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

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

Offset

0,3

Comments

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

Links

Alois P. Heinz, Table of n, a(n) for n = 0..150

Henry Bottomley, Partition and composition calculator

Formula

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.

Example

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

Maple

b := proc(n, i, t) option remember;
`if`(n<t*(t+1)/2 or n>t*(2*i-t+1)/2, 0,
`if`(n=0, 1, b(n, i-1, t) + `if`(n<i, 0, b(n-i, i-1, t-1)))) end:
a := 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

Mathematica

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 *)

Crossrefs

Cf. A000142, A052456.

Sequence in context: A360820 A362828 A197927 * A112693 A239162 A136384

Adjacent sequences: A100565 A100566 A100567 * A100569 A100570 A100571

Keyword

nonn

Author

Henry Bottomley, Nov 28 2004

Status

approved