%I #65 Feb 19 2024 01:49:19
%S 1,0,1,1,1,2,1,5,1,6,3,5,1,7,7,21,1,8,8,4,28,28,14,1,9,9,9,36,72,12,
%T 84,1,10,10,10,5,45,90,45,45,120,180,42,1,11,11,11,11,55,110,110,55,
%U 55,165,495,165,330,1,12,12,12,12,6,66,132,132,66,66,132,22,220,660,330,660,55,495,990,132
%N Coefficients of reduced partition polynomials of A134264 for computing Lagrange compositional inversion.
%C Coefficients of reduced partition polynomials of A134264 for computing the complete partition polynomials for the Lagrange compositional inversion of A134264 (see Oct 2014 comment by Copeland there). Umbrally,
%C e^(x*t) * exp[Prt(.;1,0,h_2,..) * t] = exp[Prt(.;1,x,h_2,..) * t], where Prt(n;1,0,h_2,..,h_n) are the reduced (h_0 = 1 and h_1 = 0) partition polynomials of the complete polynomials Prt(n;h_0,h_1,h_2,..,h_n) of A134264.
%C Partitions are given in the order of those on page 831 of Abramowitz and Stegun. Formulas for the coefficients of the partitions are given in A134264.
%C Row sums are the Motzkin sums or Riordan numbers A005043. - _Tom Copeland_, Nov 09 2014
%C From _Tom Copeland_, Jul 03 2018: (Start)
%C The matrix and operator formalism for Sheffer Appell sequences leads to the following relations with D = d/dh_1.
%C Exp[Prt(.;1,0,h_2,..) * D] (h_1)^n = [h_1 + Prt(.;1,0,h_2,...)]^n = Prt(n;1,h_1,h_2,...), the partition polynomials of A134264 for g(t)/t with h_0 = 1.
%C For the umbral compositional inverses described below,
%C Exp[UPrt(.;1,0,h_2,..) * D] (h_1)^n = [h_1 + UPrt(.;1,0,h_2,...)]^n = UPrt(n;1,h_1,h_2,...).
%C The respective e.g.f.s are multiplicative inverses; that is, exp[Prt(.;1,0,h_2,..) * t] = 1/exp[UPrt(.;1,0,h_2,..) * t], so the formalism of A133314 applies.
%C The raising operator R such that R Prt(n;1,h_1,h_2,...) = Prt(n+1;1,h_1,h_2,...) is R = exp[Prt(.;1,0,h_2,...)*D] h_1 exp[UPrt(.;1,0,h_2,..)*D] since R Prt(n+1;1,h_1,h_2,...) = exp[Prt(.;1,0,h_2,...)*D] h_1 (h_1)^n = Prt(n+1;h_1,h_2,...) from the definition of the umbral compositional inverse. This may also be expressed as R = h_1 + d/dD log[exp[Prt(.;1,0,h_2,...) * D]], so, using A127671, 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 - 9 h_2 h_4 + 5 h_2^3 - 7 h_3^2) D^5/5! + (h_7 - 28 h_3 h_4 - 14 h_2 h_5 + 56 h_2^2 h_3) D^6/6! + ... . (End)
%H 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].
%F From _Tom Copeland_, Nov 10 2014: (Start)
%F Terms may be computed symbolically up to order n by using an iterated derivative evaluated at t=0:
%F with g(t) = 1/{d/dt [t/(1 + 0 t + h_2 t^2 + h_3 t^3 + ... + h_n t^n)]},
%F evaluate 1/n! * [g(t) d/dt]^n t at t=0, i.e., ask a symbolic math app for the first term in a series expansion of this iterated derivative, to obtain Prt(n-1).
%F Alternatively, the explicit formula in A134264 for the numerical coefficients of each partition can be used. (End)
%F From _Tom Copeland_, Nov 12 2014: (Start)
%F The first few partitions polynomials formed by taking the reciprocal of the e.g.f. of this entry's e.g.f. (cf. A133314) are
%F UPrt(0) = 1
%F UPrt(1;1,0) = 0
%F UPrt(2;1,0,h_2) = -1 h_2
%F UPrt(3;1,0,h_2,h_3) = -1 h_3
%F UPrt(4;1,0,h_2,..,h_4) = -1 h_4 + 4 (h_2)^2
%F UPrt(5;1,0,h_2,..,h_5) = -1 h_5 + 15 h_2 h_3
%F UPrt(6;1,0,h_2,..,h_6) = -1 h_6 + 24 h_2 h_4 + 17 (h_3)^2 + -35 (h_2)^3
%F ...
%F Therefore, umbrally, [Prt(.;1,0,h_2,...) + UPrt(.;1,0,h_2,...)]^n = 0 for n>0 and unity for n=0.
