A304462 -id:A304462 - cp's OEIS frontend

A111785 T(n,k) are coefficients used for power series inversion (sometimes called reversion), n >= 0, k = 1..A000041(n), read by rows.

1, -1, -1, 2, -1, 5, -5, -1, 6, 3, -21, 14, -1, 7, 7, -28, -28, 84, -42, -1, 8, 8, 4, -36, -72, -12, 120, 180, -330, 132, -1, 9, 9, 9, -45, -90, -45, -45, 165, 495, 165, -495, -990, 1287, -429, -1, 10, 10, 10, 5, -55, -110, -110, -55, -55, 220, 660, 330, 660, 55, -715, -2860, -1430, 2002, 5005, -5005, 1430, -1, 11, 11

Offset: 0

Wolfdieter Lang, Aug 23 2005

Coefficients are listed in Abramowitz and Stegun order (A036036).
The formula for the inversion of the power series y = F(x) = x*G(x) = x*(1 + Sum_{k>=1} g[k]*(x^k)) is obtained as a corollary of Lagrange's inversion theorem. The result is F^{(-1)}(y)= Sum_{n>=1} P(n-1)*y^n, where P(n):=sum over partitions of n of a(n,k)* G[k], with G[k]:=g[1]^e(k,1)*g[2]^e(k,2)*...*g[n]^e(k,n) if the k-th partition of n, in Abramowitz-Stegun order(see the given ref, pp. 831-2), is [1^e(k,1),2^e(k,2),...,n^e(k,n)], for k=1..p(n):= A000041(n) (partition numbers).
The sequence of row lengths is A000041(n) (partition numbers).
The signs are given by (-1)^m(n,k), with the number of parts m(n,k) = Sum_{j=1..n} e(k,j) of the k-th partition of n. For m(n,k) see A036043.
The proof that the unsigned row sums give Schroeder's little numbers A001003(n) results from their formula ((d^(n-1)/dx^(n-1)) ((1-x)/(1-2*x))^n)/n!|_{x=0}, n >= 1. This formula for A001003 can be proved starting with the compositional inverse of the g.f. of A001003 (which is given there in a comment) and using Lagrange's inversion theorem to recover the original sequence A001003.
For alternate formulations and relation to the geometry of associahedra or Stasheff polytopes (and other combinatorial objects) see A133437. [Tom Copeland, Sep 29 2008]
The coefficients of the row polynomials P(n) with monomials in lexicographically descending order e.g. P(6) = -1*g[6] + 8*g[5]*g[1] + 8*g[4]*g[2] - 36*g[4]*g[1]^2 + 4*g[3]^2 - 72*g[3]*g[2]*g[1] - 12*g[2]^3 + 120*g[3]*g[1]^3 + 180*g[2]^2*g[1]^2 - 330*g[2]*g[1]^4 + 132*g[1]^6 are given in A304462. [Herbert Eberle, Aug 16 2018]

			[ +1];
[ -1];
[ -1, 2];
[ -1, 5, -5];
[ -1, 6,  3, -21,  14];
[ -1, 7,  7, -28, -28, 84, -42];
[ -1, 8,  8,   4, -36, -72, -12, 120, 180, -330, 132];  ...
The seventh row, [ -1, 8, 8, 4, -36, -72, -12, 120, 180, -330, 132], stands for the row polynomial P(6) with monomials in lexicographically ascending order P(6) = -1*g[0]^5*g[6] + 8*g[0]^4*g[1]*g[5] + 8*g[0]^4*g[2]*g[4] + 4*g[0]^4*g[3]^2 - 36*g[0]^3*g[1]^2*g[4] - 72*g[0]^3*g[1]*g[2]*g[3] - 12*g[0]^3*g[2]^3 + 120*g[0]^2*g[1]^3*g[3] + 180*g[0]^2*g[1]^2*g[2]^2 - 330*g[0]*g[1]^4*g[2] + 132*g[1]^6 = (1/7!)*(differentiate 1/G(x)^7 six times and evaluate at x = 0). This gives the coefficient of y^7 of F^{(-1)}(y).

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 150, Table 4.1 (unsigned).

Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..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. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 16, 3.6.25.
Bartomeu Fiol and Alan Rios Fukelman, On the planar free energy of matrix models, arXiv:2111.14783 [hep-th], 2021. See also J. High Energy Phys. (2022) Iss. 2.
Wolfdieter Lang, First 10 rows and a formula.
Jin Wang, Nonlinear Inverse Relations for Bell Polynomials via the Lagrange Inversion Formula, J. Int. Seq., Vol. 22 (2019), Article 19.3.8.
Eric Weisstein's World of Mathematics, Series Reversion

Row sums give (-1)^n. Unsigned row sums are A001003(n) (little Schroeder numbers). Inversion triangle with leading quadratic term: A276738. Conjectured simplification: A283298.

