A124324 -id:A124324 - cp's OEIS frontend

A000295 Eulerian numbers (Euler's triangle: column k=2 of A008292, column k=1 of A173018).

0, 0, 1, 4, 11, 26, 57, 120, 247, 502, 1013, 2036, 4083, 8178, 16369, 32752, 65519, 131054, 262125, 524268, 1048555, 2097130, 4194281, 8388584, 16777191, 33554406, 67108837, 134217700, 268435427, 536870882, 1073741793, 2147483616, 4294967263, 8589934558

Offset: 0

N. J. A. Sloane

There are 2 versions of Euler's triangle:
* A008292 Classic version of Euler's triangle used by Comtet (1974).
* A173018 Version of Euler's triangle used by Graham, Knuth and Patashnik in Concrete Math. (1990).
Euler's triangle rows and columns indexing conventions:
* A008292 The rows and columns of the Eulerian triangle are both indexed starting from 1. (Classic version: used in the classic books by Riordan and Comtet.)
* A173018 The rows and columns of the Eulerian triangle are both indexed starting from 0. (Graham et al.)
Number of Dyck paths of semilength n having exactly one long ascent (i.e., ascent of length at least two). Example: a(4)=11 because among the 14 Dyck paths of semilength 4, the paths that do not have exactly one long ascent are UDUDUDUD (no long ascent), UUDDUUDD and UUDUUDDD (two long ascents). Here U=(1,1) and D=(1,-1). Also number of ordered trees with n edges having exactly one branch node (i.e., vertex of outdegree at least two). - Emeric Deutsch, Feb 22 2004
Number of permutations of {1,2,...,n} with exactly one descent (i.e., permutations (p(1),p(2),...,p(n)) such that #{i: p(i)>p(i+1)}=1). E.g., a(3)=4 because the permutations of {1,2,3} with one descent are 132, 213, 231 and 312.
a(n+1) is the convolution of nonnegative integers (A001477) and powers of two (A000079). - Graeme McRae, Jun 07 2006
Partial sum of main diagonal of A125127. - Jonathan Vos Post, Nov 22 2006
Number of partitions of an n-set having exactly one block of size > 1. Example: a(4)=11 because, if the partitioned set is {1,2,3,4}, then we have 1234, 123|4, 124|3, 134|2, 1|234, 12|3|4, 13|2|4, 14|2|3, 1|23|4, 1|24|3 and 1|2|34. - Emeric Deutsch, Oct 28 2006
k divides a(k+1) for k in A014741. - Alexander Adamchuk, Nov 03 2006
(Number of permutations avoiding patterns 321, 2413, 3412, 21534) minus one. - Jean-Luc Baril, Nov 01 2007, Mar 21 2008
The chromatic invariant of the prism graph P_n for n >= 3. - Jonathan Vos Post, Aug 29 2008
Decimal integer corresponding to the result of XORing the binary representation of 2^n - 1 and the binary representation of n with leading zeros. This sequence and a few others are syntactically similar. For n > 0, let D(n) denote the decimal integer corresponding to the binary number having n consecutive 1's. Then D(n).OP.n represents the n-th term of a sequence when .OP. stands for a binary operator such as '+', '-', '*', 'quotentof', 'mod', 'choose'. We then get the various sequences A136556, A082495, A082482, A066524, A000295, A052944. Another syntactically similar sequence results when we take the n-th term as f(D(n)).OP.f(n). For example if f='factorial' and .OP.='/', we get (A136556)(A000295) ; if f='squaring' and .OP.='-', we get (A000295)(A052944). - K.V.Iyer, Mar 30 2009
Chromatic invariant of the prism graph Y_n.
Number of labelings of a full binary tree of height n-1, such that each path from root to any leaf contains each label from {1,2,...,n-1} exactly once. - Michael Vielhaber (vielhaber(AT)gmail.com), Nov 18 2009
Also number of nontrivial equivalence classes generated by the weak associative law X((YZ)T)=(X(YZ))T on words with n open and n closed parentheses. Also the number of join (resp. meet)-irreducible elements in the pruning-grafting lattice of binary trees with n leaves. - Jean Pallo, Jan 08 2010
Nonzero terms of this sequence can be found from the row sums of the third sub-triangle extracted from Pascal's triangle as indicated below by braces:
                     1;
                  1,    1;
              {1},   2,    1;
           {1,    3},   3,    1;
        {1,    4,    6},   4,    1;
     {1,    5,   10,   10},   5,    1;
  {1,    6,   15,   20,   15},   6,    1;
  ... - L. Edson Jeffery, Dec 28 2011
For integers a, b, denote by a<+>b the least c >= a, such that the Hamming distance D(a,c) = b (note that, generally speaking, a<+>b differs from b<+>a). Then for n >= 3, a(n) = n<+>n. This has a simple explanation: for n >= 3 in binary we have a(n) = (2^n-1)-n = "anti n". - Vladimir Shevelev, Feb 14 2012
a(n) is the number of binary sequences of length n having at least one pair 01. - Branko Curgus, May 23 2012
