) StirlingS1[n + 3, k + 3], {k, 0, n}], {n, 1, 16}] (* Zerinvary Lajos, Jul 08 2009 *)
a[n_]:=(-1)*((2*n+3)*(n+1)!-Abs[StirlingS1[n+3,2]]);Flatten[Table[a[n],{n,1,21}]] (* Detlef Meya, Apr 09 2024 *)

N=66; x='x+O('x^66); Vec(serlaplace((x+2*log(1-x))/(x-1)^3)) \\ Joerg Arndt, Apr 09 2016

From Vladeta Jovovic, Sep 15 2003: (Start)
a(n) = Sum_{k=1..n} k * |Stirling1(n+2, k+2)|.
E.g.f.: (x+2*log(1-x))/(x-1)^3. (End)
With alternating signs: Ramanujan polynomials psi_2(n, x) evaluated at -1. - Ralf Stephan, Apr 16 2004
a(n) = (n+2)*a(n-1) + n*n!, n>=1, a(0):=0.
a(n) = (n+2)!*HarmonicSum(n+2) + (n+1)! - 2(n+2)! where HarmonicSum(n) = 1 + 1/2 + 1/3 + ... + 1/n. - David Callan, Mar 07 2006
a(n) = (n+1)!*((n+2)*h(n+2)-2*n-3) where h(n) = Sum_{k=1..n} 1/k. - Gary Detlefs, Mar 25 2011
Conjecture: a(n) + 2*(-n-2)*a(n-1) + (n^2+4*n+1)*a(n-2) - n*(n-1)*a(n-3) = 0. - R. J. Mathar, Oct 27 2014
a(n) = (-1)*((2*n + 3)*(n + 1)! - abs(Stirling1(n + 3, 2))). - Detlef Meya, Apr 09 2024

More terms from Joerg Arndt, Apr 09 2016

A340556 E2(n, k), the second-order Eulerian numbers with E2(0, k) = δ_{0, k}. Triangle read by rows, E2(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 8, 6, 0, 1, 22, 58, 24, 0, 1, 52, 328, 444, 120, 0, 1, 114, 1452, 4400, 3708, 720, 0, 1, 240, 5610, 32120, 58140, 33984, 5040, 0, 1, 494, 19950, 195800, 644020, 785304, 341136, 40320, 0, 1, 1004, 67260, 1062500, 5765500, 12440064, 11026296, 3733920, 362880

Offset: 0

Peter Luschny, Feb 05 2021

The second-order Eulerian number E2(n, k) is the number of Stirling permutations of order n with exactly k descents; here the last index is defined to be a descent. More formally, let Q_n denote the set of permutations of the multiset {1,1,2,2, ..., n,n} in which, for all j, all entries between two occurrences of j are larger than j, then E2(n, k) = card({s in Q_n with des(s) = k}), where des(s) = card({j: s(j) > s(j+1)}) is the number of descents of s.
Also the number of Riordan trapezoidal words of length n with k distinct letters (see Riordan 1976, p. 9).
Also the number of rooted plane trees on n + 1 vertices with k leaves (see Janson 2008, p. 543).
Let b(n) = (1/2)*Sum_{k=0..n-1} (-1)^k*E2(n-1, k+1) / C(2*n-1, k+1). Apparently b(n) = Bernoulli(n, 1) = -n*Zeta(1 - n) = Integral_{x=0..1}F_n(x) for n >= 1. Here F_n(x) are the signed Fubini polynomials (A278075). (See Rzadkowski and Urlinska, example 4.)

