OFFSET
0,12
COMMENTS
This sets an upper bound on the second order term of the complexity measure introduced by Franco, 2001 for symmetric Boolean functions. The sum of the terms for a given N is equal to 2^(N-1).
LINKS
L. Franco, A measure for the complexity of Boolean functions related to their implementation in neural networks, arXiv:cond-mat/0111169 [cond-mat.dis-nn], 2001.
L. Franco and S. A. Cannas, Non-glassy ground-state in a long-range antiferromagnetic frustrated model in the hypercubic cell, arXiv:cond-mat/0302412 [cond-mat.stat-mech], 2003; Phys. A 332 (2004), no. 1-4, 337-348.
FORMULA
T(n, N) = ((N-n)^2 + n^2 - N) * C(N, n) / (N^2 - N) n is the term for the series containing N+1 terms
From Peter Bala, Mar 20 2018: (Start)
Except for (n,k) = (1,0) the formula T(n,k) = C(n,k) - 2*C(n-1,n-k-1) + 2*C(n-2,n-k-2), where C(n,k) = n!/(k!*(n-k)!) for 0 <= k <= n, otherwise 0, appears to give the correct table entries.
Appears to equal A159853, the Riordan array ((1-2*x+2*x^2)/(1-x), x/(1-x)), except for the entry T(1,0). If this is correct then provided n =! 1 we have exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(1 + x + x^2/2! + x^3/3!) = 1 + 2*x + 2*x^2/2! + 4*x^3/3! + 8*x^4/4! + 15*x^5/5! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). (End)
EXAMPLE
Triangle begins:
1 N=0
1 1 N=1
1 0 1 N=2
1 1 1 1 N=3
1 2 2 2 1 N=4
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Leonardo Franco (Leonardo.Franco(AT)psy.ox.ac.uk), Sep 24 2003
STATUS
approved