A000311 -id:A000311 - cp's OEIS frontend

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

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

A318813 Number of balanced reduced multisystems with n atoms all equal to 1.

Original entry on oeis.org

1, 1, 2, 6, 20, 90, 468, 2910, 20644, 165874, 1484344, 14653890, 158136988, 1852077284, 23394406084, 317018563806, 4587391330992, 70598570456104, 1151382852200680, 19835976878704628, 359963038816096924, 6863033015330999110, 137156667020252478684, 2867083618970831936826

Offset: 1

Gus Wiseman, Sep 04 2018

nonn

For n > 1, also the number of balanced reduced multisystems whose atoms are an integer partition of n with at least one part > 1. A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem. - Gus Wiseman, Dec 31 2019

			The a(5) = 20 balanced reduced multisystems (with n written in place of 1^n):
  5  (14)  (23)  (113)      (122)      (1112)
                 ((1)(13))  ((1)(22))  ((1)(112))
                 ((3)(11))  ((2)(12))  ((2)(111))
                                       ((11)(12))
                                       ((1)(1)(12))
                                       ((1)(2)(11))
                                       (((1))((1)(12)))
                                       (((1))((2)(11)))
                                       (((2))((1)(11)))
                                       (((12))((1)(1)))
                                       (((11))((1)(2)))

Andrew Howroyd, Table of n, a(n) for n = 1..200

The maximum-depth case is A000111.

Cf. A000311, A001055, A002846, A005121, A213427, A281118, A281119, A317145, A318812, A318846, A320154, A330474, A330679.

normize[m_]:=m/.Rule@@@Table[{Union[m][[i]],i},{i,Length[Union[m]]}];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
totfact[n_]:=totfact[n]=1+Sum[totfact[Times@@Prime/@normize[f]],{f,Select[facs[n],1

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(v=vector(n, i, i==1), u=vector(n)); for(r=1, #v, u += v*sum(j=r, #v, (-1)^(j-r)*binomial(j-1, r-1)); v=EulerT(v)); u} \\ Andrew Howroyd, Dec 30 2019

a(n > 1) = A330679(n)/2. - Gus Wiseman, Dec 31 2019

Terms a(14) and beyond from Andrew Howroyd, Dec 30 2019

Terminology corrected by Gus Wiseman, Dec 31 2019

A320160 Number of series-reduced balanced rooted trees whose leaves form an integer partition of n.

Original entry on oeis.org

1, 2, 3, 6, 9, 19, 31, 63, 110, 215, 391, 773, 1451, 2879, 5594, 11173, 22041, 44136, 87631, 175155, 348186, 694013, 1378911, 2743955, 5452833, 10853541, 21610732, 43122952, 86192274, 172753293, 347114772, 699602332, 1414033078, 2866580670, 5826842877, 11874508385

Offset: 1

Gus Wiseman, Oct 06 2018

nonn

A rooted tree is series-reduced if every non-leaf node has at least two branches, and balanced if all leaves are the same distance from the root.

Also the number of balanced unlabeled phylogenetic rooted trees with n leaves.

			The a(1) = 1 through a(6) = 19 rooted trees:
  1  2     3      4           5            6
     (11)  (12)   (13)        (14)         (15)
           (111)  (22)        (23)         (24)
                  (112)       (113)        (33)
                  (1111)      (122)        (114)
                  ((11)(11))  (1112)       (123)
                              (11111)      (222)
                              ((11)(12))   (1113)
                              ((11)(111))  (1122)
                                           (11112)
                                           (111111)
                                           ((11)(13))
                                           ((11)(22))
                                           ((12)(12))
                                           ((11)(112))
                                           ((12)(111))
                                           ((11)(1111))
                                           ((111)(111))
                                           ((11)(11)(11))

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A000081, A000311, A000669, A001678, A005804, A048816, A079500, A119262, A120803, A141268, A244925, A319312.

