A007785 -id:A007785 - 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

A321282 Number of set partitions of [n^2] into n subsets having the same sum.

Original entry on oeis.org

1, 1, 1, 9, 2650, 100664383, 808087012923418

Offset: 0

Alois P. Heinz, Nov 01 2018

			a(3) = 9: 12345|69|78, 1239|456|78, 1248|357|69, 1257|348|69, 1347|258|69, 1356|249|78, 159|2346|78, 168|249|357, 159|267|348.

Wikipedia, Partition of a set

Cf. A007785, A275714, A317807, A321230.

b:= proc(l, n) option remember; `if`(n=0, 1, add(`if`(n>l[j],
       0, b(sort(subsop(j=l[j]-n, l)), n-1)), j=1..nops(l)))
    end:
a:= n-> b([n*(1+n^2)/2$n], n^2)/n!:
seq(a(n), n=0..5);

b[l_, n_] := b[l, n] = If[n == 0, 1, Sum[If[n > l[[j]], 0, b[Sort[ ReplacePart[l, j -> l[[j]] - n]], n-1]], {j, 1, Length[l]}]];
a[n_] := b[Table[n(1+n^2)/2, {n}], n^2]/n!;
a /@ Range[0, 5] (* Jean-François Alcover, May 04 2020, after Maple *)

a(n) = A275714(n^2,n).

cp's OEIS Frontend

A052456 Number of magic series of order n.

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