%F Example of the umbral operation:
%F (a. + b.)^2 = a.^2 + 2 a.* b. + b.^2 = a_2 + 2 a_1 * b_1 + b_2.
%F This implies that the umbral compositional inverses (see below) of the partition polynomials of the Lagrange inversion formula (LIF) of A134264 with h_0=1 are given by UPrt(n;1,h_1,h_2,...,h_n) = [UPrt(.;1,0,h_2,...,h_n) + h_1]^n, so that the sequence of polynomials UPrt(n;1,h_1,h_2,...,h_n) is an Appell sequence in the indeterminate h_1. So, if one calculates UPrt(n;1,h_1,...,h_n), the lower order UPrt(n-1;1,h_1,...,h_(n-1)) can be found by taking the derivative w.r.t. h_1 and dividing by n. Same applies for Prt(n;1,h_1,h_2,...,h_n).
%F This connects the combinatorics of the permutohedra through A133314 and A049019, or their duals, to the noncrossing partitions, Dyck lattice paths, etc. that are isomorphic with the LIF of A134264.
%F An Appell sequence P(.,x) with the e.g.f. e^(x*t)/w(t) possesses an umbral inverse sequence UP(.,x) with the e.g.f. w(t)e^(x*t), i.e., polynomials such that P(n,UP(.,x))= x^n = UP(n,P(.,x)) through umbral substitution, as in the binomial example. The Bernoulli polynomials with w(t) = t/(e^t - 1) are a good example with the umbral compositional inverse sequence UP(n,x) = [(x+1)^(n+1)-x^(n+1)] / (n+1) (cf. A074909 and A135278). (End)
%e Prt(0) = 1
%e Prt(1;1,0) = 0
%e Prt(2;1,0,h_2) = 1 h_2
%e Prt(3;1,0,h_2,h_3) = 1 h_3
%e Prt(4;1,0,h_2,..,h_4) = 1 h_4 + 2 (h_2)^2
%e Prt(5;1,0,h_2,..,h_5) = 1 h_5 + 5 h_2 h_3
%e Prt(6;1,0,h_2,..,h_6) = 1 h_6 + 6 h_2 h_4 + 3 (h_3)^2 + 5 (h_2)^3
%e Prt(7;1,0,h_2,..,h_7) = 1 h_7 + 7 h_3 h_4 + 7 h_2 h_5 + 21 h_3 (h_2)^2
%e ...
%e ------------
%e With h_n denoted by (n'), the first seven partition polynomials of A134264 with h_0=1 are given by the first seven coefficients of the truncated Taylor series expansion of the Euler binomial transform
%e e^[(1') * t] * {1 + 1 (2') * t^2/2! + 1 (3') * t^3/3! + [1 (4') + 2 (2')^2] * t^4/4! + [1 (5') + 5 (2')(3')] * t^5/5! + [1 (6') + 6 (2')(4') + 3 (3')^2 + 5 (2')^3] * t^6/6!}, giving the truncated expansion
%e 1 + 1 (1') * t + [1 (2') + 1 (1')^2] * t^2/2! + ... + [1 (6') + 6 (1')(5') + 6 (2')(4') + 3 (3')^2 + 15 (1')^2(4') + 30 (1')(2')(3') + 5 (2')^3 + 20 (1')^3(3') + 30 (1')^2(2')^2 + 15 (1')^4(2') + 1 (1')^6] * t^6/6!.
%e Extending the number of reduced partition polynomials of the transform allows for further complete polynomials of A134264 to be computed.
%t rows[n_] := {{1}, {0}}~Join~Module[
%t {g = 1 / D[t / (1 + Sum[h[k] t^k, {k, 2, n}] + O[t]^(n+1)), t], p = t, r},
%t r = Reap[Do[p = g D[p, t]/k; Sow[Expand[Normal@p /. {t -> 0}]], {k, n+1}]][[2, 1, 2 ;;]];
%t Table[Coefficient[r[[k]], Product[h[t], {t, p}]], {k, 2, n}, {p, Sort[Sort /@ IntegerPartitions[k, k, Range[2, k]]]}]];
%t rows[12] // Flatten (* _Andrey Zabolotskiy_, Feb 18 2024 *)
%Y Cf. A134264, A005043, A133314, A049019, A074909, A135278.
%Y Cf. A127671.
%Y Rows lengths are given by A002865 (except for row 1).
%K nonn,tabf
%O 0,6
%A _Tom Copeland_, Oct 31 2014
%E Formula for Prt(7,..) and a(12)-a(15) added by _Tom Copeland_, Jul 22 2016
%E Rows 8-12 added by _Andrey Zabolotskiy_, Feb 18 2024