Cf. A316624, A320154, A320155, A320169, A320173, A320179.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
phy2[labs_]:=If[Length[labs]==1,labs,Union@@Table[Sort/@Tuples[phy2/@ptn],{ptn,Select[mps[Sort[labs]],Length[#1]>1&]}]];
Table[Sum[Length[Select[phy2[ptn],SameQ@@Length/@Position[#,_Integer]&]],{ptn,IntegerPartitions[n]}],{n,8}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(u=vector(n, n, 1), v=vector(n)); while(u, v+=u; u=EulerT(u)-u); v} \\ Andrew Howroyd, Oct 25 2018

Terms a(14) and beyond from Andrew Howroyd, Oct 25 2018

A108919 Number of series-reduced labeled trees with n nodes.

Original entry on oeis.org

1, 0, 1, 1, 13, 51, 601, 4803, 63673, 775351, 12186061, 196158183, 3661759333, 72413918019, 1583407093633, 36916485570331, 929770285841137, 24904721121298671, 711342228666833173, 21502519995056598639, 687345492498807434461, 23135454269839313430715, 818568166383797223246601, 30357965273255025673685091

Offset: 1

Vladeta Jovovic, Jul 20 2005

"Series-reduced" means that if the tree is rooted at 1, then there is no node with just a single child.

Callan points out that A002792 is an incorrect version of this sequence. - Joerg Arndt, Jul 01 2014

Seiichi Manyama, Table of n, a(n) for n = 1..415
David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], 2014.

The rooted version is A060356.

Cf. A000311, A000669, A001678, A292504, A316651, A316652, A318231, A318813, A330465, A330624, A352410.

f[n_] := Sum[(-1)^(n-k)*n!/k!*Binomial[n-1, k-1]*k^(k-1), {k, n}]/n; Table[ f[n], {n, 20}] (* Robert G. Wilson v, Jul 21 2005 *)

a(n) = { 1/n * sum(k=1, n, (-1)^(n-k) * binomial(n,k) * (n-1)!/(k-1)! * k^(k-1) ); } \\ Joerg Arndt, Aug 28 2014

a(n) = A060356(n)/n.
1 = Sum_{n>=0} a(n+1)*(exp(x)-x)^(-n-1)*x^n/n!.
E.g.f.: A(x) = Sum_{n>=0} a(n+1)*x^n/n! satisfies A(x) = exp(x*A(x))/(1+x). - Olivier Gérard, Dec 31 2013 (edited by Gus Wiseman, Dec 31 2019)
E.g.f.: -Integral (LambertW(-x/(1 + x))/x) dx. - Ilya Gutkovskiy, Jul 01 2020

More terms from Robert G. Wilson v, Jul 21 2005
New name (from A002792) by Joerg Arndt, Aug 28 2014
Offset corrected by Gus Wiseman, Dec 31 2019

A330465 Number of non-isomorphic series-reduced rooted trees whose leaves are multisets with a total of n elements.

Original entry on oeis.org

1, 4, 14, 87, 608, 5573, 57876, 687938, 9058892, 130851823, 2048654450, 34488422057, 620046639452, 11839393796270, 238984150459124, 5079583100918338, 113299159314626360, 2644085918303683758, 64393240540265515110, 1632731130253043991252, 43013015553755764179000

Offset: 1

Gus Wiseman, Dec 21 2019

nonn

Also inequivalent leaf-colorings of phylogenetic rooted trees with n labels. A phylogenetic rooted tree is a series-reduced rooted tree whose leaves are (usually disjoint) sets.

			Non-isomorphic representatives of the a(3) = 14 trees:
  ((1)((1)(1)))  ((1)((1)(2)))  ((1)((2)(3)))  ((2)((1)(1)))
  ((1)(1)(1))    ((1)(1)(2))    ((1)(2)(3))    ((2)(1,1))
  ((1)(1,1))     ((1)(1,2))     ((1)(2,3))
  (1,1,1)        (1,1,2)        (1,2,3)

The version where leaves are atoms is A318231.
The case with sets as leaves is A330624.
The case with disjoint sets as leaves is A141268.
Labeled versions are A330467 (strongly normal) and A330469 (normal).
The singleton-reduced version is A330470.
Cf. A000311, A000669, A004114, A005804, A007716, A281118, A289501, A292504, A316651, A316652, A318812, A319312, A330471, A330474, A330625, A339645.

