0,3

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 (se16.btinternet.com), March 13 2002.

R. A. Brualdi, H. J. Ryser, Combinatorial Matrix Theory, Cambridge Univ. Press, 1992.

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 (pauldhanna(AT)juno.com), Feb 15 2007

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}.

(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 (pauldhanna(AT)juno.com), Feb 15 2007

Cf. A000569, A004250, A004251, A029889, A047993, A067059, A068607.

Sequence in context: A092031 A095341 A167123 this_sequence A073268 A073190 A066051

Adjacent sequences: A029892 A029893 A029894 this_sequence A029896 A029897 A029898

nonn

TORSTEN.SILLKE(AT)LHSYSTEMS.COM

More terms and comments from Wouter Meeussen (wouter.meeussen(AT)pandora.be), Aug 14 2001.

Edited by Henry Bottomley (se16.btinternet.com), Feb 17 2002.