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A127671 Cumulant expansion numbers: Coefficients in expansion of log(1+sum(x[k]*(t^k)/k!,k=1..infinity)). 3
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.

Wolfdieter Lang, First 10 rows of cumulant numbers and polynomials.

FORMULA

E.g.f. for multivariate row polynomials A(t):=ln(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).

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

Sequence in context: A190698 A077427 A107641 * A210797 A222220 A193815

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 April 19 21:10 EDT 2014. Contains 240777 sequences.