			Triangle starts:
  [0] 1;
  [1] 0, 1;
  [2] 0, 1, 2;
  [3] 0, 1, 8,    6;
  [4] 0, 1, 22,   58,    24;
  [5] 0, 1, 52,   328,   444,     120;
  [6] 0, 1, 114,  1452,  4400,    3708,    720;
  [7] 0, 1, 240,  5610,  32120,   58140,   33984,    5040;
  [8] 0, 1, 494,  19950, 195800,  644020,  785304,   341136,   40320;
  [9] 0, 1, 1004, 67260, 1062500, 5765500, 12440064, 11026296, 3733920, 362880.
To illustrate the generating function for row 3: The expansion of (1 - x)^7*(x*exp(-x) + 16*x^2*exp(-x)^2 + (243*x^3*exp(-x)^3)/2) gives the polynomial x + 8*x^2 + 6*x^3. The coefficients of this polynomial give row 3.
.
Stirling permutations of order 3 with exactly k descents: (When counting the descents one may assume an invisible '0' appended to the permutations.)
  T[3, k=0]:
  T[3, k=1]: 112233;
  T[3, k=2]: 331122; 223311; 221133; 133122; 122331; 122133; 113322; 112332;
  T[3, k=3]: 332211; 331221; 233211; 221331; 133221; 123321.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd ed. Addison-Wesley, Reading, MA, 1994, p. 270.

J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, Generalized Stirling permutations and forests: Higher-order Eulerian and Ward numbers, Electronic Journal of Combinatorics 22(3), 2015.
Kenny Barrese, Jennifer Elder, Pamela E. Harris, and Anthony Simpson, Enumerating Flat Fubini Rankings, arXiv:2504.06466 [math.CO], 2025. Conjecture 4.4. p. 14.
Brian Drake, An inversion theorem for labeled trees and some limits of areas under lattice paths (Example 1.10.1), Thesis, Brandeis Univ., Aug. 2008, p. 58.
Sergi Elizalde, Descents on quasi-Stirling permutations, arXiv:2002.00985 [math.CO], 2020.
I. Gessel and R. P. Stanley, Stirling polynomials, J. Combin. Theory, A 24, 24-33, 1978.
Svante Janson, Plane recursive trees, Stirling permutations and an urn model, Fifth Colloquium on Mathematics and Computer Science, 541-547, Discrete Math. Theor. Comput. Sci. Proc., AI, 2008.
Peter Luschny, A companion to A340556, A SageMath-Jupyter notebook, Feb. 2021.
Peter Luschny, How are the Eulerian numbers of the first-order related to the Eulerian numbers of the second-order?, MathOverflow, Feb. 2021.
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.
John Riordan, The blossoming of Schröder's fourth problem, Acta Math., 137 (1976), 1-16.
G. Rzadkowski and M. Urlinska, A Generalization of the Eulerian Numbers, arXiv:1612.06635 [math.CO], 2016.

Indexing the second-order Eulerian numbers comes in three flavors: A008517 (following Riordan and Comtet), A201637 (following Graham, Knuth, and Patashnik) and this indexing, extending the definition of Gessel and Stanley. (A008517 is the main entry of the numbers.) The corresponding triangles of the first-order Eulerian numbers are A008292, A173018, and A123125.
Row reversed: A163936 (with offset = 0).
Values: E2poly(n, 1) = A001147(n), E2poly(n, -1) ~ -A001662(n+1), E2poly(n, 2) = A112487(n), 2^n*E2poly(n, 1/2) = A000311(n+1), 2^n*E2poly(n, -1/2) = A341106(n).
Diagonals: A000142, A002538, A002539, A112008, A112485.
Columns: A000007, A000012, A005803, A004301, A006260.
Cf. A008299, A137375, A008306, A106828, A269939, A278075, A331998.

# Using the recurrence:
E2 := proc(n, k) option remember;
if k = 0 and n = 0 then return 1 fi; if n < 0 then return 0 fi;
E2(n-1, k)*k + E2(n-1, k-1)*(2*n - k) end: seq(seq(E2(n, k), k = 0..n), n = 0..9);
# Using the row generating function:
E2egf := n -> (1-x)^(2*n+1)*add(k^(n+k)/k!*(x*exp(-x))^k, k=0..n);
T := (n, k) -> coeftayl(E2egf(n), x=0, k): seq(print(seq(T(n, j),j=0..n)), n=0..7);
# Using the built-in function:
E2 := (n, k) -> `if`(k=0, k^n, combinat:-eulerian2(n, k-1)):
# Using the compositional inverse (series reversion):
E2triangle := proc(N) local r, s, C; Order := N + 2;
s := solve(y = series(x - t*(exp(x) - 1), x), x):
r := n -> -n!*(t - 1)^(2*n - 1)*coeff(s, y, n); C := [seq(expand(r(n)), n = 1..N)];
seq(print(seq(coeff(C[n+1], t, k), k = 0..n)), n = 0..N-1) end: E2triangle(10);

