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A006847
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Extreme points of set of n X n symmetric doubly-stochastic matrices.
(Formerly M1471)
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1
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1, 1, 2, 5, 14, 58, 238, 1516, 9020, 79892, 635984, 7127764, 70757968, 949723600, 11260506056, 175400319992, 2416123951952, 42776273847184, 671238787733920, 13302324582892048, 234257439470319776, 5135062189842955616, 100292619307729965152
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Contribution from Jeffrey Shallit (shallit(AT)cs.uwaterloo.ca), Dec 05 2009: (Start)
A recurrence formula for this sequence is:
A(n) = A(n-1) + (n-1)^2*A(n-2) - ((n-1)*(n-2)/2)*A(n-3) - (n-1)*(n-2)*(n-3)*A(n-4)
This is given in Stanley, 1980, p. 180, except that there is a typographical error in Stanley's formula (corrected here). (End)
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REFERENCES
| M. Katz, On the extreme points of a certain convex polytope, J. Combin. Theory, 8 (1970), 417-423.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Differentiably finite power series, European J. Combin., 1 (1980), 175-188.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.24(b).
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FORMULA
| E.g.f.: ((1+x)/(1-x))^(1/4)*exp(1/2*x+1/2*x^2).
A(n) = A(n-1) + (n-1)^2*A(n-2) - ((n-1)*(n-2)/2)*A(n-3) - (n-1)*(n-2)*(n-3)*A(n-4) [From Jeffrey Shallit (shallit(AT)cs.uwaterloo.ca), Dec 05 2009]
a(n)=n!*sum((if r=0 then 1 else sum((1/2)^k*binomial(k,r-k)/k!,k,1,r))*b(n-r),r,0,n), b(n)=if n=0 then 1 else 1/2+sum(2^m*binomial(n-1,m-1)*(-1)^(m-1)*((1/4)^m*sum(sum(binomial(j,m-1-3*k+2*j)*binomial(k,j)*3^(3*k-m+1-j)*2^(m-1-5*k+3*j)*(-1)^(m-1-3*k),j,0,k)*binomial(k+m-1,m-1),k,1,m-1))/m,m,2,n). [From Kruchinin Vladimir (kru(AT)ie.tusur.ru), Sep 09 2010]
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MATHEMATICA
| max = 22; f[x_] = ((1+x)/(1-x))^(1/4)*Exp[x/2+x^2/2]; CoefficientList[ Series[ f[x], {x, 0, max}], x]*Range[0, max]! (* From Jean-François Alcover, Nov 14 2011, after g.f. *)
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PROG
| (Other) b(n):=if n=0 then 1 else 1/2+sum(2^m*binomial(n-1, m-1)*(-1)^(m-1)*((1/4)^m*sum(sum(binomial(j, m-1-3*k+2*j)*binomial(k, j)*3^(3*k-m+1-j)*2^(m-1-5*k+3*j)*(-1)^(m-1-3*k), j, 0, k)*binomial(k+m-1, m-1), k, 1, m-1))/m, m, 2, n); a(n):=n!*sum((if r=0 then 1 else sum((1/2)^k*binomial(k, r-k)/k!, k, 1, r))*b(n-r), r, 0, n); (for Maxima) [From Kruchinin Vladimir (kru(AT)ie.tusur.ru), Sep 09 2010]
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CROSSREFS
| Cf. A053553.
Sequence in context: A047042 A174795 A110043 * A008286 A049082 A158095
Adjacent sequences: A006844 A006845 A006846 * A006848 A006849 A006850
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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