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Number of partitions of floor(n^2/2) with at most n parts and maximal height n.
8

%I #33 Sep 25 2021 04:20:35

%S 1,1,2,3,8,20,58,169,526,1667,5448,18084,61108,208960,723354,2527074,

%T 8908546,31630390,113093022,406680465,1470597342,5342750699,

%U 19499227828,71442850111,262754984020,969548468960,3589093760726,13323571588607,49596793134484

%N Number of partitions of floor(n^2/2) with at most n parts and maximal height n.

%C 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

%C Bisection of A277218 with even indexes. - _Vladimir Reshetnikov_, Oct 09 2016

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

%H Vladimir Reshetnikov, <a href="/A029895/b029895.txt">Table of n, a(n) for n = 0..190</a>

%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/q-BinomialCoefficient.html">q-Binomial Coefficient</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Q-binomial">q-binomial</a>

%F 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).

%F 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

%e 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}.

%t Table[Coefficient[Expand[FunctionExpand[QBinomial[2 n, n, q]]], q, Floor[n^2/2]], {n, 0, 30}] (* _Vladimir Reshetnikov_, Oct 09 2016 *)

%o (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

%Y Cf. A000569, A004250, A004251, A029889, A047993, diagonal of A067059, A068607, A277218.

%K nonn

%O 0,3

%A torsten.sillke(AT)lhsystems.com

%E More terms and comments from _Wouter Meeussen_, Aug 14 2001

%E Edited by _Henry Bottomley_, Feb 17 2002

%E a(27)-a(28) from _Alois P. Heinz_, Oct 31 2018