T[n_, k_] := T[n, k] = If[k == 0, Boole[n == 0], If[n < 0, 0, k T[n - 1, k] + (2 n - k) T[n - 1, k - 1]]]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten
(* Via row polynomials: *)
E2poly[n_] := If[n == 0, 1,
  Expand[Simplify[x (x - 1)^(2 n) D[((1 - x)^(1 - 2 n) E2poly[n - 1]), x]]]];
Table[CoefficientList[E2poly[n], x], {n, 0, 9}] // Flatten
(* Series reversion *)
Revert[gf_, len_] := Module[{S = InverseSeries[Series[gf, {x, 0, len + 1}], x]},
Table[CoefficientList[(n + 1)! (1 - t)^(2 n + 1) Coefficient[S, x, n + 1], t],
{n, 0, len}] // Flatten]; Revert[x + t - t Exp[x], 6]

E2poly(n) = if(n == 0, 1, x*(x-1)^(2*n)*deriv((1-x)^(1-2*n)*E2poly(n-1)));
{ for(n = 0, 9, print(Vecrev(E2poly(n)))) }

T(n, k) = sum(j=0, n-k, (-1)^(n-j)*binomial(2*n+1, j)*stirling(2*n-k-j+1, n-k-j+1, 1)); \\ Michel Marcus, Feb 11 2021

# See also link to notebook.
@cached_function
def E2(n, k):
    if n < 0: return 0
    if k == 0: return k^n
    return k * E2(n - 1, k) + (2*n - k) * E2(n - 1, k - 1)  # Peter Luschny, Mar 08 2025

E2(n, k) = E2(n-1, k)*k + E2(n-1, k-1)*(2*n - k) for n > 0 and 0 <= k <= n, and E2(0, 0) = 1; in all other cases E(n, k) = 0.
E2(n, k) = Sum_{j=0..n-k}(-1)^(n-j)*binomial(2*n+1, j)*Stirling1(2*n-k-j+1, n-k-j+1).
E2(n, k) = Sum_{j=0..k}(-1)^(k-j)*binomial(2*n + 1, k - j)*Stirling2(n + j, j).
Stirling1(x, x - n) = (-1)^n*Sum_{k=0..n} E2(n, k)*binomial(x + k - 1, 2*n).
Stirling2(x, x - n) = Sum_{k=0..n} E2(n, k)*binomial(x + n - k, 2*n).
E2poly(n, x) = Sum_{k=0..n} E2(n, k)*x^k, as row polynomials.
E2poly(n, x) = x*(x-1)^(2*n)*d_{x}((1-x)^(1-2*n)*E2poly(n-1)) for n>=1 and E2poly(0)=1.
E2poly(n, x) = (1 - x)^(2*n + 1)*Sum_{k=0..n}(k^(n + k)/k!)*(x*exp(-x))^k.
W(n, k) = [x^k] (1+x)^n*E2poly(n, x/(1 + x)) are the Ward numbers A269939.
E2(n, k) = [x^k] (1-x)^n*Wpoly(n, x/(1 - x)); Wpoly(n, x) = Sum_{k=0..n}W(n, k)*x^k.
W(n, k) = Sum_{j=0..k} E2(n, j)*binomial(n - j, n - k).
E2(n, k) = Sum_{j=0..k} (-1)^(k-j)*W(n, j)*binomial(n - j, k - j).
The compositional inverse with respect to x of x - t*(exp(x) - 1) (see B. Drake):
T(n, k) = [t^k](n+1)!*(1-t)^(2*n+1)*[x^(n+1)] InverseSeries(x - t*(exp(x)-1), x).
AS1(n, k) = Sum_{j=0..n-k} binomial(j, n-2*k)*E2(n-k, j+1), where AS1(n, k) are the associated Stirling numbers of the first kind (A008306, A106828).
E2(n, k) = Sum_{j=0..n-k+1} (-1)^(n-k-j+1)*AS1(n+j, j)*binomial(n-j, n-k-j+1), for n >= 1.
AS2(n, k) = Sum_{j=0..n-k} binomial(j, n-2*k)*E2(n-k, n-k-j) for n >=1, where AS2(n, k) are the associated Stirling numbers of the second kind (A008299, A137375).
E2(n, k) = Sum_{j=0..k} (-1)^(k-j)*AS2(n + j, j)*binomial(n - j, k - j).

