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A029895
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Number of partitions of floor(n^2/2) with at most n parts and maximal height n.
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8
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1, 1, 2, 3, 8, 20, 58, 169, 526, 1667, 5448, 18084, 61108, 208960, 723354, 2527074, 8908546, 31630390, 113093022, 406680465, 1470597342, 5342750699, 19499227828, 71442850111, 262754984020, 969548468960, 3589093760726, 13323571588607, 49596793134484
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OFFSET
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0,3
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COMMENTS
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This is the maximum value for the distribution of partitions of (0 .. n^2) that fit in an n X n box; assuming the peak of a normal distribution 1/sqrt(variance*2*Pi) approximates to these partitions and using A068606 suggests C(2n,n)*sqrt(6/(Pi*n^2*(2n+1))) could be an approximation [within 0.3% for a(100)=88064925963069745337300842293630181021718294488842002448]; using Stirling's approximation gives the simpler (sqrt(3)/Pi)*4^n/n^2 [about 0.6% away for a(100)] though experimentation suggests that something like (sqrt(3)/Pi)*4^n/(n^2+3n/5+1/5) is closer [about 0.0001% away for a(100)]. - Henry Bottomley, Mar 13 2002
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REFERENCES
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R. A. Brualdi, H. J. Ryser, Combinatorial Matrix Theory, Cambridge Univ. Press, 1992.
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LINKS
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FORMULA
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Calculated using Cor. 6.3.3, Th. 6.3.6, Cor. 6.2.5 of Brualdi-Ryser. Table[T[Floor[n^2/2], n, n], {n, 0, 36}] with T[ ] defined as in A047993. a(n)=A067059(n, n).
a(n) equals the central coefficient of q in the central q-binomial coefficients for n>0: a(n) = [q^([n^2/2])] Product_{i=1..n} (1-q^(n+i))/(1-q^i), with a(0)=1. - Paul D. Hanna, Feb 15 2007
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EXAMPLE
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a(4)=8 because the partitions of Floor[4^2 /2] that fit inside a 4 X 4 box are {4, 4}, {4, 3, 1}, {4, 2, 2}, {4, 2, 1, 1}, {3, 3, 2}, {3, 3, 1, 1}, {3, 2, 2, 1}, {2, 2, 2, 2}.
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MATHEMATICA
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Table[Coefficient[Expand[FunctionExpand[QBinomial[2 n, n, q]]], q, Floor[n^2/2]], {n, 0, 30}] (* Vladimir Reshetnikov, Oct 09 2016 *)
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PROG
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(PARI) {a(n)=if(n==0, 1, polcoeff(prod(i=1, n, (1-q^(n+i))/(1-q^i)), n^2\2, q))} \\ Paul D. Hanna, Feb 15 2007
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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torsten.sillke(AT)lhsystems.com
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EXTENSIONS
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STATUS
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approved
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