A248927 -id:A248927 - cp's OEIS frontend

A248120 Triangle read by rows: Lagrange (compositional) inversion of a function in terms of the coefficients of the Taylor series expansion of its reciprocal, scaled version of A248927, n >= 1, k = 1..A000041(n-1).

1, 2, 6, 3, 24, 36, 4, 120, 360, 60, 80, 5, 720, 3600, 1800, 1200, 300, 150, 6, 5040, 37800, 37800, 16800, 3150, 12600, 3150, 420, 630, 252, 7, 40320, 423360, 705600, 235200, 176400, 352800, 58800, 35280, 23520, 35280, 7056, 1960, 1176, 392, 8

Offset: 1

Tom Copeland, Oct 28 2014

Coefficients are listed in reverse graded colexicographic order (A228100). This is the reverse of Abramowitz and Stegun order (A036036).
Coefficients for Lagrange (compositional) inversion of a function in terms of the Taylor series expansion of its shifted reciprocal. Complementary to A134264 for formal power series and a scaled version of A248927. A refinement of A055302, which enumerates the number of labeled rooted trees with n nodes and k leaves, with row sums A000169.
Given an invertible function f(t) analytic about t=0 with f(0)=0 and df(0)/dt not 0, form h(t) = t / f(t) and denote h_n = (n') as the coefficient of t^n/n! in h(t). Then the compositional inverse of f(t), g(t), as a formal Taylor series, or e.g.f., is given up to the first few orders by
g(t) =   [ 1 (0') ] * t
       + [ 2 (0') (1') ] * t^2/2!
       + [ 6 (0') (1')^2 + 3 (0')^2 (2') ] * t^3/3!
       + [24 (0') (1')^3 + 36 (0')^2 (1') (2') + 4 (0')^3 (3')] * t^4/4!
       + [120 (0') (1')^4 + 360 (0')^2 (1')^2 (2') + (0')^3 [60 (2')^2
           + 80 (1') (3')] + 5 (0')^4 (4')] * t^5/5!
       + [720 (0')(1')^5 + 3600 (0')^2 (1')^3(2') + (0')^3 [1800 (1')(2')^2 + 1200 ( 1')^2(3')] + (0')^4 [300 (2')(3') + 150 (1')(4')] + 6 (0')^5 (5')] * t^6/6! + ... .
Operating with [1/(n*(n-1))] d/d(1') = [1/(n*(n-1))] d/d(h_1) on the n-th partition polynomial in square brackets above associated with t^n/n! generates the (n-1)-th partition polynomial.
Each n-th partition polynomial here is n times the (n-1)-th partition polynomial of A248927.
From Tom Copeland, Nov 24 2014: (Start)
The n-th row is a mapping of the homogeneous symmetric monomials generated by [x(1) + x(2) + ... + x(n)]^(n-1) under the umbral mapping x(m)^j = h_j, for any m. E.g., [a + b + c]^2 = [a^2 + b^2 + c^2] + 2 * [a*b + a*c + b*c] is mapped to [3 * h_2] + 2 * [3 * h_1 * h_1] = 3 * h_2 + 6 *  h_1^2 = A248120(3) with h_0 = 1. (Example corrected Jul 14 2015.)
For another example and relations to A134264 and A036038, see A134264. The general relation is n * A134264(n) = A248120(n) / A036038(n-1) where the arithmetic is performed on the coefficients of matching partitions in each row n.
The Abramowitz and Stegun reference in A036038 gives combinatorial interpretations of A036038 and relations to other number arrays.
This can also be related to repeated umbral composition of Appell sequences and topology with the Bernoulli numbers playing a special role. See the Todd class link. (End)
As presented above and in the Copeland link, this entry is related to exponentiation of e.g.f.s and, therefore, to discussions in the Scott and Sokal preprint (see eqn. 3.1 on p. 10 and eqn. 3.62 p. 24). - Tom Copeland, Jan 17 2017

			Triangle begins
     1;
     2;
     6,     3;
    24,    36,     4;
   120,   360,    60,    80,    5;
   720,  3600,  1800,  1200,  300,   150,    6;
  5040, 37800, 37800, 16800, 3150, 12600, 3150, 420, 630, 252, 7;
  ...
For f(t)= e^t-1, h(t)= t/f(t)= t/(e^t-1), the e.g.f. for the Bernoulli numbers, and plugging the Bernoulli numbers into the Lagrange inversion formula gives g(t)= t - t^2/2 + t^3/3 + ... = log(1+t).

Andrew Howroyd, Table of n, a(n) for n = 1..2087 (rows 1..20)
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].
Tom Copeland, The Hirzebruch criterion for the Todd class, Dec 14 2014.
A. Scott and A. Sokal, Some variants of the exponential formula, with application to the multivariate Tutte polynomial (alias Potts model), arXiv:0803.1477 [math.CO], 2009.

Cf. A145271, A000169, A000272, A139605, A055302, A133314, A049019.
Cf. A134264 and A248927, "scaled" versions of this Lagrange inversion.
Cf. A036038.

C(v)={my(n=vecsum(v), S=Set(v)); (n+1)*n!^2/((n-#v+1)!*prod(i=1, #S, my(x=S[i], c=#select(y->y==x, v)); x!^c*c!))}
row(n)=[C(Vec(p)) | p<-Vecrev(partitions(n-1))]
{ for(n=1, 7, print(row(n))) } \\ Andrew Howroyd, Feb 02 2022