A163936 Triangle related to the o.g.f.s. of the right-hand columns of A130534 (E(x,m=1,n)).

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 6, 8, 1, 0, 24, 58, 22, 1, 0, 120, 444, 328, 52, 1, 0, 720, 3708, 4400, 1452, 114, 1, 0, 5040, 33984, 58140, 32120, 5610, 240, 1, 0, 40320, 341136, 785304, 644020, 195800, 19950, 494, 1, 0, 362880, 3733920, 11026296, 12440064, 5765500, 1062500

Offset: 1

Johannes W. Meijer, Aug 13 2009

The asymptotic expansions of the higher-order exponential integral E(x,m=1,n) lead to triangle A130524, see A163931 for information on E(x,m,n). The o.g.f.s. of the right-hand columns of triangle A130534 have a nice structure: gf(p) = W1(z,p)/(1-z)^(2*p-1) with p = 1 for the first right-hand column, p = 2 for the second right-hand column, etc. The coefficients of the W1(z,p) polynomials lead to the triangle given above, n >= 1 and 1 <= m <= n. Our triangle is the same as A112007 with an extra right-hand column, see also the second Eulerian triangle A008517. The row sums of our triangle lead to A001147.
We observe that the row sums of the triangles A163936 (m=1), A163937 (m=2), A163938 (m=3) and A163939 (m=4) for z=1 lead to A001147, A001147 (minus a(0)), A001879 and A000457 which are the first four left-hand columns of the triangle of the Bessel coefficients A001497 or, if one wishes, the right-hand columns of A001498. We checked this phenomenon for a few more values of m and found that this pattern persists: m = 5 leads to A001880, m=6 to A001881, m=7 to A038121 and m=8 to A130563 which are the next left- (right-) hand columns of A001497 (A001498). An interesting phenomenon.
If one assumes the triangle not (1,1) based but (0,0) based, one has T(n, k) = E2(n, n-k), where E2(n, k) are the second-order Eulerian numbers A340556. - Peter Luschny, Feb 12 2021

			Triangle starts:
[ 1]      1;
[ 2]      1,       0;
[ 3]      2,       1,      0;
[ 4]      6,       8,      1,      0;
[ 5]     24,      58,     22,      1,      0;
[ 6]    120,     444,    328,     52,      1,     0;
[ 7]    720,    3708,   4400,   1452,    114,     1,   0;
[ 8]   5040,   33984,  58140,  32120,   5610,   240,   1,  0;
[ 9]  40320,  341136, 785304, 644020, 195800, 19950, 494,  1, 0;
The first few W1(z,p) polynomials are
W1(z,p=1) = 1/(1-z);
W1(z,p=2) = (1 + 0*z)/(1-z)^3;
W1(z,p=3) = (2 + 1*z + 0*z^2)/(1-z)^5;
W1(z,p=4) = (6 + 8*z + 1*z^2 + 0*z^3)/(1-z)^7.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Row sums equal A001147.
A000142, A002538, A002539, A112008, A112485 are the first few left hand columns.
A000007, A000012, A005803(n+2), A004301, A006260 are the first few right hand columns.
Cf. A163931 (E(x,m,n)), A048994 (Stirling1) and A008517 (Euler).
Cf. A112007, A163937 (E(x,m=2,n)), A163938 (E(x,m=3,n)) and A163939 (E(x,m=4,n)).
Cf. A001497 (Bessel), A001498 (Bessel), A001147 (m=1), A001147 (m=2), A001879 (m=3) and A000457 (m=4), A001880 (m=5), A001881 (m=6) and A038121 (m=7).
Cf. A340556.

