A052456 -id:A052456 - cp's OEIS frontend

A204459 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of k-element subsets that can be chosen from {1,2,...,kn} having element sum k(k*n+1)/2.

1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 2, 1, 1, 0, 1, 0, 3, 0, 1, 0, 1, 8, 8, 4, 1, 1, 0, 1, 0, 33, 0, 5, 0, 1, 0, 1, 58, 141, 86, 25, 6, 1, 1, 0, 1, 0, 676, 0, 177, 0, 7, 0, 1, 0, 1, 526, 3370, 3486, 1394, 318, 50, 8, 1, 1, 0, 1, 0, 17575, 0, 11963, 0, 519, 0, 9, 0, 1

Offset: 0

Alois P. Heinz, Jan 15 2012

A(n,k) is the number of partitions of k*(k*n+1)/2 into k distinct parts <=k*n.

A(n,k) = 0 if k>0 and (n = 0 or k*(k*n+1) mod 2 = 1).

			A(0,0) = 1: {}.
A(1,1) = 1: {1}.
A(5,1) = 1: {3}.
A(1,5) = 1: {1,2,3,4,5}.
A(2,2) = 2: {1,4}, {2,3}.
A(3,2) = 3: {1,6}, {2,5}, {3,4}.
A(2,3) = 0: no subset of {1,2,3,4,5,6} has element sum 3*(3*2+1)/2 = 21/2.
A(4,2) = 4: {1,8}, {2,7}, {3,6}, {4,5}.
A(3,3) = 8: {1,5,9}, {1,6,8}, {2,4,9}, {2,5,8}, {2,6,7}, {3,4,8}, {3,5,7}, {4,5,6}.
A(2,4) = 8: {1,2,7,8}, {1,3,6,8}, {1,4,5,8}, {1,4,6,7}, {2,3,5,8}, {2,3,6,7}, {2,4,5,7}, {3,4,5,6}.
Square array A(n,k) begins:
  1, 0, 0,  0,   0,    0,     0,      0, ...
  1, 1, 1,  1,   1,    1,     1,      1, ...
  1, 0, 2,  0,   8,    0,    58,      0, ...
  1, 1, 3,  8,  33,  141,   676,   3370, ...
  1, 0, 4,  0,  86,    0,  3486,      0, ...
  1, 1, 5, 25, 177, 1394, 11963, 108108, ...
  1, 0, 6,  0, 318,    0, 32134,      0, ...
  1, 1, 7, 50, 519, 5910, 73294, 957332, ...

Alois P. Heinz, Antidiagonals d=0..60

Rows n=0-10 give: A000007, A000012, A063074, A109655, A204460, A204461, A204462, A204463, A204464, A204465, A204466.
Columns k=0..10 give: A000012, A000035, A001477, A204467, A204468, A204469, A204470, A204471, A204472, A204473, A204474.
Main diagonal gives: A052456.

b:= proc(n, i, t) option remember;
      `if`(it*(2*i-t+1)/2, 0,
      `if`(n=0, 1, b(n, i-1, t) +`if`(n

b[n_, i_, t_] /; it*((2*i-t+1)/2) = 0; b[0, , ] = 1; b[n_, i_, t_] := b[n, i, t] = b[n, i-1, t] + If[n, 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]]]; Flatten[ Table[ a[n, d-n], {d, 0, 15}, {n, 0, d}]] (* Jean-François Alcover, Jun 15 2012, translated from Maple, after Alois P. Heinz *)

A067059 Square array read by antidiagonals of partitions which half fill an n*k box, i.e., partitions of floor(nk/2) or ceiling(nk/2) into up to n positive integers, each no more than k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 3, 5, 5, 3, 1, 1, 1, 1, 4, 6, 8, 6, 4, 1, 1, 1, 1, 4, 8, 12, 12, 8, 4, 1, 1, 1, 1, 5, 10, 18, 20, 18, 10, 5, 1, 1, 1, 1, 5, 13, 24, 32, 32, 24, 13, 5, 1, 1, 1, 1, 6, 15, 33, 49, 58, 49, 33, 15, 6, 1, 1, 1, 1, 6

Offset: 0

Henry Bottomley, Feb 17 2002

The number of partitions of m into up to n positive integers each no more than k is maximized for given n and k by m=floor(nk/2) or ceiling(nk/2) (and possibly some other values).