Nonzero terms are those integers k for which there exists a perfect (Hamming) error-correcting code. - L. Edson Jeffery, Nov 28 2012
a(n) is the number of length n binary words constructed in the following manner:  Select two positions in which to place the first two 0's of the word.  Fill in all (possibly none) of the positions before the second 0 with 1's and then complete the word with an arbitrary string of 0's or 1's. So a(n) = Sum_{k=2..n} (k-1)*2^(n-k). - Geoffrey Critzer, Dec 12 2013
Without first 0: a(n)/2^n equals Sum_{k=0..n} k/2^k. For example: a(5)=57, 57/32 = 0/1 + 1/2 + 2/4 + 3/8 + 4/16 + 5/32. - Bob Selcoe, Feb 25 2014
The first barycentric coordinate of the centroid of the first n rows of Pascal's triangle, assuming the numbers are weights, is A000295(n+1)/A000337(n). See attached figure. - César Eliud Lozada, Nov 14 2014
Starting (0, 1, 4, 11, ...), this is the binomial transform of (0, 1, 2, 2, 2, ...). - Gary W. Adamson, Jul 27 2015
Also the number of (non-null) connected induced subgraphs in the n-triangular honeycomb rook graph. - Eric W. Weisstein, Aug 27 2017
a(n) is the number of swaps needed in the worst case to transform a binary tree with n full levels into a heap, using (bottom-up) heapify. - Rudy van Vliet, Sep 19 2017
The utility of large networks, particularly social networks, with n participants is given by the terms a(n) of this sequence. This assertion is known as Reed's Law, see the Wikipedia link. - Johannes W. Meijer, Jun 03 2019
a(n-1) is the number of subsets of {1..n} in which the largest element of the set exceeds by at least 2 the next largest element. For example, for n = 5, a(4) = 11 and the 11 sets are {1,3}, {1,4}, {1,5}, {2,4}, {2,5}, {3,5}, {1,2,4}, {1,2,5}, {1,3,5}, {2,3,5}, {1,2,3,5}. - Enrique Navarrete, Apr 08 2020
a(n-1) is also the number of subsets of {1..n} in which the second smallest element of the set exceeds by at least 2 the smallest element. For example, for n = 5, a(4) = 11 and the 11 sets are {1,3}, {1,4}, {1,5}, {2,4}, {2,5}, {3,5}, {1,3,4}, {1,3,5}, {1,4,5}, {2,4,5}, {1,3,4,5}. - Enrique Navarrete, Apr 09 2020
a(n+1) is the sum of the smallest elements of all subsets of {1..n}. For example, for n=3, a(4)=11; the subsets of {1,2,3} are {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}, and the sum of smallest elements is 11. - Enrique Navarrete, Aug 20 2020
Number of subsets of an n-set that have more than one element. - Eric M. Schmidt, Mar 13 2021
Number of individual bets in a "full cover" bet on n-1 horses, dogs, etc. in different races. Each horse, etc. can be bet on or not, giving 2^n bets.  But, by convention, singles (a bet on only one race) are not included, reducing the total number bets by n.  It is also impossible to bet on no horses at all, reducing the number of bets by another 1.  A full cover on 4 horses, dogs, etc. is therefore 6 doubles, 4 trebles and 1 four-horse etc. accumulator.  In British betting, such a bet on 4 horses etc. is a Yankee; on 5, a super-Yankee. - Paul Duckett, Nov 17 2021
From Enrique Navarrete, May 25 2022: (Start)
Number of binary sequences of length n with at least two 1's.
a(n-1) is the number of ways to choose an odd number of elements greater than or equal to 3 out of n elements.
a(n+1) is the number of ways to split [n] = {1,2,...,n} into two (possibly empty) complementary intervals {1,2,...,i} and {i+1,i+2,...,n} and then select a subset from the first interval (2^i choices, 0 <= i <= n), and one block/cell (i.e., subinterval) from the second interval (n-i choices, 0 <= i <= n).
(End)
Number of possible conjunctions in a system of n planets; for example, there can be 0 conjunctions with one planet, one with two planets, four with three planets (three pairs of planets plus one with all three) and so on. - Wendy Appleby, Jan 02 2023
Largest exponent m such that 2^m divides (2^n-1)!. - Franz Vrabec, Aug 18 2023
It seems that a(n-1) is the number of odd r with 0 < r < 2^n for which there exist u,v,w in the x-independent beginning of the Collatz trajectory of 2^n x + r with u+v = w+1, as detailed in the link "Collatz iteration and Euler numbers?". A better understanding of this might also give a formula for A374527. - Markus Sigg, Aug 02 2024
This sequence has a connection to consecutively halved positional voting (CHPV); see Mendenhall and Switkay. - Hal M. Switkay, Feb 25 2025
a(n) is the number of subsets of size 2 and more of an n-element set. Equivalently, a(n) is the number of (hyper)edges of size 2 and more in a complete hypergraph of n vertices. - Yigit Oktar, Apr 05 2025

