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A165438 Number a(n) of alternative sets of orthogonal contrasts available to partition variation between n levels of a categorical factor in analysis of variance, with each set described by a unique general linear model. 0
1, 3, 4, 8, 15, 34, 69, 152, 332, 751, 1698, 3905, 9020, 21051, 49356, 116505, 276217, 658091, 1573835, 3778152, 9098915, 21980209, 53241777, 129294912, 314714273, 767700735, 1876437054, 4595005570, 11271747564, 27695048780 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,2

COMMENTS

Each set has n-1 orthogonal contrasts.

REFERENCES

Doncaster, C. P. & Davey, A. J. H. (2007) Analysis of Variance and Covariance: How to Choose and Construct Models for the Life Sciences. Cambridge: Cambridge University Press.

LINKS

Table of n, a(n) for n=3..32.

C. P. Doncaster, Contrast sets

C. P. Doncaster, Orthogonal contrasts

C. P. Doncaster & A. J. H. Davey, Analysis of Variance and Covariance

FORMULA

For n=5,6,7: a(n) = -mod(n,2)*a([n-mod{n,2}]/2) + sum_{k=3..n-1} a[k]

For n>7: a(n) = -mod(n,2)*a([n-mod{n,2}]/2) + 2*a(n-1) + b(n) - b(n-1)

where b(m) = 0^mod(log(m,2),1) + mod(m-1,2)*0.5*a([m-mod{m,2}]/2)*(a[{m-mod(m,2)}/2]-1)

+ sum_{k=3..(m-1-mod[m-1,2])/2} a(m-k)*(a[k]-1)

EXAMPLE

A factor 'A' with n = 5 levels, has a(5) = 4 alternative sets of orthogonal

contrasts, each with n - 1 = 4 contrasts. The corresponding alternative

general linear models describing contrasts 'B', 'C', 'D', 'E' are:

B + C(B) + D(B) + E(D B)

B + C(B) + D(C B) + E(D C B)

B + C(B) + D(C B) + E(C B)

B + C(B) + D(B) + E(B)

CROSSREFS

Sequence in context: A337438 A007486 A027977 * A293781 A202025 A227615

Adjacent sequences: A165435 A165436 A165437 * A165439 A165440 A165441

KEYWORD

nonn

AUTHOR

C. Patrick Doncaster (cpd(AT)soton.ac.uk), Sep 18 2009

EXTENSIONS

Corrected and edited by C. P. Doncaster (cpd(AT)soton.ac.uk), Mar 02 2010

STATUS

approved

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Last modified November 27 09:16 EST 2022. Contains 358367 sequences. (Running on oeis4.)