(* Graded Colex Ordering: by length, then reverse lexicographic by digit *)
ClearAll[P, L, T, c, g]
P[0] := 1
P[n_] := -Total[
   Multinomial @@ # c[Total@# - 1] Times @@
       Power[g[#] & /@ Range[0, n - 1], #] & /@
    Table[ Count[p, i], {p, Drop[IntegerPartitions[n + 1], 1]}, {i,
      n}]]
L[n_] := Join @@ GatherBy[IntegerPartitions[n], Length]
T[1] := {1}
T[n_] := Coefficient[ Do[g[i] = P[i], {i, 0, n - 1}];
    P[n - 1], #] & /@ (Times @@@ Map[c, L[n - 1], {2}])
Array[T, 9] // Flatten (* Bradley Klee and Michael Somos, Apr 14 2017 *)

sv(n)={eval(Str("'s",n))}
Trm(q,v)={my(S=Set(v)); for(i=1, #S, my(x=S[i], c=#select(y->y==x, v)); q=polcoef(q, c, sv(x))); q}
Q(n)={polcoef(serreverse(x + x*sum(k=1, n, x^k*sv(k), O(x*x^n)))/x, n)}
row(n)={my(q=Q(n)); [Trm(q,Vec(v)) | v<-partitions(n)]} \\ Andrew Howroyd, Feb 01 2022

C(v)={my(n=vecsum(v), S=Set(v)); (-1)^#v*(n+#v)!/(n+1)!/prod(i=1, #S, my(x=S[i], c=#select(y->y==x, v)); c!)}
row(n)=[C(Vec(p)) | p<-partitions(n)]
{ for(n=0, 7, print(row(n))) } \\ Andrew Howroyd, Feb 01 2022

def A111785_list(dim): # returns the first dim rows
    C = [[0 for k in range(m+1)] for m in range(dim+1)]
    C[0][0] = 1; F = [1]; i = 1
    X = lambda n: 1 if n == 1 else var('x'+str(n))
    while i <= dim: F.append(F[i-1]*X(i)); i += 1
    for m in (1..dim):
        C[m][m] = -C[m-1][m-1]/F[1]
        for k in range(m-1, 0, -1):
            C[m][k] = -(C[m-1][k-1]+sum(F[i]*C[m][k+i-1] for i in (2..m-k+1)))/F[1]
    P = [expand((-1)^m*C[m][1]) for m in (1..dim)]
    R = PolynomialRing(ZZ, [X(i) for i in (2..dim)], order='lex')
    return [R(p).coefficients()[::-1] for p in P]
A111785_list(8) # Peter Luschny, Apr 14 2017

For row n >= 1 the row polynomial in the variables g[1], ..., g[n] is P(n) = (1/(n+1)!)*(d^n/dx^n)(1/G(x)^(n+1))|{x=0}. P(0):=1. (d^k/dx^k)G(x)|{x=0} = k!*g[k], k>=1; G(0)=1.
a(n, k) is the coefficient in P(n) of g[1]^e(k, 1)*g[2]^e(k, 2)*..*g[n]^e(k, n) with the k-th partition of n written as [1^e(k, 1), 2^e(k, 2), ..., n^e(k, n)] in Abramowitz-Stegun order (e(k, j) >= 0; if e(k, j)=0 then j^0 is not recorded).
T(n,k) = (-1)^j*(n+j)!/((n+1)!*Product_{i>=1} s_i!), where (1*s_1 + 2*s_2 + ... = n) is the k-th partition of n and j = s_1 + s_2 ... is the number of parts. - Andrew Howroyd, Feb 01 2022

Name edited by Andrew Howroyd, Feb 02 2022

A306557 Numerator coefficients of the bivariate Maclaurin series ("inverse Kepler equation") developed as Lagrange inversion E=KeplerInv(e,M) of Kepler's equation M = Kepler(e,E) = E - e*sin(E).

Original entry on oeis.org

1, 1, 9, 1, 54, 225, 1, 243, 4131, 11025, 1, 1008, 50166, 457200, 893025, 1, 4077, 520218, 11708154, 70301925, 108056025, 1, 16362, 5020623, 243313164, 3274844175, 14427513450, 18261468225, 1, 65511, 46789461, 4535570691, 119537963811, 1107456067125, 3821273720775, 4108830350625

Offset: 0

Herbert Eberle, Feb 23 2019

Coefficients of the numerator polynomials of the bivariate Maclaurin series ("inverse Kepler equation") developed as Lagrange inversion E = KeplerInv(e,M) of Kepler's equation M = Kepler(e,E) = E - e*sin(E), where e=numeric eccentricity, M=mean anomaly, E=eccentric anomaly. The series is KeplerInv(e,M) = M/(1-e) + Sum_{n>=1} (-1)^n*(Sum_{j=1..n} a(n,j)*e^j)/(1-e)^(3n+1)*M^(2n+1)/(2n+1)! = M/(1-e) - (e/(1-e)^4)*M^3/3! + ((e+9*e^2)/(1-e)^7)*M^5/5! - + ... .
The element a(n,n) with highest index in each row (the diagonal element) has the form Product_{j=1..n} (2*j+1)^2.
The derivative dKepler/dE = 1 - e*cos(E) goes to zero at E = i*arccosh(1/e) in the complex plane. Thus dKeplerInv/dM goes to infinity at M = i*(arccosh(1/e) - sqrt(1-e^2)), so that the radius of convergence of KeplerInv(e,M) is arccosh(1/e) - sqrt(1-e^2). KeplerInv(e,M) converges linearly within the circle of convergence |M| < arccosh(1/e) - sqrt(1-e^2).

			Matrix (regular triangle) lexicographically ascending in the rows:
  1;
  1,     9;
  1,    54,     225;
  1,   243,    4131,     11025;
  1,  1008,   50166,    457200,     893025;
  1,  4077,  520218,  11708154,   70301925,   108056025;
  1, 16362, 5020623, 243313164, 3274844175, 14427513450, 18261468225;
  ...

Wikipedia, Kepler's equation

Generated by A111785 or A304462, diagonal elements are in A001818.

While M = E - e*sin(E) = E*(1-e) - e*Sum_{n>=1} (-1)^n*E^(2n+1)/(2n+1)! the formal power series of the compositional inverse KeplerInv(e,M) is as above according to A111785 and A304462.

cp's OEIS Frontend

A111785 T(n,k) are coefficients used for power series inversion (sometimes called reversion), n >= 0, k = 1..A000041(n), read by rows.

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