login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A127671 Cumulant expansion numbers: Coefficients in expansion of log(1 + Sum_{k>=1}x[k]*(t^k)/k!). 13
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 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

See notes on relation to Appell sequences in a differently ordered version of this array A263634. - Tom Copeland, Sep 13 2016

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]. [Link broken]

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) = 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

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, A263634.

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 8 15:08 EST 2016. Contains 278945 sequences.