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A127671 Cumulant expansion numbers: Coefficients in expansion of log(1 + Sum_{k>=1} x[k]*(t^k)/k!). 14


%S 1,1,-1,1,-3,2,1,-4,-3,12,-6,1,-5,-10,20,30,-60,24,1,-6,-15,-10,30,

%T 120,30,-120,-270,360,-120,1,-7,-21,-35,42,210,140,210,-210,-1260,

%U -630,840,2520,-2520,720,1,-8,-28,-56,-35,56,336,560,420,560,-336,-2520,-1680,-5040,-630,1680,13440,10080,-6720

%N Cumulant expansion numbers: Coefficients in expansion of log(1 + Sum_{k>=1} x[k]*(t^k)/k!).

%C Connected objects from general (disconnected) objects.

%C The row lengths of this array is p(n):=A000041(n) (partition numbers).

%C In row n the partitions of n are taken in the Abramowitz-Stegun order.

%C One could call the unsigned numbers |a(n,k)| M_5 (similar to M_0, M_1, M_2, M_3 and M_4 given in A111786, A036038, A036039, A036040 and A117506, resp.).

%C The inverse relation (disconnected from connected objects) is found in A036040.

%C (d/da(1))p_n[a(1),a(2),...,a(n)] = n b_(n-1)[a(1),a(2),...,a(n-1)], where p_n are the partition polynomials of the cumulant generator A127671 and b_n are the partition polynomials for A133314. - _Tom Copeland_, Oct 13 2012

%C See notes on relation to Appell sequences in a differently ordered version of this array A263634. - _Tom Copeland_, Sep 13 2016

%C Given a binomial Sheffer polynomial sequence defined by the e.g.f. exp[t * f(x)] = Sum_{n >= 0} p_n(t) * x^n/n!, the cumulants formed from these polynomials are the Taylor series coefficients of f(x) multipied by t. An example is the sequence of the Stirling polynomials of the first kind A008275 with f(x) = log(1+x), so the n-th cumulant is (-1)^(n-1) * (n-1)! * t. - _Tom Copeland_, Jul 25 2019

%D C. Itzykson and J.-M. Drouffe, Statistical field theory, vol. 2, p. 413, eq.(13), Cambridge University Press, (1989).

%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, pp. 831-2.

%H T. Copeland, <a href="http://tcjpn.wordpress.com/2015/11/21/the-creation-raising-operators-for-appell-sequences/">The creation / raising operators for Appell sequences</a>

%H Wolfdieter Lang, <a href="/A127671/a127671.pdf">First 10 rows of cumulant numbers and polynomials.</a>

%F E.g.f. for multivariate row polynomials A(t) := log(1 + Sum_{k>=1} x[k]*(t^k)/k!).

%F Row n polynomial p_n(x[1],...,x[n]) = [(t^n)/n!]A(t).

%F a(n,m) = A264753(n, m) * M_3(n,m) with M_3(n,m) = A036040(n,m) (Abramowitz-Stegun M_3 numbers). - corrected by _Johannes W. Meijer_, Jul 12 2016

%F p_n(x[1],...,x[n]) = -(n-1)!*F(n,x[1],x[2]/2!,..,x[n]/n!) in terms of the Faber polynomials F(n,b1,..,bn) of A263916. - _Tom Copeland_, Nov 17 2015

%F With D = d/dz and M(0) = 1, the differential operator R = z + d(log(M(D))/dD = z + d(log(1 + x[1] D + x[2] D^2/2! + ...))/dD = z + p.*exp(p.D) = z + Sum_{n>=0} p_(n+1)(x[1],..,x[n]) D^n/n! is the raising operator for the Appell sequence A_n(z) = (z + x[.])^n = Sum_{k=0..n} binomial(n,k) x[n-k] z^k with the e.g.f. M(t) e^(zt), i.e., R A_n(z) = A_(n+1)(z) and dA_n(z)/dz = n A_(n-1)(z). The operator Q = z - p.*exp(p.D) generates the Appell sequence with e.g.f. e^(zt) / M(t). - _Tom Copeland_, Nov 19 2015

%e Row n=3: [1,-3,2] stands for the polynomial 1*x[3] - 3*x[1]*x[2] + 2*x[1]^3 (the Abramowitz-Stegun order of the p(3)=3 partitions of n=3 is [3],[1,2],[1^3]).

%Y Cf. A133314, A263916, A263634.

%Y Cf. A008275.

%K sign,easy,tabf

%O 1,5

%A _Wolfdieter Lang_, Jan 23 2007

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