			Rows start:
1, 1, 1, 1, 1, 1, ...;
1, 1, 1, 1, 1, 1, ...;
1, 1, 2, 2, 3, 3, ...;
1, 1, 2, 3, 5, 6, ...;
1, 1, 3, 5, 8, 12, ...; etc.
T(4,5)=12 since 10 can be partitioned into
5+5, 5+4+1, 5+3+2, 5+3+1+1, 5+2+2+1, 4+4+2, 4+3+3,
4+4+1+1, 4+3+2+1, 4+2+2+2, 3+3+3+1, and 3+3+2+2.

As this is symmetric, rows and columns each include A000012 twice, A008619, A001971, A001973, A001975, A001977, A001979 and A001981. Diagonal is A029895. T(n, n*(n-1)) is the magic series A052456.

A067059 := proc(n,k)
    local m,a1,a2 ;
    a1 := 0 ;
    m := floor(n*k/2) ;
    for L in combinat[partition](m) do
        if nops(L) <= n then
            if max(op(L)) <= k then
                a1 := a1+1 ;
            end if ;
        end if;
    end do:
    a2 := 0 ;
    m := ceil(n*k/2) ;
    for L in combinat[partition](m) do
        if nops(L) <= n then
            if max(op(L)) <= k then
                a2 := a2+1 ;
            end if ;
        end if;
    end do:
    max(a1,a2) ;
end proc:
for d from 0 to 12 do
    for k from 0 to d do
        printf("%d,",A067059(d-k,k)) ;
    end do:
end do: # R. J. Mathar, Nov 13 2016

t[n_, k_] := Length[ IntegerPartitions[ Floor[n*k/2], n, Range[k]]]; Flatten[ Table[ t[n-k , k], {n, 0, 13}, {k, 0, n}]] (* Jean-François Alcover, Jan 02 2012 *)

def A067059(n, k):
    return Partitions((n*k)//2, max_length=n, max_part=k).cardinality()
for n in (0..9): [A067059(n,k) for k in (0..9)] # Peter Luschny, May 05 2014

A052457 Number of bimagic series for squares of order n.

Original entry on oeis.org

1, 0, 0, 2, 8, 98, 1844, 38039, 949738, 24643236, 947689757, 45828982764, 2151748695931, 123821075526032, 8131094055190149, 573957471153552576, 44010987379157415768, 3655486139293429450720, 333633403912637510806972

Offset: 1

Eric W. Weisstein

Christian Boyer, Multimagic Squares
Eric Weisstein's World of Mathematics, Multimagic Series.

Cf. A052456, A052458, A090037, A106646, A090653.

Extended Nov 08 2003, using the values of a(3) through a(11) from Christian Boyer's web site. - N. J. A. Sloane.
a(12) from Christian Boyer (cboyer(AT)club-internet.fr), Nov 05 2004
a(13)-a(15) computed by Lorenz Schlangen and later independently confirmed by Walter Trump. a(16) computed by Walter Trump. - Christian Boyer (cboyer(AT)club-internet.fr), Oct 04 2005
a(17) computed by Walter Trump and communicated by Christian Boyer (cboyer(AT)club-internet.fr), Apr 02 2007
a(18)-a(19) computed by Michael Quist Christian Boyer (cboyer(AT)club-internet.fr), Feb 06 2009

A052458 Number of trimagic series for squares of order n.