\\ See links in A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(v=vector(n), p=sEulerT(x*sv(1) + O(x*x^n))); v[1]=sv(1); for(n=2, #v, v[n] = polcoef( sEulerT(x*Ser(v[1..n])), n ) + polcoef(p,n)); x*Ser(v)}
InequivalentColoringsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Dec 13 2020

Terms a(7) and beyond from Andrew Howroyd, Dec 13 2020

A118376 Number of all trees of weight n, where nodes have positive integer weights and the sum of the weights of the children of a node is equal to the weight of the node.

Original entry on oeis.org

1, 2, 6, 24, 112, 568, 3032, 16768, 95200, 551616, 3248704, 19389824, 117021824, 712934784, 4378663296, 27081760768, 168530142720, 1054464293888, 6629484729344, 41860283723776, 265346078982144, 1687918305128448, 10771600724946944, 68941213290561536

Offset: 1

Jeremy Johnson (jjohnson(AT)cs.drexel.edu), May 15 2006

nonn

The number of trees with leaf nodes equal to 1 is counted by the sequence A001003 of super-Catalan numbers. The number of binary trees is counted by the sequence A007317 and the number of binary trees with leaf nodes equal to 1 is counted by the sequence A000108 of Catalan numbers.
Also the number of series-reduced enriched plane trees of weight n. A series-reduced enriched plane tree of weight n is either the number n itself or a finite sequence of at least two series-reduced enriched plane trees, one of each part of an integer composition of n. For example, the a(3) = 6 trees are: 3, (21), (12), (111), ((11)1), (1(11)). - Gus Wiseman, Sep 11 2018
Conjectured to be the number of permutations of length n avoiding the partially ordered pattern (POP) {1>2, 1>3, 3>4, 3>5} of length 5. That is, conjectured to be the number of length n permutations having no subsequences of length 5 in which the first element is the largest, and the third element is larger than the fourth and fifth elements. - Sergey Kitaev, Dec 13 2020
This conjecture has been proven. It can be restated as the number of size n permutations avoiding 51423, 51432, 52413, 52431, 53412, 53421, 54312, 54321. There are twelve sets of permutations avoiding eight size five permutations that are known to match this sequence. A further four are conjectured to match this sequence. - Christian Bean, Jul 24 2024

			T(3) = 6 because there are six trees
  3    3      3     3     3       3
      2 1    2 1   1 2   1 2    1 1 1
            1 1           1 1
From _Gus Wiseman_, Sep 11 2018: (Start)
The a(4) = 24 series-reduced enriched plane trees:
  4,
  (31), (13), (22), (211), (121), (112), (1111),
  ((21)1), ((12)1), (1(21)), (1(12)), (2(11)), ((11)2),
  ((111)1), (1(111)), ((11)(11)), ((11)11), (1(11)1), (11(11)),
  (((11)1)1), ((1(11))1), (1((11)1)), (1(1(11))).
(End)

G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael H. Albert, Christian Bean, Anders Claesson, Émile Nadeau, Jay Pantone, and Henning Ulfarsson, PermPAL database.
Paul Barry, Generalized Eulerian Triangles and Some Special Production Matrices, arXiv:1803.10297 [math.CO], 2018.
Christian Bean, Émile Nadeau, Jay Pantone, and Henning Ulfarsson, Permutations avoiding bipartite partially ordered patterns have a regular insertion encoding, The Electronic Journal of Combinatorics, Volume 31, Issue 3 (2024); arXiv preprint, arXiv:2312.07716 [math.CO], 2023.
Hadrien Cambazard and Nicolas Catusse, Fixed-Parameter Algorithms for Rectilinear Steiner tree and Rectilinear Traveling Salesman Problem in the Plane, arXiv preprint arXiv:1512.06649 [cs.DS], 2015.
S. B. Ekhad and M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017).
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
Pawel Hitczenko, Jeremy R. Johnson, and Hung-Jen Huang, Distribution of a class of divide and conquer recurrences arising from the computation of the Walsh-Hadamard transform, Theoretical Computer Science, Vol. 352, 2006, pp. 8-30.
J. R. Johnson and M. Püschel, In search for the optimal Walsh-Hadamard transform, Proc. ICASSP, Vol. 4, 2000, pp. 3347-3350.
Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Cf. A000108, A000311, A001003, A005804, A007317, A007853, A032171, A032200, A143363, A289501, A317852.

