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A134685 Irregular triangle read by rows: coefficients C(j,k) of a partition transform for direct Lagrange inversion.

%I A134685 #163 Aug 28 2025 18:36:10
%S A134685 1,-1,3,-1,-15,10,-1,105,-105,10,15,-1,-945,1260,-280,-210,35,21,-1,
%T A134685 10395,-17325,6300,3150,-280,-1260,-378,35,56,28,-1,-135135,270270,
%U A134685 -138600,-51975,15400,34650,6930,-2100,-1575,-2520,-630,126,84,36,-1
%N A134685 Irregular triangle read by rows: coefficients C(j,k) of a partition transform for direct Lagrange inversion.
%C A134685 Let f(t) = u(t) - u(0) = Ev[exp(u.* t) - u(0)] = log{Ev[(exp(z.* t))/z_0]} = Ev[-log(1- a.* t)], where the operator Ev denotes umbral evaluation of the umbral variables u., z. or a., e.g., Ev[a.^n + a.^m] = a_n + a_m . The relation between z_n and u_n is given in reference in A127671 and u_n = (n-1)! * a_n .
%C A134685 If u_1 is not equal to 0, then the compositional inverse for these expressions is given by g(t) = Sum_{j>=1} P(j,t) where, with u_n denoted by (n') for brevity,
%C A134685 P(1,t) = (1')^(-1) * [ 1 ] * t
%C A134685 P(2,t) = (1')^(-3) * [ -(2') ] * t^2 / 2!
%C A134685 P(3,t) = (1')^(-5) * [ 3 (2')^2 - (1')(3') ] * t^3 / 3!
%C A134685 P(4,t) = (1')^(-7) * [ -15 (2')^3 + 10 (1')(2')(3') - (1')^2 (4') ] * t^4 / 4!
%C A134685 P(5,t) = (1')^(-9) * [ 105 (2')^4 - 105 (1') (2')^2 (3') + 10 (1')^2 (3')^2 + 15 (1')^2 (2') (4') - (1')^3 (5') ] * t^5 / 5!
%C A134685 P(6,t) = (1')^(-11) * [ -945 (2')^5 + 1260 (1') (2')^3 (3') - 280 (1')^2 (2') (3')^2 - 210 (1')^2 (2')^2 (4') + 35 (1')^3 (3')(4') + 21 (1')^3 (2')(5') - (1')^4 (6') ] * t^6 / 6!
%C A134685 P(7,t) = (1')^(-13) * [ 10395 (2')^6 - 17325 (1') (2')^4 (3') +  (1')^2 [ 6300 (2')^2 (3')^2 + 3150  (2')^3 (4')] - (1')^3 [280 (3')^3 + 1260  (2')(3')(4') + 378  (2')^2(5')] + (1')^4 [35 (4')^2 + 56 (3')(5') + 28 (2')(6')]  - (1')^5 (7') ] * t^7 / 7!
%C A134685 P(8,t) = (1')^(-15) * [ -135135 (2')^7 + 270270 (1') (2')^5 (3') -  (1')^2 [ 138600 (2')^3 (3')^2 +  51975 (2')^4 (4')] + (1')^3 [15400 (2')(3')^3 + 34650  (2')^2(3')(4') + 6930  (2')^3(5')] - (1')^4 [2100 (3')^2(4') + 1575 (2')(4')^2 + 2520 (2')(3')(5') + 630 (2')^2(6') ]  + (1')^5 [126 (4')(5') + 84 (3')(6') + 36 (2')(7')] - (1')^6 (8') ] * t^8 / 8!
%C A134685 ...
%C A134685 Substituting ((m-1)') for (m') in each partition and ignoring the (0') factors, the partitions in the brackets of P(n,t) become those of n-1 listed in Abramowitz and Stegun on page 831 (in the reversed order) and the number of partitions in P(n,t) is given by A000041(n-1).
%C A134685 Combinatorial interpretations are given in the link.