Original entry on oeis.org

1, 0, 0, 2, 2, 0, 0, 121, 126, 0, 31187, 2226896, 17265701, 0, 69303997733

Offset: 1

Eric W. Weisstein

Asymptotic results are presented in Quist for magic cube and hypercube series, bimagic series, and trimagic series. - Jonathan Vos Post, Jun 04 2013

M. Kraitchik, Mathematical Recreations, 1942, see Section 7.10.

Christian Boyer, Multimagic Squares
Michael Quist, Asymptotic enumeration of magic series, arXiv:1306.0616v1 [math.CO], June 3, 2013.
Eric Weisstein's World of Mathematics, Multimagic Series

Cf. A052456, A052457, A090653, A092312, A090037.

Corrected and extended Nov 15 2003, using the values of a(3) through a(12) from Christian Boyer's web site. - N. J. A. Sloane
One more term from Christian Boyer (cboyer(AT)club-internet.fr), Nov 05 2004
One further term from Christian Boyer (cboyer(AT)club-internet.fr), May 30 2005
a(15) computed by Michael Quist, and communicated by Christian Boyer (cboyer(AT)club-internet.fr), Feb 06 2009

A090037 Number of tetramagic (4-multimagic) series for squares of order n.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 106, 555, 0, 0, 235275

Offset: 1

Eric W. Weisstein, Nov 19 2003

Christian Boyer, Multimagic Squares
Eric Weisstein's World of Mathematics, Magic Series

Cf. A052456, A052457, A052458, A090653, A092312, A106646.

One more term from Christian Boyer (cboyer(AT)club-internet.fr), Nov 05 2004
One further term from Christian Boyer (cboyer(AT)club-internet.fr), May 30 2005
a(15)=0 proved by Robert Gerbitz and communicated by Christian Boyer (cboyer(AT)club-internet.fr), Apr 02 2007
a(16) computed by Michael Quist, and communicated by Christian Boyer (cboyer(AT)club-internet.fr), Feb 06 2009

A007785 Number of sets of positive integers <= n^2 whose sum is (n^3 + n)/2.

Original entry on oeis.org

1, 1, 2, 17, 306, 10828, 654857, 63019177, 9183937890, 1953896126383, 589909767142505, 247074213707554144, 140902072248206260266, 107704589610917073318533, 108877374411946899963718973, 143864444783939220165210185294, 245934054410000090878614435736720

Offset: 0

Hidetoshi MINO [ mino(AT)hep.esb.yamanashi.ac.jp, mino(AT)mino.scri.fsu.edu ]

nonn

			a(2) = 2: {1,4}, {2,3}.
a(3) = 17: {6,9}, {7,8}, {1,5,9}, {1,6,8}, {2,4,9}, {2,5,8}, {2,6,7}, {3,4,8}, {3,5,7}, {4,5,6}, {1,2,3,9}, {1,2,4,8}, {1,2,5,7}, {1,3,4,7}, {1,3,5,6}, {2,3,4,6}, {1,2,3,4,5}.

Alois P. Heinz, Table of n, a(n) for n = 0..40 (terms n = 1..18 from Sean A. Irvine)

Cf. A052456.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i*(i+1)/2n, 0, b(n-i, min(n-i, i-1)))))
    end:
a:= n-> (s-> b(n*(1+s)/2, s))(n^2):
seq(a(n), n=0..16);  # Alois P. Heinz, Nov 02 2018

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i*(i + 1)/2 < n, 0, b[n, i - 1] + If[i > n, 0, b[n - i, Min[n - i, i - 1]]]]];
a[n_] := With[{s = n^2}, b[n*(1 + s)/2, s]];
Table[a[n], {n, 0, 16}] (* Jean-François Alcover, May 20 2022, after Alois P. Heinz *)

Corrected and extended by David W. Wilson
a(12) corrected and more terms from Sean A. Irvine, Jan 27 2018
a(0)=1 prepended by Alois P. Heinz, Nov 02 2018

A100568 Number of compositions of n(n^2+1)/2 into n distinct parts each no more than n^2.

