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A029895 Number of partitions of floor(n^2/2) with at most n parts and maximal height n. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
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
Bisection of A277218 with even indexes. - Vladimir Reshetnikov, Oct 09 2016
REFERENCES
R. A. Brualdi, H. J. Ryser, Combinatorial Matrix Theory, Cambridge Univ. Press, 1992.
LINKS
Vladimir Reshetnikov, Table of n, a(n) for n = 0..190
Eric W. Weisstein, q-Binomial Coefficient
Wikipedia, q-binomial
FORMULA
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
EXAMPLE
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}.
MATHEMATICA
Table[Coefficient[Expand[FunctionExpand[QBinomial[2 n, n, q]]], q, Floor[n^2/2]], {n, 0, 30}] (* Vladimir Reshetnikov, Oct 09 2016 *)
PROG
(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
CROSSREFS
Sequence in context: A254533 A095341 A167123 * A073268 A073190 A066051
KEYWORD
nonn
AUTHOR
torsten.sillke(AT)lhsystems.com
EXTENSIONS
More terms and comments from Wouter Meeussen, Aug 14 2001
Edited by Henry Bottomley, Feb 17 2002
a(27)-a(28) from Alois P. Heinz, Oct 31 2018
STATUS
approved

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