%C A134685 From _Tom Copeland_, Jul 10 2018: (Start)
%C A134685 Coefficients occurring in prolongation for the special Euclidean group SE(2) and special affine group SA(2) in the Olver presentation on moving frames (MFP) in slides 33 and 42. These are a result of applying an iterated derivative of the form h(x)d/dx = d/dy as in this entry (more generally as g(x) d/dx as discussed in A145271). See also p. 6 of Olver's paper on contact forms, but note that the 12 should be a 15 in the formula for the compositional inverse of S(t).
%C A134685 Change variables in the MFP to obtain connections to the partition polynomials Prt_n = n! * P(n,1) above. Let delta and beta in the formulas for the equi-affine curves in MFP be L and B, respectively, and D_y = (1/(L-B*u_x)) d/dx = (1/w_x) d/dx. Then v_(yy) = (1/B) [-w_(xx)/(w_x)^3] in MFP (there is an overall sign error in MFP for v_(yy) and higher derivatives w.r.t. y), and (d/dy)^n v = v_n = (1/B)* [(1/w_1)*(d/dx)]^(n-2) [-w_2/(w_1)^3] for n > 1, with w_n = (d/dx)^n w. Consequently, in the partition polynomials Prt_n for n > 1 here substitute (n') = -B*u_n = w_n for n > 1 and (1')  = L-B*u_1 = w_1, where u_n = (d/dx)^n u, and then divide by B. For example, v_4 = (1/B)*Prt_4 = (1/B)*4!*P(4,1) = (1/B) (L-B*u_n)^(-7) [-15*(-B*u_2)^3 + 10 (L-B*u_1)(-B*u_2)(-B*u_3) - (L-B*u_1)^2 (-B*u_4)], agreeing with v_4 in MFP except for the overall sign.
%C A134685 For the SE(2) transformation formulas in MFP, let w_x = cos(phi) + sin(phi)*u_x, and then the same transformations apply as above with cos(phi) and sin(phi) substituted for L and -B, respectively. (End)
%D A134685 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 831.
%D A134685 D. S. Alexander, A History of Complex Dynamics: From Schröder to Fatou to Julia, Friedrich Vieweg & Sohn, 1994, p. 10.
%D A134685 J. Riordan, Combinatorial Identities, Robert E. Krieger Pub. Co., 1979, (unsigned partition polynomials in Table 5.2 on p. 181, but may have errors).
%H A134685 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 A134685 O. Arnaldsson, <a href="https://web.archive.org/web/20221026042028/https://notendur.hi.is/~ornarnalds/files/malstofaglaerur.pdf">Élie Cartan’s Theory of Moving Frames</a>, slide presentation, 2014.
%H A134685 Tom Copeland, <a href="http://tcjpn.wordpress.com/2015/12/21/generators-inversion-and-matrix-binomial-and-integral-transforms/">Generators, Inversion, and Matrix, Binomial, and Integral Transforms</a>, 2015.
%H A134685 Tom Copeland, <a href="http://mathoverflow.net/questions/214927/important-formulas-in-combinatorics/215203#215203">Important formulas in combinatorics</a>, MathOverflow answer, 2015.
%H A134685 Tom Copeland, <a href="http://mathoverflow.net/questions/181284/compositional-inversion-and-generating-functions-in-algebraic-geometry">Compositional inversion and generating functions in algebraic geometry</a>, MathOverflow question, 2014.
%H A134685 Tom Copeland, <a href="https://tcjpn.wordpress.com/2014/09/17/compositional-inverse-pairs-the-inviscid-burgers-hopf-equation-and-the-stasheff-associahedra/">Compositional inverse pairs, the inviscid Burgers-Hopf equation, and the Stasheff associahedra</a>, 2014.
%H A134685 Tom Copeland, <a href="http://tcjpn.wordpress.com/2011/04/11/lagrange-a-la-lah/">Lagrange a la Lah</a>, 2011.
%H A134685 Tom Copeland, <a href="http://tcjpn.wordpress.com/2008/09/13/short-note-on-lagrange-inversion/">Short Note on Lagrange Inversion</a>, 2008.