T := proc(n) option remember; local C, s, p, tp, k, i; if n = 1 then return 1; else s := 1; for k from 2 to n do C := combinat[composition](n,k); for p in C do tp := map(T,p); s := s + mul(tp[i],i=1..nops(tp)); end do; end do; end if; return s; end;

Rest[CoefficientList[Series[(Sqrt[1-8*x+8*x^2]-1)/(4*x-4), {x, 0, 20}], x]] (* Vaclav Kotesovec, Feb 03 2014 *)
a[n_] := 1+Sum[Binomial[n-1, k-1]*Hypergeometric2F1[2-k, k+1, 2, -1], {k, 2, n}]; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Apr 03 2015, after Vladimir Kruchinin *)
urp[n_]:=Prepend[Join@@Table[Tuples[urp/@ptn],{ptn,Join@@Permutations/@Select[IntegerPartitions[n],Length[#]>1&]}],n];
Table[Length[urp[n]],{n,7}] (* Gus Wiseman, Sep 11 2018 *)

a(n):=sum((-1)^j*2^(n-j-1)*binomial(n,j)*binomial(2*n-2*j-2,n-2*j-1),j,0,(n-1)/2)/n; /* Vladimir Kruchinin, Sep 29 2020 */

x='x+O('x^25); Vec((sqrt(1-8*x+8*x^2) - 1)/(4*x-4)) \\ G. C. Greubel, Feb 08 2017

Recurrence: T(1) = 1; For n > 1, T(n) = 1 + Sum_{n=n1+...+nt} T(n1)*...*T(nt).
G.f.: (-1+(1-8*z+8*z^2)^(1/2))/(-4+4*z).
From Vladimir Kruchinin, Sep 03 2010: (Start)
O.g.f.: A(x) = A001003(x/(1-x)).
a(n) = Sum_{k=1..n} binomial(n-1,k-1)*A001003(k), n>0. (End)
D-finite with recurrence: n*a(n) + 3*(-3*n+4)*a(n-1) + 4*(4*n-9)*a(n-2) + 8*(-n+3)*a(n-3) = 0. - R. J. Mathar, Sep 27 2013
a(n) ~ sqrt(sqrt(2)-1) * 2^(n-1/2) * (2+sqrt(2))^(n-1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 03 2014
From Peter Bala, Jun 17 2015: (Start)
With offset 0, binomial transform of A001003.
O.g.f. A(x) = series reversion of x*(2*x - 1)/(2*x^2 - 1); 2*(1 - x)*A^2(x) - A(x) + x = 0.
A(x) satisfies the differential equation (x - 9*x^2 + 16*x^3 - 8*x^4)*A'(x) + x*(3 - 4*x)*A(x) + x*(2*x - 1) = 0. Extracting coefficients gives Mathar's recurrence above. (End)
a(n) = Sum_{j=0..(n-1)/2} (-1)^j*2^(n-j-1)*C(n,j)*C(2*n-2*j-2,n-2*j-1)/n. - Vladimir Kruchinin, Sep 29 2020

A119262 Number of B-trees of order infinity with n leaves, where a(n) = Sum_{k=1..floor(n/2)} a(k)C(n-k-1,n-2k) for n >= 2, with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 5, 8, 14, 25, 46, 85, 158, 294, 548, 1022, 1908, 3567, 6683, 12556, 23669, 44781, 85046, 162122, 310157, 595322, 1146057, 2212004, 4278908, 8292738, 16097018, 31286456, 60873574, 118543329, 231009934, 450434739, 878687665

Offset: 0

Paul D. Hanna, May 11 2006

nonn

A B-tree of order m is an ordered tree such that every node has at most m children, the root has at least 2 children, every node except the root has 0 or at least m/2 children, all end-nodes are at the same level. This sequence is the limit of the B-trees as m --> infinity.
Starting with offset 2, the eigensequence of triangle A011973. - Gary W. Adamson, Jul 08 2012
Number of balanced series-reduced rooted plane trees with n leaves. A rooted tree is series-reduced if every non-leaf node has at least two branches, and balanced if all leaves are the same distance from the root. - Gus Wiseman, Oct 07 2018