			G.f. = x^2 + 4*x^3 + 11*x^4 + 26*x^5 + 57*x^6 + 120*x^7 + 247*x^8 + 502*x^9 + ...

O. Bottema, Problem #562, Nieuw Archief voor Wiskunde, 28 (1980) 115.
L. Comtet, "Permutations by Number of Rises; Eulerian Numbers." Section 6.5 in Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, pp. 51 and 240-246, 1974.
F. N. David and D. E. Barton, Combinatorial Chance. Hafner, NY, 1962, p. 151.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990.
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 3, p. 34.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 215.
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..500
Joerg Arndt and N. J. A. Sloane, Counting Words that are in "Standard Order"
Dillan Agrawal, Selena Ge, Jate Greene, Tanya Khovanova, Dohun Kim, Rajarshi Mandal, Tanish Parida, Anirudh Pulugurtha, Gordon Redwine, Soham Samanta, and Albert Xu, Chip-Firing on Infinite k-ary Trees, arXiv:2501.06675 [math.CO], 2025. See pp. 9, 17, 18.
E. Banaian, S. Butler, C. Cox, J. Davis, J. Landgraf and S. Ponce A generalization of Eulerian numbers via rook placements, arXiv:1508.03673 [math.CO], 2015.
Jean-Luc Baril and J. M. Pallo, The pruning-grafting lattice of binary trees, Theoretical Computer Science, 409, 2008, 382-393.
Jean-Luc Baril and José L. Ramírez, Some distributions in increasing and flattened permutations, arXiv:2410.15434 [math.CO], 2024. See p. 7.
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Peter J. Cameron, Maximilien Gadouleau, James D. Mitchell, and Yann Peresse, Chains of subsemigroups, arXiv preprint arXiv:1501.06394 [math.GR], 2015. See Table 4.
Matteo Cervetti and Luca Ferrari, Pattern avoidance in the matching pattern poset, arXiv:2009.01024 [math.CO], 2020.
Shelby Cox, Pratik Misra, and Pardis Semnani, Homaloidal Polynomials and Gaussian Models of Maximum Likelihood Degree One, arXiv:2402.06090 [math.AG], 2024.
Benjamin Hellouin de Menibus and Yvan Le Borgne, Asymptotic behaviour of the one-dimensional "rock-paper-scissors" cyclic cellular automaton, arXiv:1903.12622 [math.PR], 2019.
Karl Dilcher and Maciej Ulas, Arithmetic properties of polynomial solutions of the Diophantine equation P(x)x^(n+1)+Q(x)(x+1)^(n+1) = 1, arXiv:1909.11222 [math.NT], 2019.
Filippo Disanto, Some Statistics on the Hypercubes of Catalan Permutations, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.2.
J. M. Dusel, Balanced parabolic quotients and branching rules for Demazure crystals, J Algebr Comb (2016) 44: 363. DOI: 10.1007/s10801-016-0673-y.
Pascal Floquet, Serge Domenech and Luc Pibouleau, Combinatorics of Sharp Separation System synthesis : Generating functions and Search Efficiency Criterion, Industrial Engineering and Chemistry Research, 33, pp. 440-443, 1994.
Pascal Floquet, Serge Domenech, Luc Pibouleau and Said Aly, Some Complements in Combinatorics of Sharp Separation System Synthesis, American Institute of Chemical Engineering Journal, 39(6), pp. 975-978, 1993.
E. T. Frankel, A calculus of figurate numbers and finite differences, American Mathematical Monthly, 57 (1950), 14-25. [Annotated scanned copy]
Joël Gay, Representation of Monoids and Lattice Structures in the Combinatorics of Weyl Groups, Doctoral Thesis, Discrete Mathematics [cs.DM], Université Paris-Saclay, 2018.
R. K. Guy, Letter to N. J. A. Sloane.
Ryota Inagaki, Tanya Khovanova, and Austin Luo, On Chip-Firing on Undirected Binary Trees, Ann. Comb. (2025). See pp. 24-25.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 388.
Wayne A. Johnson, An Euler operator approach to Ehrhart series, arXiv:2303.16991 [math.CO], 2023.
Lucas Kang, Investigation of Rule 73 as Case Study of Class 4 Long-Distance Cellular Automata, arXiv preprint arXiv:1310.3311 [nlin.CG], 2013.
Oliver Kullmann and Xishun Zhao, Bounds for variables with few occurrences in conjunctive normal forms, arXiv preprint arXiv:1408.0629 [math.CO], 2014.
César Eliud Lozada, Centroids of Pascal triangles
Candice A. Marshall, Construction of Pseudo-Involutions in the Riordan Group, Dissertation, Morgan State University, 2017.
Peter Charles Mendenhall and Hal M. Switkay, Consecutively Halved Positional Voting: A Special Case of Geometric Voting, Social Sciences vol. 12 no. 2 (2023), 47-59.
J. C. P. Miller, Letter to N. J. A. Sloane, Mar 26 1971
J. W. Moon, A problem on arcs without bypasses in tournaments, J. Combinatorial Theory Ser. B 21 (1976), no. 1, 71--75. MR0427129(55 #165).
Agustín Moreno Cañadas, Hernán Giraldo, and Gabriel Bravo Rios, On the Number of Sections in the Auslander-Reiten Quiver of Algebras of Dynkin Type, Far East Journal of Mathematical Sciences (FJMS), Vol. 101, No. 8 (2017), pp. 1631-1654.
Nagatomo Nakamura, Pseudo-Normal Random Number Generation via the Eulerian Numbers, Josai Mathematical Monographs, vol 8, p 85-95, 2015.
Emily Norton, Symplectic Reflection Algebras in Positive Characteristic as Ore Extensions, arXiv preprint arXiv:1302.5411 [math.RA], 2013.
Ronald Orozco López, Solution of the Differential Equation y^(k)= e^(a*y), Special Values of Bell Polynomials and (k,a)-Autonomous Coefficients, Universidad de los Andes (Colombia 2021).
J. M. Pallo, Weak associativity and restricted rotation, Information Processing Letters, 109, 2009, 514-517.
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260.
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260. [Annotated scanned copy]
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
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D. P. Roselle, Permutations by number of rises and successions, Proc. Amer. Math. Soc., 19 (1968), 8-16. [Annotated scanned copy]
Markus Sigg, Collatz iteration and Euler numbers?
Eric Weisstein's World of Mathematics, Chromatic Invariant
Eric Weisstein's World of Mathematics, Prism Graph
Wikipedia, Reed's Law
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Anssi Yli-Jyra, On Dependency Analysis via Contractions and Weighted FSTs, in Shall We Play the Festschrift Game?, Springer, 2012, pp. 133-158. - _N. J. A. Sloane_, Dec 25 2012
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A008292 (classic version of Euler's triangle used by Comtet (1974)).
Cf. A173018 (version of Euler's triangle used by Graham, Knuth and Patashnik in Concrete Math. (1990)).
Cf. A002662 (partial sums).
Partial sums of A000225.
Row sums of A014473 and of A143291.
Second column of triangles A112493 and A112500.
Sequences A125128 and A130103 are essentially the same.
Column k=1 of A124324.
Cf. A000079, A000108, A000217, A000325, A000975, A001511, A002663.
Cf. A002664, A007814, A008292, A008949, A014741, A016031, A028366.
Cf. A035039, A035040, A035041, A035042, A079583, A107907, A130321.
Cf. A130128, A130330, A131768, A131816.

a000295 n = 2^n - n - 1  -- Reinhard Zumkeller, Nov 25 2013

[2^n-n-1: n in [0..40]]; // Vincenzo Librandi, Jul 29 2015

[EulerianNumber(n, 1): n in [0..40]]; // G. C. Greubel, Oct 02 2024

[ seq(2^n-n-1, n=1..50) ];
A000295 := -z/(2*z-1)/(z-1)**2; # Simon Plouffe in his 1992 dissertation
# Grammar specification:
spec := [S, { B = Set(Z, 1 <= card), C = Sequence(B, 2 <= card), S = Prod(B, C) }, unlabeled]:
struct := n -> combstruct[count](spec, size = n+1);
seq(struct(n), n = 0..33); # Peter Luschny, Jul 22 2014

a[n_] = If[n==0, 0, n*(HypergeometricPFQ[{1, 1-n}, {2}, -1] - 1)];
Table[a[n], {n,0,40}] (* Olivier Gérard, Mar 29 2011 *)
LinearRecurrence[{4, -5, 2}, {0, 0, 1}, 40] (* Vincenzo Librandi, Jul 29 2015 *)
Table[2^n -n-1, {n,0,40}] (* Eric W. Weisstein, Nov 16 2017 *)

