

A127671


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


16



1, 1, 1, 1, 3, 2, 1, 4, 3, 12, 6, 1, 5, 10, 20, 30, 60, 24, 1, 6, 15, 10, 30, 120, 30, 120, 270, 360, 120, 1, 7, 21, 35, 42, 210, 140, 210, 210, 1260, 630, 840, 2520, 2520, 720, 1, 8, 28, 56, 35, 56, 336, 560, 420, 560, 336, 2520, 1680, 5040, 630, 1680, 13440, 10080, 6720
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OFFSET

1,5


COMMENTS

Connected objects from general (disconnected) objects.
The row lengths of this array is p(n):=A000041(n) (partition numbers).
In row n the partitions of n are taken in the AbramowitzStegun order.
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.).
The inverse relation (disconnected from connected objects) is found in A036040.
(d/da(1))p_n[a(1),a(2),...,a(n)] = n b_(n1)[a(1),a(2),...,a(n1)], 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
See notes on relation to Appell sequences in a differently ordered version of this array A263634.  Tom Copeland, Sep 13 2016
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 nth cumulant is (1)^(n1) * (n1)! * t.  Tom Copeland, Jul 25 2019
From Tom Copeland, Oct 15 2020: (Start)
With a_n = n! * b_n = (n1)! * c_n for n > 0, represent a function with f(0) = a_0 = b_0 = 1 as an
A) exponential generating function (e.g.f), or formal Taylor series: f(x) = e^{a.x} = 1 + Sum_{n > 0} a_n * x^n/n!
B) ordinary generating function (o.g.f.), or formal power series: f(x) = 1/(1b.x) = 1 + Sum_{n > 0} b_n * x^n
C) logarithmic generating function (l.g.f): f(x) = 1  log(1  c.x) = 1 + Sum_{n > 0} c_n * x^n /n.
Expansions of log(f(x)) are given in
I) A127671 and A263634 for the e.g.f: log[ e^{a.*x} ] = e^{L.(a_1,a_2,...)x} = Sum_{n > 0} L_n(a_1,...,a_n) * x^n/n!, the logarithmic polynomials, cumulant expansion polynomials
II) A263916 for the o.g.f.: log[ 1/(1b.x) ] = log[ 1  F.(b_1,b_2,...)x ] = Sum_{n > 0} F_n(b_1,...,b_n) * x^n/n, the Faber polynomials.
Expansions of exp(f(x)1) are given in
III) A036040 for an e.g.f: exp[ e^{a.x}  1 ] = e^{BELL.(a_1,...)x}, the Bell/Touchard/exponential partition polynomials, a.k.a. the Stirling partition polynomials of the second kind
IV) A130561 for an o.g.f.: exp[ b.x/(1b.x) ] = e^{LAH.(b.,...)x}, the Lah partition polynomials
V) A036039 for an l.g.f.: exp[ log(1c.x) ] = e^{CIP.(c_1,...)x}, the cycle index polynomials of the symmetric groups S_n, a.k.a. the Stirling partition polynomials of the first kind.
Since exp and log are a compositional inverse pair, one can extract the indeterminates of the log set of partition polynomials from the exp set and vice versa. For a discussion of the relations among these polynomials and the combinatorics of connected and disconnected graphs/maps, see Novak and LaCroix on classical moments and cumulants and the two books on statistical mechanics referenced in A036040. (End)


REFERENCES

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


LINKS

Table of n, a(n) for n=1..63.
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, pp. 8312.
T. Copeland, The creation / raising operators for Appell sequences
Wolfdieter Lang, First 10 rows of cumulant numbers and polynomials.


FORMULA

E.g.f. for multivariate row polynomials A(t) := log(1 + Sum_{k>=1} x[k]*(t^k)/k!).
Row n polynomial p_n(x[1],...,x[n]) = [(t^n)/n!]A(t).
a(n,m) = A264753(n, m) * M_3(n,m) with M_3(n,m) = A036040(n,m) (AbramowitzStegun M_3 numbers).  corrected by Johannes W. Meijer, Jul 12 2016
p_n(x[1],...,x[n]) = (n1)!*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
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[nk] 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_(n1)(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


EXAMPLE

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


CROSSREFS

Cf. A133314, A263916, A263634.
Cf. A008275.
Cf. A036039, A036040, A130561.
Sequence in context: A077427 A107641 A299352 * A271724 A247641 A336886
Adjacent sequences: A127668 A127669 A127670 * A127672 A127673 A127674


KEYWORD

sign,easy,tabf


AUTHOR

Wolfdieter Lang, Jan 23 2007


STATUS

approved



