A077047 Doubly restricted composition numbers: number of compositions of floor(n(n+2)/2) into exactly n positive integers each no more than n+1.

1, 1, 3, 12, 85, 780, 9331, 134512, 2306025, 45433800, 1018872811, 25506741084, 707972099627, 21518492021208, 712601187601395, 25491847538274240, 981272544393935569, 40392787067756440272, 1772592132899627652691

Offset: 0

Author

Henry Bottomley, Oct 22 2002

Keywords

• nonn

Comments

a(n) is the maximum number of ordered partitions when using n numbers each ranging from 0 to n. This maximum occurs when partitioning n^2/2 for n even, or (n^2 - 1)/2 or (n^2 + 1)/2 for n odd. Example for a(3)=12: the partitions of 4 are (1,1,2) and (0,2,2), each having 3 ordered arrangements, and (0,1,3) having 6 arrangements; hence 3+3+6=12. For 5 the partitions are (1,2,2) and (1,1,3), with 3 ordered arrangements each, and (0,2,3) having 6 arrangements. - J. M. Bergot, Jul 11 2015

Largest coefficient of (1 + x + x^2 + ... + x^n)^n. - Vaclav Kotesovec, Mar 26 2016

Examples

			a(3)=12 since the compositions of [3*5/2]=7 into exactly 3 positive integers each no more than 4 are 1+2+4, 1+3+3, 1+4+2, 2+1+4, 2+2+3, 2+3+2, 2+4+1, 3+1+3, 3+2+2, 3+3+2, 4+1+2, 4+2+1.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..375
• Index entries for sequences related to compositions

Crossrefs

Cf. A077042, A077045, A077046, A077048, A270918.

Programs

Maple

f:= proc(n) if n::odd then coeff(add(x^i,i=0..n)^n,x,(n^2-1)/2)
else coeff(add(x^i,i=0..n)^n,x,n^2/2) fi end proc:
map(f, [$0..40]); # Robert Israel, Nov 16 2016

Mathematica

Table[Max[CoefficientList[Expand[Sum[x^k, {k, 0, n}]^n], x]], {n, 0, 20}] (* Vaclav Kotesovec, Mar 26 2016 *)
Table[Max[CoefficientList[((x^(n+1)-1)/(x-1))^n, x]], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 16 2016 *)

Formula

a(n) = A077042(n+1, n).

a(n) ~ exp(1) * sqrt(6/Pi) * n^(n-3/2). - Vaclav Kotesovec, Mar 26 2016