a(n)=2^n-n-1 \\ Charles R Greathouse IV, Jun 10 2011

[2^n -(n+1) for n in range(41)] # G. C. Greubel, Oct 02 2024

a(n) = 2^n - n - 1.
G.f.: x^2/((1-2*x)*(1-x)^2).
A107907(a(n+2)) = A000079(n+2). - Reinhard Zumkeller, May 28 2005
E.g.f.: exp(x)*(exp(x)-1-x). - Emeric Deutsch, Oct 28 2006
a(0)=0, a(1)=0, a(n) = 3*a(n-1) - 2*a(n-2) + 1. - Miklos Kristof, Mar 09 2005
a(0)=0, a(n) = 2*a(n-1) + n - 1 for all n in Z.
a(n) = Sum_{k=2..n} binomial(n, k). - Paul Barry, Jun 05 2003
a(n+1) = Sum_{i=1..n} Sum_{j=1..i} C(i, j). - Benoit Cloitre, Sep 07 2003
a(n+1) = 2^n*Sum_{k=0..n} k/2^k. - Benoit Cloitre, Oct 26 2003
a(0)=0, a(1)=0, a(n) = Sum_{i=0..n-1} i+a(i) for i > 1. - Gerald McGarvey, Jun 12 2004
a(n+1) = Sum_{k=0..n} (n-k)*2^k. - Paul Barry, Jul 29 2004
a(n) = Sum_{k=0..n} binomial(n, k+2); a(n+2) = Sum_{k=0..n} binomial(n+2, k+2). - Paul Barry, Aug 23 2004
a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-k-1, k+1)*2^(n-k-2)*(-1/2)^k. - Paul Barry, Oct 25 2004
a(0) = 0; a(n) = Stirling2(n,2) + a(n-1) = A000225(n-1) + a(n-1). - Thomas Wieder, Feb 18 2007
a(n) = A000325(n) - 1. - Jonathan Vos Post, Aug 29 2008
a(0) = 0, a(n) = Sum_{k=0..n-1} 2^k - 1. - Doug Bell, Jan 19 2009
a(n) = A000217(n-1) + A002662(n) for n>0. - Geoffrey Critzer, Feb 11 2009
a(n) = A000225(n) - n. - Zerinvary Lajos, May 29 2009
a(n) = n*(2F1([1,1-n],[2],-1) - 1). - Olivier Gérard, Mar 29 2011
Column k=1 of A173018 starts a'(n) = 0, 1, 4, 11, ... and has the hypergeometric representation n*hypergeom([1, -n+1], [-n], 2). This can be seen as a formal argument to prefer Euler's A173018 over A008292. - Peter Luschny, Sep 19 2014
E.g.f.: exp(x)*(exp(x)-1-x); this is U(0) where U(k) = 1 - x/(2^k - 2^k/(x + 1 - x^2*2^(k+1)/(x*2^(k+1) - (k+1)/U(k+1)))); (continued fraction, 3rd kind, 4-step). - Sergei N. Gladkovskii, Dec 01 2012
a(n) = A079583(n) - A000225(n+1). - Miquel Cerda, Dec 25 2016
a(0) = 0; a(1) = 0; for n > 1: a(n) = Sum_{i=1..2^(n-1)-1} A001511(i). - David Siegers, Feb 26 2019
a(n) = A007814(A028366(n)). - Franz Vrabec, Aug 18 2023
a(n) = Sum_{k=1..floor((n+1)/2)} binomial(n+1, 2*k+1). - Taras Goy, Jan 02 2025

A124323 Triangle read by rows: T(n,k) is the number of partitions of an n-set having k singleton blocks (0<=k<=n).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 3, 0, 1, 4, 4, 6, 0, 1, 11, 20, 10, 10, 0, 1, 41, 66, 60, 20, 15, 0, 1, 162, 287, 231, 140, 35, 21, 0, 1, 715, 1296, 1148, 616, 280, 56, 28, 0, 1, 3425, 6435, 5832, 3444, 1386, 504, 84, 36, 0, 1, 17722, 34250, 32175, 19440, 8610, 2772, 840, 120, 45, 0, 1

Offset: 0

Emeric Deutsch, Oct 28 2006

Row sums are the Bell numbers (A000110). T(n,0)=A000296(n). T(n,k) = binomial(n,k)*T(n-k,0). Sum(k*T(n,k),k=0..n) = A052889(n) = n*B(n-1), where B(q) are the Bell numbers (A000110).
Exponential Riordan array [exp(exp(x)-1-x),x]. - Paul Barry, Apr 23 2009
Sum_{k=0..n} T(n,k)*2^k = A000110(n+1) is the number of binary relations on an n-set that are both symmetric and transitive. - Geoffrey Critzer, Jul 25 2014
Also the number of set partitions of {1, ..., n} with k cyclical adjacencies (successive elements in the same block, where 1 is a successor of n). Unlike A250104, we count {{1}} as having 1 cyclical adjacency. - Gus Wiseman, Feb 13 2019

			T(4,2)=6 because we have 12|3|4, 13|2|4, 14|2|3, 1|23|4, 1|24|3 and 1|2|34 (if we take {1,2,3,4} as our 4-set).
Triangle starts:
     1
     0    1
     1    0    1
     1    3    0    1
     4    4    6    0    1
    11   20   10   10    0    1
    41   66   60   20   15    0    1
   162  287  231  140   35   21    0    1
   715 1296 1148  616  280   56   28    0    1
  3425 6435 5832 3444 1386  504   84   36    0    1
From _Gus Wiseman_, Feb 13 2019: (Start)
Row n = 5 counts the following set partitions by number of singletons:
  {{1234}}    {{1}{234}}  {{1}{2}{34}}  {{1}{2}{3}{4}}
  {{12}{34}}  {{123}{4}}  {{1}{23}{4}}
  {{13}{24}}  {{124}{3}}  {{12}{3}{4}}
  {{14}{23}}  {{134}{2}}  {{1}{24}{3}}
                          {{13}{2}{4}}
                          {{14}{2}{3}}
... and the following set partitions by number of cyclical adjacencies:
  {{13}{24}}      {{1}{2}{34}}  {{1}{234}}  {{1234}}
  {{1}{24}{3}}    {{1}{23}{4}}  {{12}{34}}
  {{13}{2}{4}}    {{12}{3}{4}}  {{123}{4}}
  {{1}{2}{3}{4}}  {{14}{2}{3}}  {{124}{3}}
                                {{134}{2}}
                                {{14}{23}}
(End)
From _Paul Barry_, Apr 23 2009: (Start)
Production matrix is
0, 1,
1, 0, 1,
1, 2, 0, 1,
1, 3, 3, 0, 1,
1, 4, 6, 4, 0, 1,
1, 5, 10, 10, 5, 0, 1,
1, 6, 15, 20, 15, 6, 0, 1,
1, 7, 21, 35, 35, 21, 7, 0, 1,
1, 8, 28, 56, 70, 56, 28, 8, 0, 1 (End)

Alois P. Heinz, Rows n = 0..140, flattened
David Callan, On conjugates for set partitions and integer compositions, arXiv:math/0508052 [math.CO], 2005.
T. Mansour, A. O. Munagi, Set partitions with circular successions, European Journal of Combinatorics, 42 (2014), 207-216.