%H A134685 Tom Copeland, <a href="https://tcjpn.wordpress.com/2018/01/23/formal-group-laws-and-binomial-sheffer-sequences/">Formal group laws and binomial Sheffer sequences</a>, 2018.
%H A134685 Bishal Deb and Alan D. Sokal, <a href="https://arxiv.org/abs/2507.18959">Higher-order Stirling cycle and subset triangles: Total positivity, continued fractions and real-rootedness</a>, arXiv:2507.18959 [math.CO], 2025. See p. 32.
%H A134685 Hector Figueroa and Jose M. Gracia-Bondia, <a href="https://arxiv.org/abs/hep-th/0408145">Combinatorial Hopf algebras in quantum field theory I</a>, arXiv:0408145 [hep-th], 2005, (p. 38).
%H A134685 Ezra Getzler, <a href="http://arxiv.org/abs/alg-geom/9612005">The semi-classical approximation for modular operads</a>, arXiv:alg-geom/9612005, 1996 (see p. 2).
%H A134685 Peter J. Olver, <a href="http://www.math.umn.edu/~olver/di_/contact.pdf">The canonical contact form</a>.
%H A134685 Peter J. Olver, <a href="http://www-users.math.umn.edu/~olver/t_/benasque1.pdf">Moving Frames</a>, slide presentation, 2009.
%H A134685 Andrew Elvey Price and Alan D. Sokal, <a href="https://arxiv.org/abs/2001.01468">Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials</a>, arXiv:2001.01468 [math.CO], 2020.
%H A134685 Jair Patrick Taylor, <a href="https://digital.lib.washington.edu/researchworks/handle/1773/36757">Formal group laws and hypergraph colorings</a>, doctoral thesis, Univ. of Wash., 2016, p. 95.
%F A134685 The bracketed partitions of P(n,t) are of the form (u_1)^e(1) (u_2)^e(2) ... (u_n)^e(n) with coefficients given by (-1)^(n-1+e(1)) * [2*(n-1)-e(1)]! / [2!^e(2)*e(2)!*3!^e(3)*e(3)! ... n!^e(n)*e(n)! ].
%F A134685 From _Tom Copeland_, Sep 05 2011: (Start)
%F A134685 Let h(t) = 1/(df(t)/dt)
%F A134685   = 1/Ev[u.*exp(u.*t)]
%F A134685   = 1/(u_1+(u_2)*t+(u_3)*t^2/2!+(u_4)*t^3/3!+...),
%F A134685   an e.g.f. for the partition polynomials of A133314
%F A134685   (signed A049019) with an index shift.
%F A134685   Then for the partition polynomials of A134685,
%F A134685   n!*P(n,t) = ((t*h(y)*d/dy)^n) y evaluated at y=0,
%F A134685   and the compositional inverse of f(t) is
%F A134685   g(t) = exp(t*h(y)*d/dy) y evaluated at y=0.
%F A134685   Also, dg(t)/dt = h(g(t)). (Cf. A000311 and A134991)(End)
%F A134685 From _Tom Copeland_, Oct 30 2011: (Start)
%F A134685 With exp[x* PS(.,t)] = exp[t*g(x)]=exp[x*h(y)d/dy] exp(t*y) eval. at y=0, the raising/creation and lowering/annihilation operators
%F A134685 defined by R PS(n,t)=PS(n+1,t) and L PS(n,t)= n*PS(n-1,t) are
%F A134685 R = t*h(d/dt) = t * 1/[u_1+(u_2)*d/dt+(u_3)*(d/dt)^2/2!+...],  and
%F A134685 L = f(d/dt)=(u_1)*d/dt+(u_2)*(d/dt)^2/2!+(u_3)*(d/dt)^3/3!+....
