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

%I A111785 #85 Feb 16 2025 08:32:58
%S A111785 1,-1,-1,2,-1,5,-5,-1,6,3,-21,14,-1,7,7,-28,-28,84,-42,-1,8,8,4,-36,
%T A111785 -72,-12,120,180,-330,132,-1,9,9,9,-45,-90,-45,-45,165,495,165,-495,
%U A111785 -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
%N A111785 T(n,k) are coefficients used for power series inversion (sometimes called reversion), n >= 0, k = 1..A000041(n), read by rows.
%C A111785 Coefficients are listed in Abramowitz and Stegun order (A036036).
%C A111785 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).
%C A111785 The sequence of row lengths is A000041(n) (partition numbers).
%C A111785 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.
%C A111785 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.
%C A111785 For alternate formulations and relation to the geometry of associahedra or Stasheff polytopes (and other combinatorial objects) see A133437. [_Tom Copeland_, Sep 29 2008]
%C A111785 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]
%D A111785 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 150, Table 4.1 (unsigned).
%H A111785 Andrew Howroyd, <a href="/A111785/b111785.txt">Table of n, a(n) for n = 0..2713</a> (rows 0..20)
%H A111785 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A111785 A. M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 16, 3.6.25.
%H A111785 Bartomeu Fiol and Alan Rios Fukelman, <a href="https://arxiv.org/abs/2111.14783">On the planar free energy of matrix models</a>, arXiv:2111.14783 [hep-th], 2021. See also <a href="https://doi.org/10.1007/JHEP02(2022)078">J. High Energy Phys.</a> (2022) Iss. 2.
%H A111785 Wolfdieter Lang, <a href="/A111785/a111785.pdf">First 10 rows and a formula.</a>
%H A111785 Jin Wang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Wang/wang53.html">Nonlinear Inverse Relations for Bell Polynomials via the Lagrange Inversion Formula</a>, J. Int. Seq., Vol. 22 (2019), Article 19.3.8.
%H A111785 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SeriesReversion.html">Series Reversion</a>
%F A111785 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.
%F A111785 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).
%F A111785 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
%e A111785 [ +1];
%e A111785 [ -1];
%e A111785 [ -1, 2];
%e A111785 [ -1, 5, -5];
%e A111785 [ -1, 6,  3, -21,  14];
%e A111785 [ -1, 7,  7, -28, -28, 84, -42];
%e A111785 [ -1, 8,  8,   4, -36, -72, -12, 120, 180, -330, 132];  ...
%e A111785 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).
%t A111785 (* Graded Colex Ordering: by length, then reverse lexicographic by digit *)
%t A111785 ClearAll[P, L, T, c, g]
%t A111785 P[0] := 1
%t A111785 P[n_] := -Total[
%t A111785    Multinomial @@ # c[Total@# - 1] Times @@
%t A111785        Power[g[#] & /@ Range[0, n - 1], #] & /@
%t A111785     Table[ Count[p, i], {p, Drop[IntegerPartitions[n + 1], 1]}, {i,
%t A111785       n}]]
%t A111785 L[n_] := Join @@ GatherBy[IntegerPartitions[n], Length]
%t A111785 T[1] := {1}
%t A111785 T[n_] := Coefficient[ Do[g[i] = P[i], {i, 0, n - 1}];
%t A111785     P[n - 1], #] & /@ (Times @@@ Map[c, L[n - 1], {2}])
%t A111785 Array[T, 9] // Flatten (* _Bradley Klee_ and _Michael Somos_, Apr 14 2017 *)
%o A111785 (Sage)
%o A111785 def A111785_list(dim): # returns the first dim rows
%o A111785     C = [[0 for k in range(m+1)] for m in range(dim+1)]
%o A111785     C[0][0] = 1; F = [1]; i = 1
%o A111785     X = lambda n: 1 if n == 1 else var('x'+str(n))
%o A111785     while i <= dim: F.append(F[i-1]*X(i)); i += 1
%o A111785     for m in (1..dim):
%o A111785         C[m][m] = -C[m-1][m-1]/F[1]
%o A111785         for k in range(m-1, 0, -1):
%o A111785             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]
%o A111785     P = [expand((-1)^m*C[m][1]) for m in (1..dim)]
%o A111785     R = PolynomialRing(ZZ, [X(i) for i in (2..dim)], order='lex')
%o A111785     return [R(p).coefficients()[::-1] for p in P]
%o A111785 A111785_list(8) # _Peter Luschny_, Apr 14 2017
%o A111785 (PARI)
%o A111785 sv(n)={eval(Str("'s",n))}
%o A111785 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}
%o A111785 Q(n)={polcoef(serreverse(x + x*sum(k=1, n, x^k*sv(k), O(x*x^n)))/x, n)}
%o A111785 row(n)={my(q=Q(n)); [Trm(q,Vec(v)) | v<-partitions(n)]} \\ _Andrew Howroyd_, Feb 01 2022
%o A111785 (PARI)
%o A111785 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!)}
%o A111785 row(n)=[C(Vec(p)) | p<-partitions(n)]
%o A111785 { for(n=0, 7, print(row(n))) } \\ _Andrew Howroyd_, Feb 01 2022
%Y A111785 Row sums give (-1)^n. Unsigned row sums are A001003(n) (little Schroeder numbers). Inversion triangle with leading quadratic term: A276738. Conjectured simplification: A283298.
%K A111785 sign,look,tabf
%O A111785 0,4
%A A111785 _Wolfdieter Lang_, Aug 23 2005
%E A111785 Name edited by _Andrew Howroyd_, Feb 02 2022