			A(x) = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 5*x^6 + 8*x^7 + 14*x^8 + ...
Series form:
A(x) = x + x^2/(1-x) + x^4/((1-x)*((1-x)-x^2)) + x^8/((1-x)*((1-x)-x^2)*((1-x)*((1-x)-x^2)-x^4)) + ... + x^(2^n)/D(n,x) + x^(2^(n+1))/[D(n,x)*(D(n,x)-x^(2^n))] + ...
Terms also satisfy the series:
x = x*(1-x) + x^2*(1-x^2)/(1+x) + x^3*(1-x^3)/(1+x)^2 + 2*x^4*(1-x^4)/(1+x)^3 + 3*x^5*(1-x^5)/(1+x)^4 + 5*x^6*(1-x^6)/(1+x)^5 + 8*x^7*(1-x^7)/(1+x)^6 + 14*x^8*(1-x^8)/(1+x)^7 + 25*x^9*(1-x^9)/(1+x)^8 + ... + a(n)*x^n*(1-x^n)/(1+x)^(n-1) + ...
From _Gus Wiseman_, Oct 07 2018: (Start)
The a(1) = 1 through a(7) = 8 balanced series-reduced rooted plane trees:
  o  (oo)  (ooo)  (oooo)      (ooooo)      (oooooo)        (ooooooo)
                  ((oo)(oo))  ((oo)(ooo))  ((oo)(oooo))    ((oo)(ooooo))
                              ((ooo)(oo))  ((ooo)(ooo))    ((ooo)(oooo))
                                           ((oooo)(oo))    ((oooo)(ooo))
                                           ((oo)(oo)(oo))  ((ooooo)(oo))
                                                           ((oo)(oo)(ooo))
                                                           ((oo)(ooo)(oo))
                                                           ((ooo)(oo)(oo))
(End)

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, B-Tree.

Cf. A092684 (similar recurrence); B-trees: A014535 (order 3), A037026 (order 4), A058521 (order 5).
Cf. A011973.
Cf. A000081, A000311, A001678, A048816, A079500, A120803, A316624, A320154, A320160, A320169.

nn=38;f[x_]:=Sum[a[n]x^n,{n,0,nn}];a[0]=0;sol=SolveAlways[0==Series[f[x]-x-f[x^2/(1-x)],{x,0,nn}],x];Table[a[n],{n,0,nn}]/.sol  (* Geoffrey Critzer, Mar 28 2013 *)

a(n)=if(n==0,0,if(n==1,1,sum(k=1,n\2,a(k)*binomial(n-k-1,n-2*k))))
for(n=1, 20, print1(a(n), ", "))

/* From: A(x) = x + A(x^2/(1-x)) */
{a(n)=local(A=x);for(i=1,n,A=x+subst(A,x,x^2/(1-x+x*O(x^n))));polcoeff(A,n)}
for(n=1, 20, print1(a(n), ", "))

/* From: x = Sum_{n>=1} a(n)*x^n*(1-x^n)/(1+x)^(n-1) */
a(n)=if(n==1, 1, -polcoeff(sum(k=1, n-1, a(k)*x^k*(1-x^k)/(1+x+x*O(x^n))^(k-1)), n))
for(n=1, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Jul 31 2013

G.f. A(x) satisfies: A(x) = x + A(x^2/(1-x)).
G.f.: Sum_{n>=0} x^(2^n)/D(n,x) where D(0,x)=1, D(n+1,x) = D(n,x)*[D(n,x) - x^(2^n)].
G.f.: x = Sum_{n>=1} a(n) * x^n * (1-x^n) / (1+x)^(n-1). - Paul D. Hanna, Jul 31 2013
Conjecture: Let M_n be an n X n matrix whose elements are m_ij = 0 for i < j - 1, m_ij = -1 for i = j - 1, and m_ij = binomial(i - j, n - i) otherwise. Then a(n + 1) = Det(M_n). - Benedict W. J. Irwin, Apr 19 2017

A292505 Number of complete orderless tree-factorizations of n >= 2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 9, 1, 1, 2, 3, 1, 4, 1, 12, 1, 1, 1, 12, 1, 1, 1, 9, 1, 4, 1, 3, 3, 1, 1, 29, 1, 3, 1, 3, 1, 9, 1, 9, 1, 1, 1, 17, 1, 1, 3, 33, 1, 4, 1, 3, 1, 4, 1, 44, 1, 1, 3, 3, 1, 4, 1, 29, 5, 1, 1, 17, 1

