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A249548
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Coefficients of reduced partition polynomials of A134264 for computing Lagrange compositional inversion.
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5
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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, 84, 1, 10, 10, 10, 5, 45, 90, 45, 45, 120, 180, 42, 1, 11, 11, 11, 11, 55, 110, 110, 55, 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
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OFFSET
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0,6
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COMMENTS
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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,
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.
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.
The matrix and operator formalism for Sheffer Appell sequences leads to the following relations with D = d/dh_1.
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.
For the umbral compositional inverses described below,
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,...).
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.
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)
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LINKS
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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].
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FORMULA
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Terms may be computed symbolically up to order n by using an iterated derivative evaluated at t=0:
with g(t) = 1/{d/dt [t/(1 + 0 t + h_2 t^2 + h_3 t^3 + ... + h_n t^n)]},
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).
Alternatively, the explicit formula in A134264 for the numerical coefficients of each partition can be used. (End)
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
UPrt(0) = 1
UPrt(1;1,0) = 0
UPrt(2;1,0,h_2) = -1 h_2
UPrt(3;1,0,h_2,h_3) = -1 h_3
UPrt(4;1,0,h_2,..,h_4) = -1 h_4 + 4 (h_2)^2
UPrt(5;1,0,h_2,..,h_5) = -1 h_5 + 15 h_2 h_3
UPrt(6;1,0,h_2,..,h_6) = -1 h_6 + 24 h_2 h_4 + 17 (h_3)^2 + -35 (h_2)^3
...
Therefore, umbrally, [Prt(.;1,0,h_2,...) + UPrt(.;1,0,h_2,...)]^n = 0 for n>0 and unity for n=0.
Example of the umbral operation:
(a. + b.)^2 = a.^2 + 2 a.* b. + b.^2 = a_2 + 2 a_1 * b_1 + b_2.
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).
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.
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)
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EXAMPLE
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Prt(0) = 1
Prt(1;1,0) = 0
Prt(2;1,0,h_2) = 1 h_2
Prt(3;1,0,h_2,h_3) = 1 h_3
Prt(4;1,0,h_2,..,h_4) = 1 h_4 + 2 (h_2)^2
Prt(5;1,0,h_2,..,h_5) = 1 h_5 + 5 h_2 h_3
Prt(6;1,0,h_2,..,h_6) = 1 h_6 + 6 h_2 h_4 + 3 (h_3)^2 + 5 (h_2)^3
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
...
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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^[(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
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!.
Extending the number of reduced partition polynomials of the transform allows for further complete polynomials of A134264 to be computed.
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MATHEMATICA
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rows[n_] := {{1}, {0}}~Join~Module[
{g = 1 / D[t / (1 + Sum[h[k] t^k, {k, 2, n}] + O[t]^(n+1)), t], p = t, r},
r = Reap[Do[p = g D[p, t]/k; Sow[Expand[Normal@p /. {t -> 0}]], {k, n+1}]][[2, 1, 2 ;; ]];
Table[Coefficient[r[[k]], Product[h[t], {t, p}]], {k, 2, n}, {p, Sort[Sort /@ IntegerPartitions[k, k, Range[2, k]]]}]];
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CROSSREFS
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Rows lengths are given by A002865 (except for row 1).
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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Formula for Prt(7,..) and a(12)-a(15) added by Tom Copeland, Jul 22 2016
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STATUS
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approved
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