For j>1, there are P(j,m;a...) = j! / [ (j-m)! (a_1)! (a_2)! ... (a_(j-1))! ] permutations of h_0 through h_(j-1) in which h_0 is repeated (j-m) times; h_1, repeated a_1 times; and so on with a_1 + a_2 + ... + a_(j-1) = m.
If, in addition, a_1 + 2 * a_2 + ... + (j-1) * a_(j-1) = j-1, then each distinct combination of these arrangements is correlated with a partition of j-1.
T(j,k) is (j-1)! P(j,m;a...) / [(2!)^a_2 (3!)^a_3 ... ((j-1)!)^a_(j-1) ] for the k-th partition of j-1. The partitions are in reverse order--from bottom to top--from the order in Abramowitz and Stegun (page 831).
For example, from g(t) above, T(6,3) = 5! * [6!/(3!*2!)]/(2!)^2 = 1800 for the 3rd partition from the bottom under n=6-1=5 with m=3 parts, and T(6,5) = 5! * [6!/4!]/(2!*3!) = 300.
If the initial factorial and final denominator of T(n,k) are removed and the expression divided by j and the partitions reversed in order, then A134264 is obtained, a refinement of the Narayana numbers.
For f(t) = t*e^(-t), g(t) = T(t), the Tree function, which is the e.g.f. of A000169, and h(t) = t/f(t) = e^t, so h_n = 1 for all n in this case; therefore, the row sums are A000169(n) = n^(n-1) = n* A000272(n).
Let W(x) = 1/(df(x)/dx)= 1/{d[x/h(x)]/dx}=1/[d{x/[h_0+h_1*x+ ...]/dx]. Then the partition polynomials above are given by (W(x)*d/dx)^n x, evaluated at x=0, and the compositional inverse of f(t) is g(t)=exp(t*W(x)*d/dx) x, evaluated at x=0. Also, dg(t)/dt = W(g(t)). See A145271.
With exp[x* PS(.,t)] = exp[t*g(x)]=exp[x*W(y)d/dy] exp(t*y) eval. at y=0, the raising (creation) and lowering (annihilation) operators defined by R PS(n,t) = PS(n+1,t) and L PS(n,t)= n * PS(n-1,t) are R = t * W(d/dt) and L =(d/dt)/h(d/dt)=(d/dt) 1/[(h_0)+(h_1)*d/dt+(h_2)*(d/dt)^2/2!+...], which will give a lowering operator associated to the refined f-vectors of permutohedra (cf. A133314 and A049019).
Then [dPS(n,z)/dz]/n eval. at z=0 are the row partition polynomials of this entry. (Cf. A139605, A145271, and link therein to Mathemagical Forests for relation to planted trees on p. 13.)
Following the notes connected to the Lagrange reversion theorem in A248927, a generator for the n-th partition polynomial P_n of this entry is (d/dx)^(n-1) (h (x))^n, and -log(1-t*P.) = (t*Q.) / (1 - t*Q.), umbrally, where (Q.)^n = Q_n is the n-th partition polynomial of A248927. - Tom Copeland, Nov 25 2016
With h_0 = 1, the n-th partition polynomial is obtained as the n-th element (with initial index 0) of the first column of M^{n+1}, where M is the matrix with M_{i,j}= binomial(i,j) h_{i-j}, i.e., the lower triangular Pascal matrix with its n-th diagonal multiplied by h_n. This follows from the Lagrange inversion theorem and the relation between powers of matrices such as M and powers of formal Taylor series discussed in A133314. This is equivalent to repeated binomial convolution of the coefficients of the Taylor series with itself. - Tom Copeland, Nov 13 2019
T(n,k) = n*A248927(n,k). - Andrew Howroyd, Feb 02 2022

Terms a(31) and beyond from Andrew Howroyd, Feb 02 2022

A036038 Triangle of multinomial coefficients.

Original entry on oeis.org

1, 1, 2, 1, 3, 6, 1, 4, 6, 12, 24, 1, 5, 10, 20, 30, 60, 120, 1, 6, 15, 20, 30, 60, 90, 120, 180, 360, 720, 1, 7, 21, 35, 42, 105, 140, 210, 210, 420, 630, 840, 1260, 2520, 5040, 1, 8, 28, 56, 70, 56, 168, 280, 420, 560, 336, 840, 1120, 1680, 2520, 1680, 3360, 5040, 6720

Offset: 1

N. J. A. Sloane

The number of terms in the n-th row is the number of partitions of n, A000041(n). - Amarnath Murthy, Sep 21 2002
For each n, the partitions are ordered according to A-St: first by length and then lexicographically (arranging the parts in nondecreasing order), which is different from the usual practice of ordering all partitions lexicographically. - T. D. Noe, Nov 03 2006
For this ordering of the partitions, for n >= 1, see the remarks and the C. F. Hindenburg link given in A036036. - Wolfdieter Lang, Jun 15 2012
The relation (n+1) * A134264(n+1) = A248120(n+1) / a(n) where the arithmetic is performed for matching partitions in each row n connects the combinatorial interpretations of this array to some topological and algebraic constructs of the two other entries. Also, these seem (cf. MOPS reference, Table 2) to be the coefficients of the Jack polynomial J(x;k,alpha=0). - Tom Copeland, Nov 24 2014
The conjecture on the Jack polynomials of zero order is true as evident from equation a) on p. 80 of the Stanley reference, suggested to me by Steve Kass. The conventions for denoting the more general Jack polynomials J(n,alpha) vary. Using Stanley's convention, these Jack polynomials are the umbral extensions of the multinomial expansion of (s_1*x_1 + s_2*x_2 + ... + s_(n+1)*x_(n+1))^n in which the subscripts of the (s_k)^j in the symmetric monomial expansions are finally ignored and the exponent dropped to give s_j(alpha) = j-th row polynomial of A094638 or |A008276| in ascending powers of alpha. (The MOPS table has some inconsistency between n = 3 and n = 4.) - Tom Copeland, Nov 26 2016

			1;
1, 2;
1, 3,  6;
1, 4,  6, 12, 24;
1, 5, 10, 20, 30, 60, 120;
1, 6, 15, 20, 30, 60,  90, 120, 180, 360, 720;

Abramowitz and Stegun, Handbook, p. 831, column labeled "M_1".

David W. Wilson, Table of n, a(n) for n = 1..11731 (rows 1 through 26).
Milton Abramowitz and Irene A. Stegun, editors, Multinomials: M_1, M_2 and M_3, Handbook of Mathematical Functions, December 1972, pp. 831-2.
T. Copeland, Witt Differential Generator for Special Jack Symmetric Functions / Polynomials, 2016.
I. Dumitriu, A. Edelman, G. Schuman, MOPS: Multivariate orthogonal polynomials (symbolically), arxiv:0409066 [math-ph], 2004.
Wolfdieter Lang, First 10 rows and more.
R. Stanley, Some combinatorial properties of Jack symmetric functions, Adv. in Math., 77, p. 76-115, 1989.