Offset: 2

Gus Wiseman, Sep 17 2017

nonn

An orderless tree-factorization (see A292504 for definition) is complete if all leaves are prime numbers. This sequence first differs from A281119 at a(64)=33.

a(n) depends only on the prime signature of n. - Andrew Howroyd, Nov 18 2018

			The a(60)=17 complete orderless tree-factorizations are: (2(2(35))), (2(3(25))), (2(5(23))), (2(235)), (3(2(25))), (3(5(22))), (3(225)), (5(2(23))), (5(3(22))), (5(223)), ((22)(35)), ((23)(25)), (22(35)), (23(25)), (25(23)), (35(22)), (2235).

Andrew Howroyd, Table of n, a(n) for n = 2..10000

Cf. A000311, A000669, A001055, A050336, A281119, A292504.

postfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[postfacs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
oltfacs[n_]:=If[n<=1,{{}},Prepend[Union@@Function[q,Sort/@Tuples[oltfacs/@q]]/@DeleteCases[postfacs[n],{n}],n]];
Table[Length[Select[oltfacs[n],FreeQ[#,_Integer?(!PrimeQ[#]&)]&]],{n,2,100}]

seq(n)={my(v=vector(n), w=vector(n)); v[1]=1; for(k=2, n, w[k]=v[k]+isprime(k); forstep(j=n\k*k, k, -k, my(i=j, e=0); while(i%k==0, i/=k; e++; v[j]+=binomial(e+w[k]-1, e)*v[i]))); w[2..n]} \\ Andrew Howroyd, Nov 18 2018

a(p^n) = A000669(n) for prime p. - Andrew Howroyd, Nov 18 2018

A320154 Number of series-reduced balanced rooted trees whose leaves form a set partition of {1,...,n}.

Original entry on oeis.org

1, 2, 5, 18, 92, 588, 4328, 35920, 338437, 3654751, 45105744, 625582147, 9539374171, 157031052142, 2757275781918, 51293875591794, 1007329489077804, 20840741773898303, 453654220906310222, 10380640686263467204, 249559854371799622350, 6301679967177242849680

Offset: 1

Gus Wiseman, Oct 06 2018

nonn

A rooted tree is series-reduced if every non-leaf node has at least two branches, and balanced if all leaves are the same distance from the root.

Also the number of balanced phylogenetic rooted trees on n distinct labels.

			The a(1) = 1 through a(4) = 18 rooted trees:
  (1)  (12)      (123)        (1234)
       ((1)(2))  ((1)(23))    ((1)(234))
                 ((2)(13))    ((12)(34))
                 ((3)(12))    ((13)(24))
                 ((1)(2)(3))  ((14)(23))
                              ((2)(134))
                              ((3)(124))
                              ((4)(123))
                              ((1)(2)(34))
                              ((1)(3)(24))
                              ((1)(4)(23))
                              ((2)(3)(14))
                              ((2)(4)(13))
                              ((3)(4)(12))
                              ((1)(2)(3)(4))
                              (((1)(2))((3)(4)))
                              (((1)(3))((2)(4)))
                              (((1)(4))((2)(3)))

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A000081, A000311, A000669, A001678, A005804, A048816, A079500, A119262, A120803, A141268, A244925, A292504, A300660, A319312.

Cf. A320155, A320160, A316624, A320169, A320173, A320176, A320179.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
gug[m_]:=Prepend[Join@@Table[Union[Sort/@Tuples[gug/@mtn]],{mtn,Select[sps[m],Length[#]>1&]}],m];
Table[Length[Select[gug[Range[n]],SameQ@@Length/@Position[#,_Integer]&]],{n,9}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
b(n,k)={my(u=vector(n), v=vector(n)); u[1]=k; u=EulerT(u); while(u, v+=u; u=EulerT(u)-u); v}
seq(n)={my(M=Mat(vectorv(n,k,b(n,k)))); vector(n, k, sum(i=1, k, binomial(k,i)*(-1)^(k-i)*M[i,k]))} \\ Andrew Howroyd, Oct 26 2018

Terms a(9) and beyond from Andrew Howroyd, Oct 26 2018

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).

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