with(combinat): a := proc(n, m): add((-1)^(n+k+1)*binomial(2*n-1, k)*stirling1(m+n-k-1, m-k), k=0..m-1) end: seq(seq(a(n, m), m=1..n), n=1..9);  # Johannes W. Meijer, revised Nov 27 2012

Table[Sum[(-1)^(n + k + 1)*Binomial[2*n - 1, k]*StirlingS1[m + n - k - 1, m - k], {k, 0, m - 1}], {n, 1, 10}, {m, 1, n}] // Flatten (* G. C. Greubel, Aug 13 2017 *)

for(n=1,10, for(m=1,n, print1(sum(k=0,m-1,(-1)^(n+k+1)* binomial(2*n-1,k)*stirling(m+n-k-1,m-k, 1)), ", "))) \\ G. C. Greubel, Aug 13 2017

\\ assuming offset = 0:
E2poly(n,x) = if(n == 0, 1, x*(x-1)^(2*n)*deriv((1-x)^(1-2*n)*E2poly(n-1,x)));
{ for(n = 0, 9, print(Vec(E2poly(n,x)))) } \\ Peter Luschny, Feb 12 2021

a(n, m) = Sum_{k=0..(m-1)} (-1)^(n+k+1)*binomial(2*n-1,k)*Stirling1(m+n-k-1,m-k), for 1 <= m <= n.

Assuming offset = 0 the T(n, k) are the coefficients of recursively defined polynomials. T(n, k) = [x^k] x^n*E2poly(n, 1/x), where E2poly(n, x) = x*(x - 1)^(2*n)*d_{x}((1 - x)^(1 - 2*n)*E2poly(n - 1, x))) for n >= 1 and E2poly(0, x) = 1. - Peter Luschny, Feb 12 2021

A008788 a(n) = n^(n+2).

Original entry on oeis.org

0, 1, 16, 243, 4096, 78125, 1679616, 40353607, 1073741824, 31381059609, 1000000000000, 34522712143931, 1283918464548864, 51185893014090757, 2177953337809371136, 98526125335693359375, 4722366482869645213696

Offset: 0

N. J. A. Sloane

			G.f. = x + 16*x^2 + 243*x^3 + 4096*x^4 + 78125*x^5 + 1679616*x^6 + ...

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, A008789, A008790, A008791.

List([0..20], n-> n^(n+2)); # G. C. Greubel, Sep 11 2019

[n^(n+2): n in [0..20]]; // Vincenzo Librandi, Jun 11 2013

Table[n^(n+2), {n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Dec 26 2010 *)
CoefficientList[Series[LambertW[-x] * (2*LambertW[-x]-1) / (1 + LambertW[-x])^5, {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Dec 20 2014 *)

vector(20, n, (n-1)^(n+1)) \\ G. C. Greubel, Nov 14 2017

[n^(n+2) for n in (0..20)] # G. C. Greubel, Sep 11 2019

E.g.f.(x): T*(1 + 2*T)*(1-T)^(-5); where T=T(x) is Euler's tree function (see A000169). - Len Smiley, Nov 17 2001
See A008517 and A134991 for similar e.g.f.s. and A048993. - Tom Copeland, Oct 03 2011
E.g.f.: d^2/dx^2 {x^2/(T(x)^2*(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

A094262 Triangle read by rows: T(n,k) is the number of rooted trees with k nodes which are disjoint sets of labels with union {1..n}. If a node has an empty set of labels then it must have at least two children.