Cf. A036036-A036040. Different from A078760. Row sums give A005651.
Cf. A183610 is a table of sums of powers of terms in rows.
Cf. A134264 and A248120.
Cf. A008275, A008276, A094638, A248927.
Cf. A096162 for connections to A130561.

nmax:=7: with(combinat): for n from 1 to nmax do P(n):=sort(partition(n)): for r from 1 to numbpart(n) do B(r):=P(n)[r] od: for m from 1 to numbpart(n) do s:=0: j:=0: while sA036038(n, m) := n!/ (mul((t!)^q(t), t=1..n)); od: od: seq(seq(A036038(n, m), m=1..numbpart(n)), n=1..nmax); # Johannes W. Meijer, Jul 14 2016

Flatten[Table[Apply[Multinomial, Reverse[Sort[IntegerPartitions[i],  Length[ #1]>Length[ #2]&]], {1}], {i,9}]] (* T. D. Noe, Nov 03 2006 *)

def ASPartitions(n, k):
    Q = [p.to_list() for p in Partitions(n, length=k)]
    for q in Q: q.reverse()
    return sorted(Q)
def A036038_row(n):
    return [multinomial(p) for k in (0..n) for p in ASPartitions(n, k)]
for n in (1..10): print(A036038_row(n))
# Peter Luschny, Dec 18 2016, corrected Apr 30 2022

The n-th row is the expansion of (x_1 + x_2 + ... + x_(n+1))^n in the basis of the monomial symmetric polynomials (m.s.p.). E.g., (x_1 + x_2 + x_3 + x_4)^3 = m[3](x_1,..,x_4) + 3*m[1,2](x_1,..,x_4) + 6*m[1,1,1](x_1,..,x_4) = (Sum_{i=1..4} x_i^3) + 3*(Sum_{i,j=1..4;i != j} x_i^2 x_j) + 6*(Sum_{i,j,k=1..4;i < j < k} x_i x_j x_k). The number of indeterminates can be increased indefinitely, extending each m.s.p., yet the expansion coefficients remain the same. In each m.s.p., unique combinations of exponents and subscripts appear only once with a coefficient of unity. Umbral reduction by replacing x_k^j with x_j in the expansions gives the partition polynomials of A248120. - Tom Copeland, Nov 25 2016
From Tom Copeland, Nov 26 2016: (Start)
As an example of the umbral connection to the Jack polynomials: J(3,alpha) = (Sum_{i=1..4} x_i^3)*s_3(alpha) + 3*(Sum_{i,j=1..4;i!=j} x_i^2 x_j)*s_2(alpha)*s_1(alpha)+ 6*(Sum_{i,j,k=1..4;i < j < k} x_i x_j x_k)*s_1(alpha)*s_1(alpha)*s_1(alpha) = (Sum_{i=1..4} x_i^3)*(1+alpha)*(1+2*alpha)+ 3*(sum_{i,j=1..4;i!=j} x_i^2 x_j)*(1+alpha) + 6*(Sum_{i,j,k=1..4;i < j < k} x_i x_j x_k).
See the Copeland link for more relations between the multinomial coefficients and the Jack symmetric functions. (End)

More terms from David W. Wilson and Wouter Meeussen

A134264 Coefficients T(j, k) of a partition transform for Lagrange compositional inversion of a function or generating series in terms of the coefficients of the power series for its reciprocal. Enumeration of noncrossing partitions and primitive parking functions. T(n,k) for n >= 1 and 1 <= k <= A000041(n-1), an irregular triangle read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 4, 2, 6, 1, 1, 5, 5, 10, 10, 10, 1, 1, 6, 6, 3, 15, 30, 5, 20, 30, 15, 1, 1, 7, 7, 7, 21, 42, 21, 21, 35, 105, 35, 35, 70, 21, 1, 1, 8, 8, 8, 4, 28, 56, 56, 28, 28, 56, 168, 84, 168, 14, 70, 280, 140, 56, 140, 28, 1, 1, 9, 9, 9, 9, 36, 72