Cf. A000110, A052889, A124324.
A250104 is an essentially identical triangle, differing only in row 1.
For columns see A000296, A250105, A250106, A250107.
Cf. A000126, A001610, A032032, A052841, A066982, A080107, A169985, A187784, A324011, A324014, A324015.

G:=exp(exp(z)-1+(t-1)*z): Gser:=simplify(series(G,z=0,14)): for n from 0 to 11 do P[n]:=sort(n!*coeff(Gser,z,n)) od: for n from 0 to 11 do seq(coeff(P[n],t,k),k=0..n) od; # yields sequence in triangular form
# Program from R. J. Mathar, Jan 22 2015:
A124323 := proc(n,k)
    binomial(n,k)*A000296(n-k) ;
end proc:

Flatten[CoefficientList[Range[0,10]! CoefficientList[Series[Exp[x y] Exp[Exp[x] - x - 1], {x, 0,10}], x], y]] (* Geoffrey Critzer, Nov 24 2011 *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],Count[#,{}]==k&]],{n,0,9},{k,0,n}] (* _Gus Wiseman, Feb 13 2019 *)

T(n,k) = binomial(n,k)*[(-1)^(n-k)+sum((-1)^(j-1)*B(n-k-j), j=1..n-k)], where B(q) are the Bell numbers (A000110).
E.g.f.: G(t,z) = exp(exp(z)-1+(t-1)*z).
G.f.: 1/(1-xy-x^2/(1-xy-x-2x^2/(1-xy-2x-3x^2/(1-xy-3x-4x^2/(1-... (continued fraction). - Paul Barry, Apr 23 2009

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

Original entry on oeis.org

1, 1, 1, 1, 4, 3, 1, 11, 25, 15, 1, 26, 130, 210, 105, 1, 57, 546, 1750, 2205, 945, 1, 120, 2037, 11368, 26775, 27720, 10395, 1, 247, 7071, 63805, 247555, 460845, 405405, 135135, 1, 502, 23436, 325930, 1939630, 5735730, 8828820, 6756750, 2027025, 1

Offset: 0

Wolfdieter Lang, Oct 14 2005

Previous name was: Coefficient triangle of polynomials used for e.g.f.s of Stirling2 diagonals.
For the o.g.f. of diagonal k of the Stirling2 triangle one has a similar result. See A008517 (second-order Eulerian triangle).
A(m,x), the o.g.f. for column m, satisfies the recurrence A(m,x) = x*(x*(d/dx)A(m-1,x) + m*A(m-1,x))/(1-(m+1)*x), for m >= 1 and A(0,x) = 1/(1-x).
The e.g.f. for the sequence in column k+1, k >= 0, of A008278, i.e., for the diagonal k >= 0 of the Stirling2 triangle A048993, is exp(x)*Sum_{m=0..k} a(k,m)*(x^(m+k))/(m+k)!.
It appears that the triangles in this sequence and A124324 have identical columns, except for shifts. - Jörgen Backelin, Jun 20 2022
A refined version of this triangle is given in A356145, which contains a link providing the precise relationship between A124324 and this entry, confirming Jörgen Backelin's observation above. - Tom Copeland, Sep 24 2022

			Triangle starts:
  [1]
  [1, 1]
  [1, 4,  3]
  [1, 11, 25,  15]
  [1, 26, 130, 210,  105]
  [1, 57, 546, 1750, 2205, 945]
  ...
The e.g.f. of [0,0,1,7,25,65,...], the k=3 column of A008278, but with offset n=0, is exp(x)*(1*(x^2)/2! + 4*(x^3)/3! + 3*(x^4)/4!).
Third row [1,4,3]: There are three plane increasing trees on 3 vertices. The number of colors are shown to the right of a vertex.
...................................................
....1o.(1+t)...........1o.t*(1+t).....1o.t*(1+t)...
....|................. /.\............/.\..........
....|................ /...\........../...\.........
....2o.(1+t)........2o.....3o......3o....2o........
....|..............................................
....|..............................................
....3o.............................................
...................................................
The total number of trees is (1+t)^2 + t*(1+t) + t*(1+t) = 1+4*t+3*t^2 = R(2,t).

F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1992, pp. 24-48.
D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions. arXiv:math/0501052v2 [math.CA], 2005.
Wolfdieter Lang, First ten rows.
MathOverflow, Recursion for row polynomials of A112493, (2025).
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.

Row sums give A006351(k+1), k>=0.
The column sequences start with A000012 (powers of 1), A000295 (Eulerian numbers), A112495, A112496, A112497.
Cf. A008278, A008517, A048993, A054589, A075856, A134991, A201637, A124324.
Antidiagonal sums give A000110.
Cf. A356145.

T := (n, k) -> add(combinat:-eulerian2(n, j)*binomial(n-j, n-k), j=0..n):
seq(seq(T(n, k), k=0..n), n=0..9); # Peter Luschny, Apr 11 2016

max = 11; f[x_, t_] := -1 - (1 + t)/t*ProductLog[-t/(1 + t)*Exp[(x - t)/(1 + t)]]; coes = CoefficientList[ Series[f[x, t], {x, 0, max}, {t, 0, max}], {x, t}]* Range[0, max]!; Table[coes[[n, k]], {n, 0, max}, {k, 1, n - 1}] // Flatten (* Jean-François Alcover, Nov 22 2012, from e.g.f. *)