%F A134685 Then  P(n,t) = (t^n/n!) dPS(n,z)/dz  eval. at z=0. (Cf. A139605, A145271, and link therein to Mathemagical Forests for relation to planted trees on p. 13.) (End)
%F A134685 The bracketed partition polynomials of P(n,t) are also given by (d/dx)^(n-1) 1/[u_1 + u_2 * x/2! + u_3 * x^2/3! + ... + u_n * x^(n-1)/n!]^n evaluated at x=0. - _Tom Copeland_, Jul 07 2015
%F A134685 Equivalent matrix computation: Multiply the m-th diagonal (with m=1 the index of the main diagonal) of the lower triangular Pascal matrix by u_m = (d/dx)^m f(x) evaluated at x=0 to obtain the matrix UP with UP(n,k) = binomial(n,k) u_{n+1-k}. Then P(n,t) = (1, 0, 0, 0, ...) [UP^(-1) * S]^(n-1) FC * t^n/n!, where S is the shift matrix A129185, representing differentiation in the basis x^n//n!, and FC is the first column of UP^(-1), the inverse matrix of UP. These results follow from A145271 and A133314. - _Tom Copeland_, Jul 15 2016
%F A134685 Also, P(n,t) = (1, 0, 0, 0, ...) [UP^(-1) * S]^n (0, 1, 0, ..)^T * t^n/n! in agreement with A139605. - _Tom Copeland_, Aug 27 2016
%F A134685 From _Tom Copeland_, Sep 20 2016: (Start)
%F A134685 Let PS(n,u1,u2,...,un) = P(n,t) / (t^n/n!), i.e., the square-bracketed part of the partition polynomials in the expansion for the inverse in the comment section, with u_k = uk.
%F A134685 Also let PS(n,u1=1,u2,...,un) = PB(n,b1,b2,...,bK,...) where each bK represents the partitions of PS, with u1 = 1, that have K components or blocks, e.g., PS(5,1,u2,...,u5) = PB(5,b1,b2,b3,b4) = b1 + b2 + b3 + b4 with b1 = -u5, b2 = 15 u2 u4 + 10 u3^2, b3 = -105 u2^2 u3,  and b4 = 105 u2^4.
%F A134685 The relation between solutions of the inviscid Burgers' equation and compositional inverse pairs (cf. link and A086810) implies, for n > 2,  PB(n, 0 * b1, 1 * b2,..., (K-1) * bK, ...) = (1/2) * Sum_{k = 2..n-1} binomial(n+1,k) * PS(n-k+1,u_1=1,u_2,...,u_(n-k+1)) * PS(k,u_1=1,u_2,...,u_k).
%F A134685 For example, PB(5,0 * b1, 1 * b2, 2 * b3, 3 * b4) = 3 * 105 u2^4 - 2 * 105 u2^2 u3 + 1 * 15 u2 u4 + 1 * 10 u3^2 - 0 * u5 = 315 u2^4 - 210 u2^2 u3 + 15 u2 u4 + 10 u3^2 = (1/2) [2 * 6!/(4!*2!) * PS(2,1,u2) * PS(4,1,u2,...,u4) + 6!/(3!*3!) * PS(3,1,u2,u3)^2] = (1/2) * [ 2 * 6!/(4!*2!)  * (-u2) (-15 u2^3 + 10 u2 u3 - u4) + 6!/(3!*3!) * (3 u2^2 - u3)^2].
%F A134685 Also, PB(n,0*b1,1*b2,...,(K-1)*bK,...) =  d/dt t^(n-2)*PS(n,u1=1/t,u2,...,un)|_{t=1} = d/dt (1/t)*PS(n,u1=1,t*u2,...,t*un)|_{t=1}.
%F A134685 (End)
%F A134685 A recursion relation for computing each partition polynomial of this entry from the lower order polynomials and the coefficients of the Bell polynomials of A036040 is presented in the blog entry "Formal group laws and binomial Sheffer sequences." - _Tom Copeland_, Feb 06 2018
%e A134685 Examples and checks:
%e A134685 1) Let u_1 = -1 and u_n = 1 for n>1,
%e A134685 then f(t) = exp(u.*t) - u(0) = exp(t)-2t-1
%e A134685 and g(t) = [e.g.f. of signed A000311];
%e A134685 therefore, the row sums of unsigned [C(j,k)] are A000311 =
%e A134685 (0,1,1,4,26,236,2752,...) = (0,-P(1,1),2!*P(2,1),-3!*P(3,1),4!*P(4,1),...).