Original entry on oeis.org

1, 1, 4, 48, 2064, 167280, 23136480, 4824953280, 1417422988800, 557894688341760, 283527366696806400, 180770613278509900800, 141310830114906688051200, 132919668653581764822067200, 148111929489204170921816985600, 192952383265326280925512415232000

Offset: 0

Henry Bottomley, Nov 28 2004

nonn

In an n X n magic square, each row and column is a composition of type described.

			a(2)=4 since 5 can be written 1+4, 2+3, 3+2 or 4+1.

Alois P. Heinz, Table of n, a(n) for n = 0..150
Henry Bottomley, Partition and composition calculator

Cf. A000142, A052456.

b := proc(n, i, t) option remember;
`if`(nt*(2*i-t+1)/2, 0,
`if`(n=0, 1, b(n, i-1, t) + `if`(n `if`(n=0, 1, n!*b(n*(n^2+1)/2, n^2, n)): seq(a(n), n=0..12); # Peter Luschny, May 06 2014, after Alois P. Heinz

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]*n!; Table[Print[a[n]];  a[n], {n, 0,   14}] (* Jean-François Alcover, Aug 15 2013, after Alois P. Heinz *)

a(n) = A000142(n)*A052456(n). a(n) is close to n^(2n-5/2)*sqrt(6/(pi*e)) in the sense that the ratio between the two tends to 1 as n increases. Experimentally, something like n^(2n) * sqrt(6 / (pi * e * (n^5 - 1.366...n^4 + 1.146...n^3 - 0.826...n^2 + 0.413...n + 0.115...))) seems to be even closer.

A106646 Number of pentamagic (5-multimagic) series for squares of order n.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 3, 0, 0, 13

Offset: 1

Christian Boyer (cboyer(AT)club-internet.fr), May 30 2005

Christian Boyer, Multimagic Squares
Eric Weisstein's World of Mathematics, Magic Series

Cf. A090037.

Cf. A052456, A052457, A052458, A090037.

a(15)=0 proved by Robert Gerbitz and communicated by Christian Boyer (cboyer(AT)club-internet.fr), Apr 02 2007

a(16) computed by Michael Quist, and communicated by Christian Boyer (cboyer(AT)club-internet.fr), Feb 06 2009

A321230 Number of set partitions of [n^2] into n n-element subsets having the same sum.

Original entry on oeis.org

1, 1, 1, 2, 392, 3245664, 6534071578530

Offset: 0

Alois P. Heinz, Oct 31 2018

			a(0) = 1: empty.
a(1) = 1: 1.
a(2) = 1: 14|23.
a(3) = 2: 168|249|357, 159|267|348.

Wikipedia, Partition of a set

Main diagonal of A203986.

Cf. A006003, A037270, A052456, A104185, A321282.

a(n) = A203986(n,n).

cp's OEIS Frontend

A204459 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of k-element subsets that can be chosen from {1,2,...,k*n} having element sum k*(k*n+1)/2.

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A067059 Square array read by antidiagonals of partitions which half fill an n*k box, i.e., partitions of floor(nk/2) or ceiling(nk/2) into up to n positive integers, each no more than k.

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A052457 Number of bimagic series for squares of order n.

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A052458 Number of trimagic series for squares of order n.

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A090037 Number of tetramagic (4-multimagic) series for squares of order n.

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A007785 Number of sets of positive integers <= n^2 whose sum is (n^3 + n)/2.

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Maple

Mathematica

Extensions

A100568 Number of compositions of n(n^2+1)/2 into n distinct parts each no more than n^2.

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Examples

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Crossrefs

Programs

Maple

Mathematica

Formula

A106646 Number of pentamagic (5-multimagic) series for squares of order n.

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Author

Keywords

Links

Crossrefs

Extensions

A321230 Number of set partitions of [n^2] into n n-element subsets having the same sum.

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Author

Keywords

A204459 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of k-element subsets that can be chosen from {1,2,...,kn} having element sum k(k*n+1)/2.