a(k, m) = 0 if k < m, a(k, -1):=0, a(0, 0)=1, a(k, m)=(m+1)*a(k-1, m) + (k+m-1)*a(k-1, m-1) else.
From Peter Bala, Sep 30 2011: (Start)
E.g.f.: A(x,t) = -1-((1+t)/t)*LambertW(-(t/(1+t))*exp((x-t)/(1+t))) = x + (1+t)*x^2/2! + (1+4*t+3*t^2)*x^3/3! + .... A(x,t) is the inverse function of (1+t)*log(1+x)-t*x.
A(x,t) satisfies the partial differential equation (1-x*t)*dA/dx = 1 + A + t*(1+t)*dA/dt. It follows that the row generating polynomials R(n,t) satisfy the recurrence R(n+1,t) =(n*t+1)*R(n,t) + t*(1+t)*dR(n,t)/dt. Cf. A054589 and A075856. The polynomials t/(1+t)*R(n,t) are the row polynomials of A134991.
The generating function A(x,t) satisfies the autonomous differential equation dA/dx = (1+A)/(1-t*A). Applying [Bergeron et al., Theorem 1] gives a combinatorial interpretation for the row generating polynomials R(n,t): R(n,t) counts plane increasing trees on n+1 vertices where the non-leaf vertices of outdegree k come in t^(k-1)*(1+t) colors. An example is given below. Cf. A006351, which corresponds to the case t = 1. Applying [Dominici, Theorem 4.1] gives the following method for calculating the row polynomials R(n,t): Let f(x) = (1+x)/(1-x*t). Then R(n,t) = (f(x)*d/dx)^n(f(x)) evaluated at x = 0. (End)
Sum_{j=0..n} T(n-j,j) = A000110(n). - Alois P. Heinz, Jun 20 2022
From Mikhail Kurkov, Apr 01 2025: (Start)
E.g.f.: B(y) = -w/(x*(1+w)) where w = LambertW(-x/(1+x)*exp((y-x)/(1+x))) satisfies the first-order ordinary differential equation (1+x)*B'(y) = B(y)*(1+x*B(y))^2, hence row polynomials are P(n,x) = P(n-1,x) + x*Sum_{j=0..n-1} binomial(n, j)*P(j,x)*P(n-j-1,x) for n > 0 with P(0,x) = 1 (see MathOverflow link).
Conjecture: row polynomials are P(n,x) = Sum_{i=0..n} Sum_{j=0..i} Sum_{k=0..j} (n+i)!*Stirling1(n+j-k,j-k)*x^k*(x+1)^(j-k)*(-1)^(j+k)/((n+j-k)!*(i-j)!*k!). (End)
Conjecture: g.f. satisfies 1/(1 - x - x*y/(1 - 2*x - 2*x*y/(1 - 3*x - 3*x*y/(1 - 4*x - 4*x*y/(1 - 5*x - 5*x*y/(1 - ...)))))) (see A383019 for conjectures about combinatorial interpretation and algorithm for efficient computing). - Mikhail Kurkov, Apr 21 2025

New name from Peter Luschny, Apr 11 2016

A136394 Triangle read by rows: T(n,k) is the number of permutations of an n-set having k cycles of size > 1 (0<=k<=floor(n/2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 1, 20, 3, 1, 84, 35, 1, 409, 295, 15, 1, 2365, 2359, 315, 1, 16064, 19670, 4480, 105, 1, 125664, 177078, 56672, 3465, 1, 1112073, 1738326, 703430, 74025, 945, 1, 10976173, 18607446, 8941790, 1346345, 45045, 1, 119481284, 216400569, 118685336

Offset: 0

Vladeta Jovovic, May 03 2008

			Triangle (n,k) begins:
  1;
  1;
  1,    1;
  1,    5;
  1,   20,    3;
  1,   84,   35;
  1,  409,  295,  15;
  1, 2365, 2359, 315;
  ...

Alois P. Heinz, Rows n = 0..200, flattened
Jean-Luc Baril and Sergey Kirgizov, Transformation à la Foata for special kinds of descents and excedances, arXiv:2101.01928 [math.CO], 2021. See Theorem 2. p. 5.
FindStat - Combinatorial Statistic Finder, The number of nontrivial cycles of a permutation pi in its cycle decomposition
Bin Han, Jianxi Mao, and Jiang Zeng, Equidistributions around special kinds of descents and excedances, arXiv:2103.13092 [math.CO], 2021, see page 2.

Columns k=0-10 give: A000012, A006231, A289950, A289951, A289952, A289953, A289954, A289955, A289956, A289957, A289958.
Row sums give A000142.
Cf. A000276, A000483, A001147, A001705, A051577, A124324, A159964.

egf:= proc(k::nonnegint) option remember; x-> exp(x)* ((-x-ln(1-x))^k)/k! end; T:= (n,k)-> coeff(series(egf(k)(x), x=0, n+1), x, n) *n!; seq(seq(T(n,k), k=0..n/2), n=0..30); # Alois P. Heinz, Aug 14 2008
# second Maple program:
b:= proc(n) option remember; expand(`if`(n=0, 1, add(b(n-i)*
      `if`(i>1, x, 1)*binomial(n-1, i-1)*(i-1)!, i=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):
seq(T(n), n=0..15);  # Alois P. Heinz, Sep 25 2016
# third Maple program:
T:= proc(n, k) option remember; `if`(k<0 or k>2*n, 0,
      `if`(n=0, 1, add(T(n-i, k-`if`(i>1, 1, 0))*
       mul(n-j, j=1..i-1), i=1..n)))
    end:
seq(seq(T(n,k), k=0..n/2), n=0..15);  # Alois P. Heinz, Jul 16 2017

max = 12; egf = Exp[x*(1-y)]/(1-x)^y; s = Series[egf, {x, 0, max}, {y, 0, max}] // Normal; t[n_, k_] := SeriesCoefficient[s, {x, 0, n}, {y, 0, k}]*n!; t[0, 0] = t[1, 0] = 1; Table[t[n, k], {n, 0, max}, {k, 0, n/2}] // Flatten (* Jean-François Alcover, Jan 28 2014 *)

E.g.f.: exp(x*(1-y))/(1-x)^y. Binomial transform of triangle A008306. exp(x)*((-x-log(1-x))^k)/k! is e.g.f. of k-th column.
From Alois P. Heinz, Jul 13 2017: (Start)
T(2n,n) = A001147(n).
T(2n+1,n) = A051577(n) = (2*n+3)!!/3 = A001147(n+2)/3. (End)
From Alois P. Heinz, Aug 17 2023: (Start)
Sum_{k=0..floor(n/2)} k * T(n,k) = A001705(n-1) for n>=1.
Sum_{k=0..floor(n/2)} (-1)^k * T(n,k) = A159964(n-1) for n>=1. (End)

A112495 Third column of triangle A112493 used for e.g.f.s of Stirling2 diagonals.

Original entry on oeis.org

3, 25, 130, 546, 2037, 7071, 23436, 75328, 237127, 735813, 2260518, 6896046, 20933673, 63325051, 191088976, 575625900, 1731858075, 5206059585, 15640198410, 46966732090, 140996664733, 423191320215, 1269993390420

Offset: 0

Wolfdieter Lang, Oct 14 2005

2*a(n-4) is the number of ternary words of length n where two of the letters are used at least twice. For example, for n=5 the 50 words that use 0 and 1 at least twice are 00011 (10 of this type), 00111 (10 of this type) and 00112 (30 of this type). - Enrique Navarrete, Feb 14 2025

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-40,82,-91,52,-12).

Cf. A000295 (second column).

Column k=2 of A124324 (shifted).

