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A127671 Cumulant expansion numbers: Coefficients in expansion of log(1 + Sum_{k>=1}x[k]*(t^k)/k!). 8
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 (list; graph; refs; listen; history; text; internal format)
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 nr. n the partitions of n are taken in the Abramowitz-Stegun 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_(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

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 [alternative scanned copy].

A. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, pp. 831-2.

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(x[k]*(t^k)/k!,k=1..infinity)).

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

a(n,m) = ((-1)^(m-1))*(m-1)!*M_3(n,m) with M_3(n,m) = A036040(n,m) (Abramowitz-Stegun M_3 numbers).

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

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 to n, binom(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

EXAMPLE

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]).

CROSSREFS

Cf. A133314, A263916.

Sequence in context: A190698 A077427 A107641 * A271724 A247641 A261876

Adjacent sequences:  A127668 A127669 A127670 * A127672 A127673 A127674

KEYWORD

sign,easy,tabf

AUTHOR

Wolfdieter Lang, Jan 23 2007

STATUS

approved

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Last modified June 27 14:33 EDT 2016. Contains 274260 sequences.