Original entry on oeis.org

1, 1, 2, 1, 1, 6, 12, 10, 3, 1, 14, 61, 124, 131, 70, 15, 1, 30, 240, 890, 1830, 2226, 1600, 630, 105, 1, 62, 841, 5060, 16990, 35216, 47062, 40796, 22225, 6930, 945, 1, 126, 2772, 25410, 127953, 401436, 836976, 1196532, 1182195, 795718, 349020, 90090, 10395

Offset: 1

André F. Labossière, Jun 01 2004

The original name for this sequence was "Triangle read by rows giving the coefficients of formulas generating each variety of S2(n,k) (Stirling numbers of 2nd kind). The p-th row (p>=1) contains T(i,p) for i=1 to 2*p-1, where T(i,p) satisfies Sum_{i=1..2*p-1} T(i,p) * C(n-p,i-1)".
The terms of the n-th diagonal sequence of the triangle of Stirling numbers of the second kind A008277, i.e., (Stirling2(N + n - 1,N)), N>=1, are given by a polynomial in N of degree 2*n - 2. This polynomial may be expressed as a linear combination of the falling factorial polynomials binomial(N - n,0), binomial(N - n,1), ... , binomial(N - n,2*n - 2). This table gives the coefficients in these expansions.
The formulas obtained are those for Stirling2(N+1,N) (A000217), Stirling2(N+2,N) (A001296), Stirling2(N+3,N) (A001297), Stirling2(N+4,N) (A001298), Stirling2(N+5,N) (A112494), Stirling2(N+6,N) (A144969) and so on.

			Row 5 contains 1,30,240,890,1830,2226,1600,630,105, so the formula generating Stirling2(n+4,n) numbers (A001298) will be the following: 1 + 30*(n-5) + 240*C(n-5,2) + 890*C(n-5,3) + 1830*C(n-5,4) + 2226*C(n-5,5) + 1600*C(n-5,6) + 630*C(n-5,7) + 105*C(n-5,8). For example, taking n = 9 gives Stirling2(13,9) = 359502.
Triangle starts:
  1;
  1,  2,   1;
  1,  6,  12,  10,    3;
  1, 14,  61, 124,  131,   70,   15;
  1, 30, 240, 890, 1830, 2226, 1600, 630, 105;
  ...
From _Peter Bala_, Jun 14 2016: (Start)
Connection with row polynomials of A134991:
  R(2,z) = (1 + z)^2*z
  R(3,z) = (1 + z)^2*(z + 3*z^2)
  R(4,z) = (1 + z)^4*(z + 10*z^2 + 15*z^3)
  R(5,z) = (1 + z)^5*(z + 25*z^2 + 105*z^3 + 105*z^4). (End)
From _Andrew Howroyd_, Mar 28 2025: (Start)
The T(3,3) = 12 trees up to relabeling have one of the following 3 forms:
     {}         {1}        {1}
    /  \       /   \        |
  {1} {2,3}   {2}  {3}     {2}
                            |
                           {3}
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..2500 (rows 1..50)
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].
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions (with Formulas, Graphs and Mathematical Tables), U.S. Dept. of Commerce, National Bureau of Standards, Applied Math. Series 55, 1964, 1046 pages (9th Printing: November 1970) - Combinatorial Analysis, Table 24.4, Stirling Numbers of the Second Kind (author: Francis L. Miksa), p. 835.
J. Fernando Barbero G., Jesús Salas, Eduardo J. S. Villaseñor, Generalized Stirling permutations and forests: Higher-order Eulerian and Ward numbers, Electronic Journal of Combinatorics 22(3) (2015), #P3.37.
M. Kazarian, KP hierarchy for Hodge integrals, p. 2, arxiv:0809.3263 [math.AG], 18 Sep 2008. [From _Tom Copeland_, Jun 12 2015]
F. R. McMorris and T. Zaslavsky, The number of cladistic characters, Math. Biosciences, 54 (1981), 3-10.
F. R. McMorris and T. Zaslavsky, The number of cladistic characters, Math. Biosciences, 54 (1981), 3-10. [Annotated scanned copy]
Eric Weisstein's World of Mathematics, Stirling numbers of the 2nd kind.