Offset: 1

Tom Copeland, Jan 14 2008

Coefficients are listed in Abramowitz and Stegun order (A036036).
Given an invertible function f(t) analytic about t=0 (or a formal power series) with f(0)=0 and Df(0) not equal to 0, form h(t) = t / f(t) and let h_n denote the coefficient of t^n in h(t).
Lagrange inversion gives the compositional inverse about t=0 as g(t) = Sum_{j>=1} ( t^j * (1/j) * Sum_{permutations s with s(1) + s(2) + ... + s(j) = j - 1} h_s(1) * h_s(2) * ... * h_s(j) ) = t * T(1,1) * h_0 + Sum_{j>=2} ( t^j * Sum_{k=1..(# of partitions for j-1)} T(j,k) * H(j-1,k ; h_0,h_1,...) ), where H(j-1,k ; h_0,h_1,...) is the k-th partition for h_1 through h_(j-1) corresponding to n=j-1 on page 831 of Abramowitz and Stegun (ordered as in A&S) with (h_0)^(j-m)=(h_0)^(n+1-m) appended to each partition subsumed under n and m of A&S.
Denoting h_n by (n') for brevity, to 8th order in t,
g(t) = t * (0')
+ t^2 * [ (0') (1') ]
+ t^3 * [ (0')^2 (2') + (0') (1')^2 ]
+ t^4 * [ (0')^3 (3') + 3 (0')^2 (1') (2') + (0') (1')^3 ]
+ t^5 * [ (0')^4 (4') + 4 (0')^3 (1') (3') + 2 (0')^3 (2')^2 + 6 (0')^2 (1')^2 (2') + (0') (1')^4 ]
+ t^6 * [ (0')^5 (5') + 5 (0')^4 (1') (4') + 5 (0')^4 (2') (3') + 10 (0')^3 (1')^2 (3') + 10 (0')^3 (1') (2')^2 + 10 (0')^2 (1')^3 (2') + (0') (1')^5 ]
+ t^7 * [ (0')^6 (6') + 6 (0')^5 (1') (5') + 6 (0')^5 (2') (4') + 3 (0')^5 (3')^2 + 15 (0')^4 (1')^2 (4') + 30 (0')^4 (1') (2') (3') + 5 (0')^4 (2')^3 + 20 (0')^3 (1')^3 (3') + 30 (0')^3 (1')^2 (2')^2 + 15 (0')^2 (1')^4 (2') + (0') (1')^6]
+ t^8 * [ (0')^7 (7') + 7 (0')^6 (1') (6') + 7 (0')^6 (2') (5') + 7  (0')^6 (3') (4') + 21 (0')^5 (1')^2* (5') + 42 (0')^5 (1') (2') (4') + 21 (0')^5 (1') (3')^2 + 21 (0')^5 (2')^2 (3') + 35 (0')^4 (1')^3 (4') + 105 (0)^4 (1')^2 (2') (3') + 35 (0')^4 (1') (2')^3  + 35 (0')^3 (1')^4 (3') + 70 (0')^3 (1')^3 (2')^2 + 21 (0')^2 (1')^5 (2') + (0') (1')^7 ]
+ ..., where from the formula section, for example, T(8,1',2',...,7') = 7! / ((8 - (1'+ 2' + ... + 7'))! * 1'! * 2'! * ... * 7'!) are the coefficients of the integer partitions (1')^1' (2')^2' ... (7')^7' in the t^8 term.
A125181 is an extended, reordered version of the above sequence, omitting the leading 1, with alternate interpretations.
If the coefficients of partitions with the same exponent for h_0 are summed within rows, A001263 is obtained, omitting the leading 1.
From identification of the elements of the inversion with those on page 25 of the Ardila et al. link, the coefficients of the irregular table enumerate non-crossing partitions on [n]. - Tom Copeland, Oct 13 2014
From Tom Copeland, Oct 28-29 2014: (Start)
Operating with d/d(1') = d/d(h_1) on the n-th partition polynomial Prt(n;h_0,h_1,..,h_n) in square brackets above associated with t^(n+1) generates n * Prt(n-1;h_0,h_1,..,h_(n-1)); therefore, the polynomials are an Appell sequence of polynomials in the indeterminate h_1 when h_0=1 (a special type of Sheffer sequence).
Consequently, umbrally, [Prt(.;1,x,h_2,..) + y]^n = Prt(n;1,x+y,h_2,..); that is, Sum_{k=0..n} binomial(n,k) * Prt(k;1,x,h_2,..) * y^(n-k) = Prt(n;1,x+y,h_2,..).
Or, e^(x*z) * exp[Prt(.;1,0,h_2,..) * z] = exp[Prt(.;1,x,h_2,..) * z]. Then with x = h_1 = -(1/2) * d^2[f(t)]/dt^2 evaluated at t=0, the formal Laplace transform from z to 1/t of this expression generates g(t), the comp. inverse of f(t), when h_0 = 1 = df(t)/dt eval. at t=0.
I.e., t / (1 - t*(x + Prt(.;1,0,h_2,..))) = t / (1 - t*Prt(.;1,x,h_2,..)) = g(t), interpreted umbrally, when h_0 = 1.
(End)
Connections to and between arrays associated to the Catalan (A000108 and A007317), Riordan (A005043), Fibonacci (A000045), and Fine (A000957) numbers and to lattice paths, e.g., the Motzkin, Dyck, and Łukasiewicz, can be made explicit by considering the inverse in x of the o.g.f. of A104597(x,-t), i.e., f(x) = P(Cinv(x),t-1) = Cinv(x) / (1 + (t-1)*Cinv(x)) = x*(1-x) / (1 + (t-1)*x*(1-x)) = (x-x^2) / (1 + (t-1)*(x-x^2)), where  Cinv(x) = x*(1-x) is the inverse of C(x) = (1 - sqrt(1-4*x)) / 2, a shifted o.g.f. for the Catalan numbers, and P(x,t) = x / (1+t*x) with inverse Pinv(x,t) = -P(-x,t) = x / (1-t*x). Then h(x,t) = x / f(x,t) = x * (1+(t-1)Cinv(x)) / Cinv(x) = 1 + t*x + x^2 + x^3 + ..., i.e., h_1=t and all other coefficients are 1, so the inverse of f(x,t) in x, which is explicitly in closed form finv(x,t) = C(Pinv(x,t-1)), is given by A091867, whose coefficients are sums of the refined Narayana numbers above obtained by setting h_1=(1')=t in the partition polynomials and all other coefficients to one. The group generators C(x) and P(x,t) and their inverses allow associations to be easily made between these classic number arrays. - Tom Copeland, Nov 03 2014
From Tom Copeland, Nov 10 2014: (Start)
Inverting in x with t a parameter, let F(x;t,n) = x - t*x^(n+1). Then h(x) = x / F(x;t,n) = 1 / (1-t*x^n) = 1 + t*x^n + t^2*x^(2n) + t^3*x^(3n) + ..., so h_k vanishes unless k = m*n with m an integer in which case h_k = t^m.
Finv(x;t,n) = Sum_{j>=0} {binomial((n+1)*j,j) / (n*j + 1)} * t^j * x^(n*j + 1), which gives the Catalan numbers for n=1, and the Fuss-Catalan sequences for n>1 (see A001764, n=2). [Added braces to disambiguate the formula. - N. J. A. Sloane, Oct 20 2015]
This relation reveals properties of the partitions and sums of the coefficients of the array. For n=1, h_k = t^k for all k, implying that the row sums are the Catalan numbers. For n = 2, h_k for k odd vanishes, implying that there are no blocks with only even-indexed h_k on the even-numbered rows and that only the blocks containing only even-sized bins contribute to the odd-row sums giving the Fuss-Catalan numbers for n=2. And so on, for n > 2.
These relations are reflected in any combinatorial structures enumerated by this array and the partitions, such as the noncrossing partitions depicted for a five-element set (a pentagon) in Wikipedia.
(End)
From Tom Copeland, Nov 12 2014: (Start)
An Appell sequence possesses an umbral inverse sequence (cf. A249548). The partition polynomials here, Prt(n;1,h_1,...), are an Appell sequence in the indeterminate h_1=u, so have an e.g.f. exp[Prt(.;1,u,h_2...)*t] = e^(u*t) * exp[Prt(.;1,0,h2,...)*t] with umbral inverses with an e.g.f  e^(-u*t) / exp[Prt(.;1,0,h2,...)*t]. This makes contact with the formalism of A133314 (cf. also A049019 and A019538) and the signed, refined face partition polynomials of the permutahedra (or their duals), which determine the reciprocal of exp[Prt(.,0,u,h2...)*t] (cf. A249548) or exp[Prt(.;1,u,h2,...)*t], forming connections among the combinatorics of permutahedra and the noncrossing partitions, Dyck paths and trees (cf. A125181), and many other important structures isomorphic to the partitions of this entry, as well as to formal cumulants through A127671 and algebraic structures of Lie algebras. (Cf. relationship of permutahedra with the Eulerians A008292.)
(End)
From Tom Copeland, Nov 24 2014: (Start)
The n-th row multiplied by n gives the number of terms in the homogeneous symmetric monomials generated by [x(1) + x(2) + ... + x(n+1)]^n under the umbral mapping x(m)^j = h_j, for any m. E.g., [a + b + c]^2 = [a^2 + b^2 + c^2] + 2 * [a*b + a*c + b*c] is mapped to [3 * h_2] + 2 * [3 * h_1^2], and 3 * A134264(3) = 3 *(1,1)= (3,3) the number of summands in the two homogeneous polynomials in the square brackets. For n=3, [a + b + c + d]^3 = [a^3 + b^3 + ...] + 3 [a*b^2 + a*c^2 + ...] + 6 [a*b*c + a*c*d + ...] maps to  [4 * h_3] + 3 [12 * h_1 * h_2] + 6 [4 * (h_1)^3], and the number of terms in the brackets is given by 4 * A134264(4) = 4 * (1,3,1) = (4,12,4).
The further reduced expression is 4 h_3 + 36 h_1 h_2 + 24 (h_1)^3 = A248120(4) with h_0 = 1. The general relation is n * A134264(n) = A248120(n) / A036038(n-1) where the arithmetic is performed on the coefficients of matching partitions in each row n.
Abramowitz and Stegun give combinatorial interpretations of A036038 and relations to other number arrays.
This can also be related to repeated umbral composition of Appell sequences and topology with the Bernoulli numbers playing a special role. See the Todd class link.
(End)
These partition polynomials are dubbed the Voiculescu polynomials on page 11 of the He and Jejjala link. - Tom Copeland, Jan 16 2015
See page 5 of the Josuat-Verges et al. reference for a refinement of these partition polynomials into a noncommutative version composed of nondecreasing parking functions. - Tom Copeland, Oct 05 2016
(Per Copeland's Oct 13 2014 comment.) The number of non-crossing set partitions whose block sizes are the parts of the n-th integer partition, where the ordering of integer partitions is first by total, then by length, then lexicographically by the reversed sequence of parts. - Gus Wiseman, Feb 15 2019
With h_0 = 1 and the other h_n replaced by suitably signed partition polynomials of A263633, the refined face partition polynomials for the associahedra of normalized A133437 with a shift in indices are obtained (cf. In the Realm of Shadows). - Tom Copeland, Sep 09 2019
Number of primitive parking functions associated to each partition of n. See Lemma 3.8 on p. 28 of Rattan. - Tom Copeland, Sep 10 2019
With h_n = n + 1, the d_k (A006013) of Table 2, p. 18, of Jong et al. are obtained, counting the n-point correlation functions in a quantum field theory. - Tom Copeland, Dec 25 2019
By inspection of the diagrams on Robert Dickau's website, one can see the relationship between the monomials of this entry and the connectivity of the line segments of the noncrossing partitions. - Tom Copeland, Dec 25 2019
Speicher has examples of the first four inversion partition polynomials on pp. 22 and 23 with his k_n equivalent to h_n = (n') here with h_0 = 1. Identifying z = t, C(z) = t/f(t) = h(t), and M(z) = f^(-1)(t)/t, then statement (3), on p. 43, of Theorem 3.26, C(z M(z)) = M(z), is equivalent to substituting f^(-1)(t) for t in t/f(t), and statement (4), M(z/C(z)) = C(z), to substituting f(t) for t in f^(-1)(t)/t. - Tom Copeland, Dec 08 2021
Given a Laurent series of the form f(z) = 1/z + h_1 + h_2 z + h_3 z^2 + ..., the compositional inverse is f^(-1)(z) = 1/z + Prt(1;1,h_1)/z^2 + Prt(2;1,h_1,h_2)/z^3 + ... = 1/z + h_1/z^2 + (h_1^2 + h_2)/z^3 + (h_1^3 + 3 h_1 h_2 + h_3)/z^4 + (h_1^4 + 6 h_1^2 h_2 + 4 h_1 h_3 + 2 h_2^2 + h_4)/z^5 + ... for which the polynomials in the numerators are the partition polynomials of this entry. For example, this formula applied to the q-expansion of Klein's j-invariant / function with coefficients A000521, related to monstrous moonshine, gives the compositional inverse with the coefficients A091406 (see He and Jejjala). - Tom Copeland, Dec 18 2021
The partition polynomials of A350499 'invert' the polynomials of this entry giving the indeterminates h_n. A multinomial formula for the coefficients of the partition polynomials of this entry, equivalent to the multinomial formula presented in the first four sentences of the formula section below, is presented in the MathOverflow question referenced in A350499. - Tom Copeland, Feb 19 2022