CoefficientList[Series[(3 - 5*x)/(((1 - x)^3)*((1 - 2*x)^2)*(1 - 3*x)), {x, 0, 50}], x] (* G. C. Greubel, Nov 13 2017 *)
Table[3^(n+4)/2 - (n+6)*2^(n+3) + n^2/2 + 9*n/2 + 21/2, {n,0,25}] (* Vaclav Kotesovec, Jul 23 2021 *)

x='x+O('x^50); Vec((3-5*x)/(((1-x)^3)*((1-2*x)^2)*(1-3*x))) \\ G. C. Greubel, Nov 13 2017

a(n) = 3*a(n-1)+ (n+3)*(2^(n+2)-(n+3)), n>=1, a(0)=3.
G.f.: (3-5*x)/(((1-x)^3)*((1-2*x)^2)*(1-3*x)).
a(n) = 3^(n+4)/2 - (n+6)*2^(n+3) + n^2/2 + 9*n/2 + 21/2. - Vaclav Kotesovec, Jul 23 2021
E.g.f.: (1/2)*exp(x)*(exp(x)-x-1)^2 (with offset 4). - Enrique Navarrete, Feb 14 2025

A124325 Number of blocks of size >1 in all partitions of an n-set.

Original entry on oeis.org

0, 0, 1, 4, 17, 76, 362, 1842, 9991, 57568, 351125, 2259302, 15288000, 108478124, 805037105, 6233693772, 50257390937, 421049519856, 3659097742426, 32931956713294, 306490813820239, 2945638599347760, 29198154161188501

Offset: 0

Emeric Deutsch, Oct 28 2006

nonn

Sum of the first entries in all blocks of all set partitions of [n-1]. a(4) = 17 because the sum of the first entries in all blocks of all set partitions of [3] (123, 12|3, 13|2, 1|23, 1|2|3) is 1+4+3+3+6 = 17. - Alois P. Heinz, Apr 24 2017

			a(3) = 4 because in the partitions 123, 12|3, 13|2, 1|23, 1|2|3 we have four blocks of size >1.

Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. A000110, A123324, A124327, A285595.

Column k=2 of A283424.

with(combinat): c:=n->bell(n+1)-bell(n)-n*bell(n-1): seq(c(n),n=0..23);

nn=22;Range[0,nn]!CoefficientList[Series[(Exp[x]-1-x)Exp[Exp[x]-1],{x,0,nn}],x]  (* Geoffrey Critzer, Mar 28 2013 *)

N = 66;  x = 'x + O('x^N);
egf = (exp(x)-1-x)*exp(exp(x)-1) + 'c0;
gf = serlaplace(egf);
v = Vec(gf);  v[1]-='c0;  v
/* Joerg Arndt, Mar 29 2013 */

a(n) = B(n+1)-B(n)-n*B(n-1), where B(q) are the Bell numbers (A000110).
E.g.f.: (exp(z)-1-z)*exp(exp(z)-1).
a(n) = Sum_{k=0..floor(n/2)} k*A124324(n,k).
a(n) = A285595(n-1,1). - Alois P. Heinz, Apr 24 2017
a(n) = Sum_{k=1..n*(n-1)/2} k * A124327(n-1,k) for n>1. - Alois P. Heinz, Dec 05 2023

A356145 Coefficients of the inverse refined Eulerian partition polynomials [E]^{-1}, partitional inverse to A145271. Irregular triangle read by row with lengths A000041.

Original entry on oeis.org

1, 1, -1, 1, 3, -4, 1, -15, 25, -4, -7, 1, 105, -210, 70, 60, -15, -11, 1, -945, 2205, -1120, -630, 70, 350, 126, -15, -26, -16, 1, 10395, -27720, 18900, 7875, -2800, -6930, -1638, 560, 455, 784, 238, -56, -42, -22, 1, -135135, 405405, -346500, -114345, 84700

Offset: 0

Tom Copeland, Jul 27 2022

These are the coefficients of the inverse refined Eulerian partitions polynomials, the substitutional inverse to the refined Eulerian partition polynomials [E] of A145271. [E] and [E]^{-1} are a conjugate dual with respect to the permutahedra polynomials [P] of A133314 (see formula section).

			The first few rows of coefficients with monomials in reverse order to the partitions of Abramowitz and Stegun (link in A000041, pp. 831-2) are
0)       1;
1)       1;
2)      -1,      1;
3)       3,     -4,       1;
4)     -15,     25,      -4,      -7,     1;
5)     105,   -210,      70,      60,   -15,    -11,     1;
6)    -945,   2205,   -1120,    -630,    70,    350,   126,   -15,    -26,    -16,      1;
7)   10395, -27720,   18900,    7875, -2800,  -6930, -1638,   560,    455,    784,    238,   -56,  -42,  -22,    1;
8) -135135, 405405, -346500, -114345, 84700, 138600, 24255, -2800, -27300, -11025, -18900, -3780, 1575, 1344, 2142, 1596, 414, -56, -98, -64, -29, 1;
    ...
The first few partition polynomials are
E_0^{(-1)} = 1,
E_1^{(-1)} = a1,
E_2^{(-1)} = -a1^2 + a2,
E_3^{(-1)} = 3 a1^3 - 4 a1 a2 + a3,
E_4^{(-1)} = -15 a1^4 + 25 a1^2 a2 - 4 a2^2 - 7 a1 a3 + a4,
E_5^{(-1)} = 105 a1^5 - 210 a1^3 a2 + 70 a1 a2^2 + 60 a1^2 a3 - 15 a2 a3 - 11 a1 a4 + a5,
E_6^{(-1)} = -945 a1^6 + 2205 a1^4 a2 - 1120 a1^2 a2^2 - 630 a1^3 a3 + 70 a2^3 + 350 a1 a2 a3 + 126 a1^2 a4 - 15 a3^2 - 26 a2 a4 - 16 a1 a5 + a6,
E_7^{(-1)} = 10395 a1^7 - 27720 a1^5 a2 + 18900 a1^3 a2^2 + 7875 a1^4 a3 - 2800 a1 a2^3 - 6930 a1^2 a2 a3 - 1638 a1^3 a4 + 560 a2^2 a3 + 455 a1 a3^2 + 784 a1 a2 a4 + 238 a1^2 a5 - 56 a3 a4 - 42 a2 a5 - 22 a1 a6 + a7,
E_8^{(-1)} = -135135 a1^8 + 405405 a1^6 a2 - 346500 a1^4 a2^2 - 114345 a1^5 a3 + 84700 a1^2 a2^3 + 138600 a1^3 a2 a3 + 24255 a1^4 a4 - 2800 a2^4 - 27300 a1 a2^2 a3 - 11025 a1^2 a3^2 - 18900 a1^2 a2 a4 - 3780 a1^3 a5 + 1575 a2 a3^2 + 1344 a2^2 a4 + 2142 a1 a3 a4 + 1596 a1 a2 a5 + 414 a1^2 a6 - 56 a4^2 - 98 a3 a5 - 64 a2 a6 - 29 a1ma7 + a8,
... .
Example substitution identities:
With the permutahedra polynomials
P_1 = -a_1,
P_2 = 2*a_1^2 - a_2,
P_3 = -6*a_1^3 + 6*a_2*a_1 - a_3,
the refined Eulerian polynomials
E_1 = a_1,
E_2 = a_1^2 + a_2,
E_3 = a_1^3 + 4*a_1*a_2 + a_3,
the reciprocal tangent polynomials
RT_1 = -a_1,
RT_2 = -a_2 + a_1^2,
RT_3 = -a_3 + 2*a_1*a_2 - a_1^3,
the Lagrange inversion polynomials
L_1 = -a_1,
L_2 = 3*a_1^2 - a_2,
L_3 = -15*a_1^3 + 10*a_1a_2 - a_3,
then
E^{-1}_3 = P_3(L_1,L_2,L_3) = -6*(-a_1)^3 + 6*(3*a_1^2 - a_2)*(-a_1) - (-15*a_1^3 + 10*a_1*a_2 - a_3) = 3*a_1^3 - 4*a_2*a_1 + a_3,
E^{-1}_3 = RT_3(P_1,P_2,P_3) = -(-6*a_1^3 + 6*a_2*a_1 - a_3) + 2*(-a_1)*(2*a_1^2 - a_2) - (-a_1)^3 = 3*a_1^3 - 4*a_2*a_1 + a_3,
E{-1}_3(E_1,E_2,E_3) = 3*a_1^3 - 4*a_1*(a_1^2 + a_2) + (a_1^3 + 4*a_1*a_2 + a_3) = a_3.