Row sums are A005172.
Columns 1..4 are A000012, A000918, A005173, A005174, A005175.
Cf. A008277, A000217, A001296, A001297, A001298, A008517, A134991, A112494, A144969.

row_poly := n -> (1+z)^(n+1)*add(z^k*add((-1)^(m+k)*binomial(n+k,n+m)*Stirling2(n+m,m), m=0..k), k=0..n): T_row := n -> seq(coeff(row_poly(n),z,j),j=1..2*n+1):
seq(T_row(n),n=0..6); # Peter Luschny, Jun 15 2016

Clear[T, q, u]; T[0] = q[1];T[n_] := Sum[m*(u^2*q[m] + 2*u*q[m+1] + q[m+2])*D[T[n-1], q[m]], {m, 1, 2*n+1}]; row[n_] := List @@ Expand[T[n-1]] /. {u -> 1, q[] -> 1}; Table[row[n], {n, 1, 7}] // Flatten (* _Jean-François Alcover, Jun 12 2015 *)

T(n)={my(g=serreverse(log(((1+1/y)*x+1)/exp(x + O(x*x^n))))); [Vecrev(p/y) | p<-Vec(serlaplace(g))]}
{ my(A=T(5)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Mar 28 2025

Apparently, a raising operator for bivariate polynomials P(n,u,z) having these coefficients is R = (u+z)^2 * z * d/dz with P(0,u,z) = z. E.g., R P(1,u,z) = R^2 P(0,u,z) = R^2 z = u^4 z + 6 u^3 z^2 + 12 u^2 z^3 + 10 u z^4 + 3 z^5 = P(2,u,z). See the Kazarian link. - Tom Copeland, Jun 12 2015
Reverse polynomials seem to be generated by 1 + exp[t*(x+1+z)^2*(1+z)d/dz]z evaluated at z = 0. - Tom Copeland, Jun 13 2015
From Peter Bala, Jun 14 2016: (Start)
T(n,k) = k*T(n,k) + 2*(k - 1)*T(n,k-1) + (k - 2)*T(n,k-2).
n-th diagonal of A008277: Stirling2(N + n - 1,N) = Sum_{k = 1..2*n - 1} T(n,k)*binomial(N - n,k - 1) for N = 1,2,3,....
Row polynomials R(n,z) = Sum_{k >= 1} k^(n+k-1)*( z/(1 + z)*exp(-z/(1 + z)) )^k/k!, n = 1,2,..., follows from the formula given in A008277 for the o.g.f.'s of the diagonals of the Stirling numbers of the second kind.
Consequently, R(n+1,z) = (1 + z)^2*z*d/dz(R(n,z)) for n >= 1 as conjectured above by Copeland.
R(n,z) = (1 + z)^n*P(n,z) where P(n,z) are the row polynomials of A134991.
R(n,z) = (1 + z)^(2*n+1)*B(n,z/(1 + z)), where B(n,z) are the row polynomials of the triangle of second-order Eulerian numbers A008517 (see Barbero et al., Section 6, equation 27). (End)
Based on the comment of Bala the row polynomials have the explicit form R(n, z) = (1+z)^(n+1)*Sum_{k=0..n}(z^k*Sum_{m=0..k}((-1)^(m+k)*binomial(n+k, n+m)* Stirling2(n+m,m))). - Peter Luschny, Jun 15 2016
E.g.f. G(x,y) satisfies G(x,y) = y*(exp(x)*exp(G(x,y)) - G(x,y) - 1). - Andrew Howroyd, Mar 28 2025

Edited and Name changed by Peter Bala, Jun 16 2016

Name changed by Andrew Howroyd, Mar 28 2025

cp's OEIS Frontend

A008306 Triangle T(n,k) read by rows: associated Stirling numbers of first kind (n >= 2, 1 <= k <= floor(n/2)).

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A134991 Triangle of Ward numbers T(n,k) read by rows.

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A201637 Triangle of second-order Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.

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A001297 Stirling numbers of the second kind S(n+3, n).

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A002538 Second-order Eulerian numbers <>.

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A340556 E2(n, k), the second-order Eulerian numbers with E2(0, k) = δ_{0, k}. Triangle read by rows, E2(n, k) for 0 <= k <= n.

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