A068607 -id:A068607 - cp's OEIS frontend

A029895 Number of partitions of floor(n^2/2) with at most n parts and maximal height n.

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

Offset: 0

torsten.sillke(AT)lhsystems.com

nonn

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

			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}.

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

Vladimir Reshetnikov, Table of n, a(n) for n = 0..190
Eric W. Weisstein, q-Binomial Coefficient
Wikipedia, q-binomial

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

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

{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

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

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

A068606 Square table by antidiagonals of T(n,k)=nk(n+k+1).

Original entry on oeis.org

0, 0, 0, 0, 3, 0, 0, 8, 8, 0, 0, 15, 20, 15, 0, 0, 24, 36, 36, 24, 0, 0, 35, 56, 63, 56, 35, 0, 0, 48, 80, 96, 96, 80, 48, 0, 0, 63, 108, 135, 144, 135, 108, 63, 0, 0, 80, 140, 180, 200, 200, 180, 140, 80, 0, 0, 99, 176, 231, 264, 275, 264, 231, 176, 99, 0, 0, 120, 216, 288

Offset: 0

Henry Bottomley, Feb 24 2002

Considering partitions with up to n positive integers each no more than k (or equivalently paths of length n+k from one corner to the opposite corner of an n*k rectangle) there are C(n+k,n) such partitions (or paths); the mean of the sums of the partitions (or mean of the areas above the paths) is nk/2; and the variance of the sums of the partitions (or variance of the areas above the paths) is a(n)/12.

			Rows start:
0,0,0,0,0,...;
0,3,8,15,24,...;
0,8,20,36,56,...;
0,15,36,63,96,...;
etc.

Cf. A068607 for the same table as a triangle.

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A029895 Number of partitions of floor(n^2/2) with at most n parts and maximal height n.

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A068606 Square table by antidiagonals of T(n,k)=nk(n+k+1).

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cp's OEIS Frontend

A029895 Number of partitions of floor(n^2/2) with at most n parts and maximal height n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A068606 Square table by antidiagonals of T(n,k)=n*k*(n+k+1).

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Author

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Comments

Examples

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A068606 Square table by antidiagonals of T(n,k)=nk(n+k+1).