%e A134685 2) Let u_1 = -1 and u_n = (n-1)! for n>1,
%e A134685 then f(t) = -log(1-t)-2t
%e A134685 and g(t) = [e.g.f. of signed (0,A032188)]
%e A134685 with (0,A032188) = (0,1,1,5,41,469,6889,...) = (0,-P(1,1),2!*P(2,1),-3!P(3,1),...).
%e A134685 3) Let u_1 = -1 and u_n = (-1)^n (n-2)! for n>1, then
%e A134685 f(t) = (1+t) log(1+t) - 2t
%e A134685 and g(t) = [e.g.f. of signed (0,A074059)]
%e A134685 with (0,A074059) = (0,1,1,2,7,34,213,...) = (0,-P(1,1),2!*P(2,1),-3!*P(3,1),...).
%e A134685 4) Let u_1 = 1, u_2 = -1 and u_n = 0 for n>2,
%e A134685 then f(t) = t(1-t/2)
%e A134685 and g(t) = [e.g.f. of (0,A001147)] = 1 - (1-2t)^(1/2)
%e A134685 with (0,A001147) = (0,1,1,3,15,105,945...) =(0,P(1,1),2!*P(2,1),3!*P(3,1),...).
%e A134685 5) Let u_1 = 1, u_2 = -2 and u_n = 0 for n>2,
%e A134685 then f(t)= t(1-t)
%e A134685 and g(t) = t * [o.g.f. of A000108] = [1 - (1-4t)^(1/2)] / 2
%e A134685 with (0,A000108) = (0,1,1,2,5,14,42,...) = (0,P(1,1),P(2,1),P(3,1),...).
%e A134685 .
%e A134685 From _Peter Luschny_, Feb 19 2021: (Start)
%e A134685 Triangle starts:
%e A134685  [1]  1;
%e A134685  [2] -1;
%e A134685  [3]  3,     -1;
%e A134685  [4] -15,     10,    -1;
%e A134685  [5]  105,   -105,   [10, 15],  -1;
%e A134685  [6] -945,    1260,  [-280, -210], [35, 21],  -1;
%e A134685  [7]  10395, -17325, [6300, 3150], [-280, -1260, -378], [35, 56, 28], -1;
%e A134685  [8] -135135, 270270, [-138600, -51975], [15400, 34650, 6930], [-2100, -1575, -2520, -630], [126, 84, 36], -1
%e A134685 The coefficients can be seen as a refinement of the Ward numbers: Let R(n, k) = Sum T(n, k), where the sum collects adjacent terms with equal sign, as indicated by the square brackets in the table, then R(n+1, k+1) = (-1)^(n-k)*W(n, k), where W(n, k) are the Ward numbers A181996, for n >= 0 and 0 <= k <= n-1.  (End)
%t A134685 rows[n_] := {{1}}~Join~Module[{h = 1/(1 + Sum[u[k] y^k/k!, {k, n-1}] + O[y]^n), g = y, r}, r = Reap[Do[g = h D[g, y]; Sow[Expand[Normal@g /. {y -> 0}]], {k, n}]][[2, 1, ;;]]; Table[Coefficient[r[[k]], Product[u[t], {t, p}]], {k, 2, n}, {p, Reverse@Sort[Sort /@ IntegerPartitions[k-1]]}]];
%t A134685 rows[8] // Flatten (* _Andrei Zabolotskii_, Feb 19 2024 *)
%Y A134685 Cf. A145271, (A134991, A019538) = (reduced array, associated g(x)).
%Y A134685 Cf. A000108, A000311, A001147, A032188, A049019, A074059, A129185, A133314, A139605, A036040.
%Y A134685 Cf. A181996 (Ward numbers).
%K A134685 sign,tabf,changed
%O A134685 1,3
%A A134685 _Tom Copeland_, Jan 26 2008, Sep 13 2008
%E A134685 P(7,t) and P(8,t) data added by _Tom Copeland_, Jan 14 2016
%E A134685 Terms in rows 5-8 reordered by _Andrei Zabolotskii_, Feb 19 2024