			1) With f(t) = t / (t-1), then h(t) = -(1-t), giving h_0 = -1, h_1 = 1 and h_n = 0 for n>1. Then g(t) = -t - t^2 - t^3 - ... = t / (t-1).
2) With f(t) = t*(1-t), then h(t) = 1 / (1-t), giving h_n = 1 for all n. The compositional inverse of this f(t) is g(t) = t*A(t) where A(t) is the o.g.f. for the Catalan numbers; therefore the sum over k of T(j,k), i.e., the row sum, is the Catalan number A000108(j-1).
3) With f(t) = (e^(-a*t)-1) / (-a), h(t) = Sum_{n>=0} Bernoulli(n) * (-a*t)^n / n! and g(t) = log(1-a*t) / (-a) = Sum_{n>=1} a^(n-1) * t^n / n. Therefore with h_n = Bernoulli(n) * (-a)^n / n!, Sum_{permutations s with s(1)+s(2)+...+s(j)=j-1} h_s(1) * h_s(2) * ... * h_s(j) = j * Sum_{k=1..(# of partitions for j-1)} T(j,k) * H(j-1,k ; h_0,h_1,...) = a^(j-1). Note, in turn, Sum_{a=1..m} a^(j-1) = (Bernoulli(j,m+1) - Bernoulli(j)) / j for the Bernoulli polynomials and numbers, for j>1.
4) With f(t,x) = t / (x-1+1/(1-t)), then h(t,x) = x-1+1/(1-t), giving (h_0)=x and (h_n)=1 for n>1. Then g(t,x) = (1-(1-x)*t-sqrt(1-2*(1+x)*t+((x-1)*t)^2)) / 2, a shifted o.g.f. in t for the Narayana polynomials in x of A001263.
5) With h(t)= o.g.f. of A075834, but with A075834(1)=2 rather than 1, which is the o.g.f. for the number of connected positroids on [n] (cf. Ardila et al., p. 25), g(t) is the o.g.f. for A000522, which is the o.g.f. for the number of positroids on [n]. (Added Oct 13 2014 by author.)
6) With f(t,x) = x / ((1-t*x)*(1-(1+t)*x)), an o.g.f. for A074909, the reverse face polynomials of the simplices, h(t,x) = (1-t*x) * (1-(1+t)*x) with h_0=1, h_1=-(1+2*t), and h_2=t*(1+t), giving as the inverse in x about 0 the o.g.f. (1+(1+2*t)*x-sqrt(1+(1+2*t)*2*x+x^2)) / (2*t*(1+t)*x) for signed A033282, the reverse face polynomials of the Stasheff polytopes, or associahedra. Cf. A248727. (Added Jan 21 2015 by author.)
7) With f(x,t) = x / ((1+x)*(1+t*x)), an o.g.f. for the polynomials (-1)^n * (1 + t + ... + t^n), h(t,x) = (1+x) * (1+t*x) with h_0=1, h_1=(1+t), and h_2=t, giving as the inverse in x about 0 the o.g.f. (1-(1+t)*x-sqrt(1-2*(1+t)*x+((t-1)*x)^2)) / (2*x*t) for the Narayana polynomials A001263. Cf. A046802. (Added Jan 24 2015 by author.)
From _Gus Wiseman_, Feb 15 2019: (Start)
Triangle begins:
   1
   1
   1   1
   1   3   1
   1   4   2   6   1
   1   5   5  10  10  10   1
   1   6   6   3  15  30   5  20  30  15   1
   1   7   7   7  21  42  21  21  35 105  35  35  70  21   1
Row 5 counts the following non-crossing set partitions:
  {{1234}}  {{1}{234}}  {{12}{34}}  {{1}{2}{34}}  {{1}{2}{3}{4}}
            {{123}{4}}  {{14}{23}}  {{1}{23}{4}}
            {{124}{3}}              {{12}{3}{4}}
            {{134}{2}}              {{1}{24}{3}}
                                    {{13}{2}{4}}
                                    {{14}{2}{3}}
(End)