Tom Copeland, Matryoshka Dolls: Iterated noncrossing partitions, the refined Narayana group, and quantum fields, 2022.
Tom Copeland, One Matrix to Rule Them All, 2022.
Tom Copeland, The reduced inverse refined Eulerian polynomials and associated arrays, 2022.

Cf. A000041, A006351, A112493, A124324, A133314, A134685, A134991, A145271, A356144.

rows[nn_] := {{1}}~Join~With[{s = 1/D[InverseSeries[x + Sum[c[k - 1] x^k/k!, {k, 2, nn}] + O[x]^(nn + 1)], x]}, Table[Coefficient[n! s, x^n Product[c[t], {t, p}]], {n, nn-1}, {p, Reverse[Sort[Sort /@ IntegerPartitions[n]]]}]];
rows[8] // Flatten (* Andrey Zabolotskiy, Feb 17 2024 *)

B. = PolynomialRing(ZZ)
A. = PowerSeriesRing(B)
f =  x + a1*x^2/factorial(2) + a2*x^3/factorial(3) + a3*x^4/factorial(4) + a4*x^5/factorial(5) + a5*x^6/factorial(6) + a6*x^7/factorial(7) + a7*x^8/factorial(8) + a8*x^9/factorial(9) + a9*x^10/factorial(10)
g = f.reverse()
w = derivative(g,x)
I = 1 / w
# Added by Peter Luschny, Feb 17 2024:
for n, c in enumerate(I.list()[:9]):
    print(f"E[{n}]", (factorial(n)*c).coefficients())

Given the formal Taylor series or e.g.f. f(x) = x + a_1 x^2/2! + a_2 x^3/3! + ...,
E_n^{-1}(a_1,a_2,...,a_n) = D_{x=0}^n 1 / (D_x f^{(-1)}(x)), where D_x is the derivative w.r.t. x and f^{(-1)}(x) is the (possibly formal) compositional inverse of f(x) about the origin.
E_n^{-1}(a_1,a_2,...,a_n) = D_{x=0}^n 1 f'(f^{(-1)}(x)) by the inverse function theorem, where the prime indicates differentiation w.r.t. the argument of the function f. Note the correspondence to the analytic definitions of the reciprocal tangents [RT] of A356144, consistent with the following algebraic identities.
[E]^{-1} = [P][L] = [P][E][P] = [RT][P], representing, e.g., the substitution of the permutahedra polynomials [P] of A133314 for the indeterminates of the reciprocal tangent polynomials [RT] of A356144. [E] are the refined Eulerian polynomials of A145271, and [L], the classic Lagrange inversion polynomials of A134685.
Since [P]^2 = [L]^2 = [RT]^2 = [I], the substitutional identity, i.e., [P], [L], and [RT] are involutive transformations, many identities follow from the basic ones above, e.g., [L] = [P][E]^{-1} gives an inversion formula for a formal e.g.f. f(x) = x + a_1 x^2/2! + a_2 x^3/3! + ..., and we can identify [E] and [E]^{-1} as a conjugate dual.
With a_n = -x, [E]^{-1} reduces to a signed version of A112493 with an additional initial row, with the row sums of the unsigned coefficients being (1, A006351). A112493 is also given by the diagonals of A124324. See my link above on the reduced polynomials and associated arrays for more detail.
The sequence of row sums of the signed coefficients, i.e., E^{-1}(1,1,...,1), is the sequence (1, 1, 0, 0, 0, 0, ...).
Conjecture: row polynomials are R(n,1) for n > 0 where R(n,k) = R(n-1,k+1) - Sum_{j=1..n-1} binomial(n-1,j-1)*R(j,k)*R(n-j,1) for n > 1, k > 0 with R(1,k) = a_k for k > 0. - Mikhail Kurkov, Mar 22 2025

cp's OEIS Frontend

A000295 Eulerian numbers (Euler's triangle: column k=2 of A008292, column k=1 of A173018).

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A124323 Triangle read by rows: T(n,k) is the number of partitions of an n-set having k singleton blocks (0<=k<=n).

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

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A136394 Triangle read by rows: T(n,k) is the number of permutations of an n-set having k cycles of size > 1 (0<=k<=floor(n/2)).

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A112495 Third column of triangle A112493 used for e.g.f.s of Stirling2 diagonals.

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A124325 Number of blocks of size >1 in all partitions of an n-set.

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A356145 Coefficients of the inverse refined Eulerian partition polynomials [E]^{-1}, partitional inverse to A145271. Irregular triangle read by row with lengths A000041.