A068606 -id:A068606 - cp's OEIS frontend

A052456 Number of magic series of order n.

1, 1, 2, 8, 86, 1394, 32134, 957332, 35154340, 1537408202, 78132541528, 4528684996756, 295011186006282, 21345627856836734, 1698954263159544138, 147553846727480002824, 13888244935445960871352, 1408407905312396429259944, 153105374581396386625831530

Offset: 0

Eric W. Weisstein

Henry Bottomley's narrowing gap could be confirmed for 2 < n <= 64. - Walter Trump, Jan 21 2005

A new algorithm was found by Robert Gerbicz. Now the enumeration of magic series of orders greater than 100 is possible. - Walter Trump, May 05 2006

			a(3) = 8 since a magic square of order 3 would require a row sum of 15=(1+2+...+9)/3 and there are 8 ways of writing 15 as the sum of three distinct positive numbers up to 9: 1+5+9, 1+6+8, 2+4+9, 2+5+8, 2+6+7, 3+4+8, 3+5+7, 4+5+6.

M. Kraitchik, Magic Series. Section 7.13.3 in Mathematical Recreations, New York, W. W. Norton, pp. 143 and 183-186, 1942.

T. D. Noe and Alois P. Heinz, Table of n, a(n) for n = 0..150 (from Gerbicz and Trump)
H. Bottomley, Partition and composition calculator
H. Bottomley and W. Trump, First 36 terms
Walter Trump, Magic Squares.
Eric Weisstein's World of Mathematics, Magic Series
Eric Weisstein's World of Mathematics, Multimagic Series
Robert Gerbicz, Walter Trump, First 150 terms
Robert Gerbicz, C-program to generate the sequence

Cf. A007785, A052457, A052458. A100568 is the same sequence times n!.

Main diagonal of A204459. - Alois P. Heinz, Jan 18 2012

$RecursionLimit = 1000; b[n_, i_, t_] /; i < t || n < t*((t + 1)/2) || n > t*((2*i - t + 1)/2) = 0; b[0, , ] = 1; b[n_, i_, t_] := b[n, i, t] = b[n, i - 1, t] + If[n < i, 0, b[n - i, i - 1, t - 1]]; a[, 0] = 1; a[0, ] = 0; a[n_, k_] :=  With[{s = k*(k*n + 1)}, If[Mod[s, 2] == 1, 0, b[s/2, k*n, k]]]; a[n_] := a[n] = a[n, n]; Table[Print[a[n]]; a[n], {n, 0, 18}] (* Jean-François Alcover, Aug 15 2013, after Alois P. Heinz *)

a(n) = A067059(n, n*(n-1)) = r(n, n*(n-1), n^2*(n-1)/2) where r(n, m, k) is a restricted partition function giving the number of partitions of k into at most n positive parts each no more than m. - Henry Bottomley, Feb 25 2002.
It seems a(n) (at least for 2A068606 and assuming the peak of a normal distribution = 1/sqrt(variance*2*Pi) - Henry Bottomley, Feb 25 2002.
a(n) ~ sqrt(3) * exp(n-1/2) * n^(n-3) / Pi. - Vaclav Kotesovec, Sep 05 2014

Edited and ten more terms from Henry Bottomley, Feb 16 2002

Terms through a(36) added to attached web page, Feb 04 2005

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

Original entry on oeis.org

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

A068607 Triangle of T(n,k)=nk(n+k+1) with n>=k>=0.

Original entry on oeis.org

0, 0, 3, 0, 8, 20, 0, 15, 36, 63, 0, 24, 56, 96, 144, 0, 35, 80, 135, 200, 275, 0, 48, 108, 180, 264, 360, 468, 0, 63, 140, 231, 336, 455, 588, 735, 0, 80, 176, 288, 416, 560, 720, 896, 1088, 0, 99, 216, 351, 504, 675, 864, 1071, 1296, 1539, 0, 120, 260, 420, 600

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.

			  0
  0   3
  0   8  20
  0  15  36  63
  0  24  56  96 144
  0  35  80 135 200 275
  0  48 108 180 264 360 468
  0  63 140 231 336 455 588 735
  0  80 176 288 416 560 720 896 1088

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A068606 for the same table as a square array.

Flatten[Table[n*k*(n+k+1),{n,0,10},{k,0,n}]] (* Harvey P. Dale, May 17 2015 *)

cp's OEIS Frontend

A052456 Number of magic series of order n.

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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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A068607 Triangle of T(n,k)=nk(n+k+1) with n>=k>=0.

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cp's OEIS Frontend

A052456 Number of magic series of order n.

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

Views

Author

Keywords

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Crossrefs

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Mathematica

PARI

Formula

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A068607 Triangle of T(n,k)=n*k*(n+k+1) with n>=k>=0.

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A068607 Triangle of T(n,k)=nk(n+k+1) with n>=k>=0.