A. Nica and R. Speicher (editors), Lectures on the Combinatorics of Free Probability, London Mathematical Society Lecture Note Series: 335, Cambridge University Press, 2006 (see in particular, Eqn. 9.14 on p. 141, enumerating noncrossing partitions).

Andrew Howroyd, Table of n, a(n) for n = 1..2087 (rows 1..20)
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].
A. Alexandrov, Enumerative geometry, tau-functions, and Heisenberg-Virasoro algebra, arXiv:hep-th/1404.3402v3, 2015 (p.22).
M. Ashelevich and W. Mlotkowski, Semigroups of distributions with linear Jacobi parameters, arxiv.org/abs/1001.1540v3 [math.CO], p. 9, 2011.
F. Ardila, F. Rincon, and L. Williams, Positroids and non-crossing partitions, arXiv preprint arXiv:1308.2698v2 [math.CO], 2013 (p. 25).
T. Banica, S. Belinschi, M. Capitaine, and B. Collins, Free Bessel laws, arXiv preprint arXiv:0710.5931 [math.PR], 2008.
Freddy Cachazo and Bruno Giménez Umbert, Connecting Scalar Amplitudes using The Positive Tropical Grassmannian, arXiv:2205.02722 [hep-th], 2022.
David Callan, Sets, Lists and Noncrossing Partitions , arXiv:0711.4841 [math.CO], 2008.
B. Collins, I. Nechita, and K. Zyczkowski, Random graph states, maximal flow and Fuss-Catalan distributions, arXiv preprint arXiv:1003.3075 [quant-ph], 2010.
Tom Copeland, Compositional inversion and Appell sequences, Nov 2, 2014.
Tom Copeland, The Hirzebruch criterion for the Todd class Dec. 14, 2014.
Tom Copeland, Appell polynomials, cumulants, noncrossing partitions, Dyck lattice paths, and inversion Dec. 23, 2014.
Tom Copeland, Formal group laws and binomial Sheffer sequences, 2018.
Tom Copeland, In the Realm of Shadows: Umbral inverses and associahedra, noncrossing partitions, symmetric polynomials, and similarity transforms, 2019.
Tom Copeland, A Taste of Moonshine in Free Moments, 2022.
R. Dickau, Non-crossing partitions, Robert Dickau's website, 2012.
K. Ebrahimi-Fard, L. Foissy, J. Kock, and F. Patras, Operads of (noncrossing) partitions, interacting bialgebras, and moment-cumulant relations, arXiv:1907.01190 [math.CO], 2019, p. 25.
K. Ebrahimi-Fard and F. Patras, Cumulants, free cumulants and half-shuffles, arXiv:1409.5664v2 [math.CO], 2015, p. 12.
K. Ebrahimi-Fard and F. Patras, The splitting process in free probability theory, arXiv:1502.02748 [math.CO], 2015, p. 3.
Y. He and V. Jejjala, Modular Matrix Models, arXiv:hep-th/0307293, 2003.
J. Jong, A. Hock, and R. Wulkenhaar Catalan tables and a recursion relation in noncommutative quantum field theory, arXiv preprint arXiv:1904.11231 [math-ph], 2019.
M. Josuat-Verges, F. Menous, J. Novelli, and J. Thibon, Noncommutative free cumulants, arxiv.org/abs/1604.04759 [math.CO], 2016.
Germain Kreweras, Sur les partitions non croisées d'un cycle, Discrete Math. 1 333-350 (1972).
C. Lenart, Lagrange inversion and Schur functions, Journal of Algebraic Combinatorics, Vol. 11, Issue 1, p. 69-78, 2000, (see p. 70, Eqn. 1.2).
M. Mastnak and A. Nica, Hopf algebras and the logarithm of the S-transform in free probability, arXiv:0807.4169v2 [math.OA], p. 28, 2008-2009.
MathOverflow, Guises of the Noncrossing Partitions (NCPs), an MO question posed by Tom Copeland, 2019.
MathOverflow, Combinatorial/diagrammatic models for free moment partition polynomials, an MO question posed by Tom Copeland, 2021.
J. McCammond, Non-crossing Partitions in Surprising Locations, American Mathematical Monthly 113 (2006) 598-610.
M. Mendez, Combinatorial differential operators in: Faà di Bruno formula, enumeration of ballot paths, enriched rooted trees and increasing rooted trees, arXiv:1610.03602 [math.CO], 2016, pp. 33-34 Example 10.
J. Pitman and R. Stanley, A polytope related to empirical distributions, plane trees, parking functions, and the associahedron, arXiv:math/9908029 [math.CO], 1999.
A. Rattan, Parking functions and related combinatorial structures, Master's thesis, Univ. of Waterloo, 2014.
A. Schuetz and G. Whieldon, Polygonal Dissections and Reversions of Series, arXiv:1401.7194 [math.CO], 2014.
R. Simion, Noncrossing partitions, Discrete Mathematics, Vol. 217, Issues 1-3, pp. 367-409, 2000.
R. Speicher, Lecture Notes on "Free Probability Theory", arXiv:1908.08125 [math.OA], 2019.
R. Stanley, Parking Functions and Noncrossing Partitions, 1996.
Jin Wang, Nonlinear Inverse Relations for Bell Polynomials via the Lagrange Inversion Formula, J. Int. Seq., Vol. 22 (2019), Article 19.3.8.
Wikipedia, Noncrossing partition
J. Zhou, On emergent geometry of the Gromov-Witten theory of quintic Calabi-Yau threefold, arXiv:2008.03407 [math-ph], 2020.
Index entries for sequences related to Łukasiewicz

(A001263,A119900) = (reduced array, associated g(x)). See A145271 for meaning and other examples of reduced and associated.
Other orderings are A125181 and A306438.
Cf. A119900 (e.g.f. for reduced W(x) with (h_0)=t and (h_n)=1 for n>0).
Cf. A248927 and A248120, "scaled" versions of this Lagrange inversion.
Cf. A091867 and A125181, for relations to lattice paths and trees.
Cf. A000045, A000108, A000957, A001764, A000522, A005043, A007317, A033282,A036038, A046802, A074909, A075834, A104597, A145271, A248727.
Cf. A249548 for use of Appell properties to generate the polynomials.
Cf. A133314, A049019, A019538, A127671, and A008292 for relations to permutahedra, Eulerians.
Cf. A263634, A263633, A000041, A000110, A001263, A016098, A124794, A133437.
Cf. A006013.
Cf. A000521, A091406.
Cf. A350499

Table[Binomial[Total[y],Length[y]-1]*(Length[y]-1)!/Product[Count[y,i]!,{i,Max@@y}],{n,7},{y,Sort[Sort/@IntegerPartitions[n]]}] (* Gus Wiseman, Feb 15 2019 *)

C(v)={my(n=vecsum(v), S=Set(v)); n!/((n-#v+1)!*prod(i=1, #S, my(x=S[i]); (#select(y->y==x, v))!))}
row(n)=[C(Vec(p)) | p<-partitions(n-1)]
{ for(n=1, 7, print(row(n))) } \\ Andrew Howroyd, Feb 01 2022

For j>1, there are P(j,m;a...) = j! / [ (j-m)! (a_1)! (a_2)! ... (a_(j-1))! ] permutations of h_0 through h_(j-1) in which h_0 is repeated (j-m) times; h_1, repeated a_1 times; and so on with a_1 + a_2 + ... + a_(j-1) = m.
If, in addition, a_1 + 2 * a_2 + ... + (j-1) * a_(j-1) = j-1, then each distinct combination of these arrangements is correlated with a partition of j-1.
T(j,k) is [ P(j,m;a...) / j ] for the k-th partition of j-1 as described in the comments.
For example from g(t) above, T(5,4) = (5! / ((5-3)! * 2!)) / 5 = 6 for the 4th partition under n=5-1=4 with m=3 parts in A&S.
From Tom Copeland, Sep 30 2011: (Start)
Let W(x) = 1/(df(x)/dx)= 1/{d[x/h(x)]/dx}
  = [(h_0)-1+:1/(1-h.*x):]^2 / {(h_0)-:[h.x/(1-h.x)]^2:}
  = [(h_0)+(h_1)x+(h_2)x^2+...]^2 / [(h_0)-(h_2)x^2-2(h_3)x^3-3(h_4)x^4-...], where :" ": denotes umbral evaluation of the expression within the colons and h. is an umbral coefficient.
Then for the partition polynomials of A134264,
  Poly[n;h_0,...,h_(n-1)]=(1/n!)(W(x)*d/dx)^n x, evaluated at x=0, and the compositional inverse of f(t) is g(t) = exp(t*W(x)*d/dx) x, evaluated at x=0. Also, dg(t)/dt = W(g(t)), and g(t) gives A001263 with (h_0)=u and (h_n)=1 for n>0 and A000108 with u=1.
(End)
From Tom Copeland, Oct 20 2011: (Start)
With exp(x* PS(.,t)) = exp(t*g(x)) = exp(x*W(y)d/dy) exp(t*y) eval. at y=0, the raising (creation) and lowering (annihilation) operators defined by R PS(n,t) = PS(n+1,t) and L PS(n,t) = n*PS(n-1,t) are
R = t*W(d/dt) = t*((h_0) + (h_1)d/dt + (h_2)(d/dt)^2 + ...)^2 / ((h_0) - (h_2)(d/dt)^2 - 2(h_3)(d/dt)^3 - 3(h_4)(d/dt)^4 + ...),  and
L = (d/dt)/h(d/dt) = (d/dt) 1/((h_0) + (h_1)*d/dt + (h_2)*(d/dt)^2 + ...)
Then P(n,t) = (t^n/n!) dPS(n,z)/dz eval. at z=0 are the row polynomials of A134264. (Cf. A139605, A145271, and link therein to Mathemagical Forests for relation to planted trees on p. 13.)
(End)
Using the formalism of A263634, the raising operator for the partition polynomials of this array with h_0 = 1 begins as R = h_1 + h_2 D + h_3 D^2/2! + (h_4 - h_2^2) D^3/3! + (h_5 - 5 h_2 h_3) D^4/4! + (h_6 + 5 h_2^3 - 7 h_3^2 - 9 h_2 h_4) D^5/5! + (h_7 - 14 h_2 h_5 + 56 h_2^2 h_3) D^6/6! + ... with D = d/d(h_1). - Tom Copeland, Sep 09 2016
Let h(x) = x/f^{-1}(x) = 1/[1-(c_2*x+c_3*x^2+...)], with c_n all greater than zero. Then h_n are all greater than zero and h_0 = 1. Determine P_n(t) from exp[t*f^{-1}(x)] = exp[x*P.(t)] with f^{-1}(x) = x/h(x) expressed in terms of the h_n (cf. A133314 and A263633). Then P_n(b.) = 0 gives a recursion relation for the inversion polynomials of this entry a_n = b_n/n! in terms of the lower order inversion polynomials and P_j(b.)P_k(b.) = P_j(t)P_k(t)|{t^n = b_n} = d{j,k} >= 0 is the coefficient of x^j/j!*y^k/k! in the Taylor series expansion of the formal group law FGL(x,y) = f[f^{-1}(x)+f^{-1}(y)]. - Tom Copeland, Feb 09 2018
A raising operator for the partition polynomials with h_0 = 1 regarded as a Sheffer Appell sequence in h_1 is described in A249548. - Tom Copeland, Jul 03 2018

Added explicit t^6, t^7, and t^8 polynomials and extended initial table to include the coefficients of t^8. - Tom Copeland, Sep 14 2016
Title modified by Tom Copeland, May 28 2018
More terms from Gus Wiseman, Feb 15 2019
Title modified by Tom Copeland, Sep 10 2019

A141618 Triangle read by rows: number of nilpotent partial transformations (of an n-element set) of height r (height(alpha) = |Im(alpha)|), 0 <= r < n.

Original entry on oeis.org

1, 1, 2, 1, 9, 6, 1, 28, 72, 24, 1, 75, 500, 600, 120, 1, 186, 2700, 7800, 5400, 720, 1, 441, 12642, 73500, 117600, 52920, 5040, 1, 1016, 54096, 571536, 1764000, 1787520, 564480, 40320, 1, 2295, 217800, 3916080, 21019824, 40007520, 27941760, 6531840, 362880, 1, 5110, 839700, 24555600, 214326000

Offset: 1

Abdullahi Umar, Aug 23 2008

The sum of each row of the sequence (as a triangular array) is A000272. Second left-downward diagonal is A058877.
From Tom Copeland, Oct 26 2014: (Start)
With T(x,t) the e.g.f. for A055302 for the number of labeled rooted trees with n nodes and k leaves, the mirror of the row polynomials of this array are given by e^T(x,t) = exp[ t * x + (2t) * x^2/2! + (6t + 3t^2) * x^3/3! + ...] = 1 + t * x + (2t + t^2) * x^2/2! + (6t + 9t^2 + t^3) * x^3/3! + ... = 1 + Nr(x,t).
Equivalently, e^x-1 = Nr[Tinv(x,t),t] = t * N[t*Tinv(x,t),1/t], where N(x,t) is the e.g.f. of this array and Tinv(x,t) is the comp. inverse in x of T(x,t). Note that Nr(x,t) = t * N(x*t,1/t), and N(x,t) = t * Nr(x*t,1/t). Also, log[1 + Nr(x,t)]= x * [t + Nr(x,t)] = T(x,t).
E.g.f. is N(x,t)= t * {exp[T(x*t,1/t)] - 1}, and log[1 + N(x,t)/t] =  T(x*t,1/t) =  x + (2t) * x^2/2! +  (3t + 6t^2) * x^3/3! + (4t + 36t^2 + 24t^3) * x^4/4! + ... = x + (t) * x^2 + (t + 2t^2) * x^3/2! + (t + 9t^2 + 6t^3) * x^4/3! +  ... is the comp. inverse in x of x / [1 + t * (e^x - 1)].
The exp/log transforms (A036040/A127671) generally give associations between enumerations of sets of connected graphs/objects (in this case, trees) and sets of disconnected (or not necessarily connected) graphs/objects (in this case, bipartite graphs of the nilpotent transformations). The transforms also relate formal cumulants and moments so that Nr(x,t) is then the e.g.f. for the formal moments associated to the formal cumulants whose e.g.f. is T(x,t). (End)
T(n,k) is the number of parking functions of length n containing exactly k+1 distinct values in its image. - Alan Kappler, Jun 08 2024

			N(J(4,2)) = 6*6*2 = 72.
From _Peter Bala_, Oct 22 2008: (Start)
Triangle begins
n\k|..0.....1.....2.....3.....4....5
=====================================
.1.|..1
.2.|..1.....2
.3.|..1.....9.....6
.4.|..1....28....72....24
.5.|..1....75...500...600...120
.6.|..1...186..2700..7800..5400...720
...
(End)

A. Laradji and A. Umar, On the number of nilpotents in the partial symmetric semigroup, Comm. Algebra 32 (2004), 3017-3023.
A. Laradji and A. Umar, On the number of nilpotents in the partial symmetric semigroup, Tech. Report TR305, King Fahd Univ. of Petroleum and Minerals, (2003).
Wikipedia, Cumulant

Cf. A000272, A007318, A036040, A048993, A055302, A058877, A127671, A198204.

A048993 := proc(n,k)
    combinat[stirling2](n,k) ;
end proc:
A141618 := proc(n,k)
    binomial(n,k)*k!*A048993(n,k+1) ;
end proc:

Flatten[CoefficientList[CoefficientList[InverseSeries[Series[Log[1 + x]/(1 + t*x),{x,0,9}]],x]*Table[n!, {n,0,9}],t]] (* Peter Luschny, Oct 24 2015, after Peter Bala *)

A055302(n,k)=n!/k!*stirling(n-1, n-k,2);
T(n,k)=A055302(n+1,n+1-k) / (n+1);
for(n=1,10,for(k=1,n,print1(T(n,k),", "));print());
\\ Joerg Arndt, Oct 27 2014

N(J(n,r)) = C(n,r)*S(n,r+1)*r! where S(n, r + 1) is a Stirling number of the second kind (given by A048993 with zeros removed); generating function = (x+1)^(n-1).
From Peter Bala, Oct 22 2008: (Start)
Define a functional I on formal power series of the form f(x) = 1 + ax + bx^2 + ... by the following iterative process. Define inductively f^(1)(x) = f(x) and f^(n+1)(x) = f(x*f^(n)(x)) for n >= 1. Then set I(f(x)) = lim_{n -> infinity} f^(n)(x) in the x-adic topology on the ring of formal power series; the operator I may also be defined by I(f(x)) := 1/x*series reversion of x/f(x).
Let f(x) = 1 + a*x + a*x^2/2! + a*x^3/3! + ... . Then the e.g.f. for this table is I(f(x)) = 1 + a*x +(a + 2*a^2)*x^2/2! + (a + 9*a^2 + 6*a^3)*x^3/3! + (a + 28*a^2 + 72*a^3 + 24*a^4)*x^4/4! + ... . Note, if we take f(x) = 1 + a*x + a*x^2 + a*x^3 + ... then I(f(x)) is the o.g.f. of the Narayana triangle A001263. (End)
A generator for this array is given by the inverse, g(x,t), of f(x,t)= x/(1 + t * (e^x-1)). Then A248927 gives h(x,t)= x / f(x,t) = 1 + t*(e^x-1)= 1 + t * (x + x^2/2! + x^3/3! + ...) and g(x,t)= x * (1 + t * x + (t + 2 t^2) * x^2/2! + (t + 9 t^2 + 6 t^3) * x^3/3! + ...), so by Bala's arguments A248927 is a refinement of A141618 with row sums A000272. The connection to Narayana numbers is reflected in the relation between A248927 and A134264. See A145271 for more relations that g(x,t) and f(x,t) must satisfy. - Tom Copeland, Oct 17 2014
T(n,k) = C(n,k-1) * A028246(n,k) = C(n,k-1) * A019538(n,k)/k = A055302(n+1,n+1-k) / (n+1). - Tom Copeland, Oct 25 2014
E.g.f. is the series reversion of log(1 + x)/(1 + t*x) with respect to x. Cf. A198204. - Peter Bala, Oct 21 2015

More terms from Joerg Arndt, Oct 27 2014

cp's OEIS Frontend

A248120 Triangle read by rows: Lagrange (compositional) inversion of a function in terms of the coefficients of the Taylor series expansion of its reciprocal, scaled version of A248927, n >= 1, k = 1..A000041(n-1).

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A036038 Triangle of multinomial coefficients.

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A141618 Triangle read by rows: number of nilpotent partial transformations (of an n-element set) of height r (height(alpha) = |Im(alpha)|), 0 <= r < n.

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