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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.

%I A077047 #31 Dec 10 2021 08:22:22
%S A077047 1,1,3,12,85,780,9331,134512,2306025,45433800,1018872811,25506741084,
%T A077047 707972099627,21518492021208,712601187601395,25491847538274240,
%U A077047 981272544393935569,40392787067756440272,1772592132899627652691
%N 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.
%C A077047 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
%C A077047 Largest coefficient of (1 + x + x^2 + ... + x^n)^n. - _Vaclav Kotesovec_, Mar 26 2016
%H A077047 Vaclav Kotesovec, <a href="/A077047/b077047.txt">Table of n, a(n) for n = 0..375</a>
%H A077047 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F A077047 a(n) = A077042(n+1, n).
%F A077047 a(n) ~ exp(1) * sqrt(6/Pi) * n^(n-3/2). - _Vaclav Kotesovec_, Mar 26 2016
%e A077047 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.
%p A077047 f:= proc(n) if n::odd then coeff(add(x^i,i=0..n)^n,x,(n^2-1)/2)
%p A077047 else coeff(add(x^i,i=0..n)^n,x,n^2/2) fi end proc:
%p A077047 map(f, [$0..40]); # _Robert Israel_, Nov 16 2016
%t A077047 Table[Max[CoefficientList[Expand[Sum[x^k, {k, 0, n}]^n], x]], {n, 0, 20}] (* _Vaclav Kotesovec_, Mar 26 2016 *)
%t A077047 Table[Max[CoefficientList[((x^(n+1)-1)/(x-1))^n, x]], {n, 0, 20}] (* _Vladimir Reshetnikov_, Nov 16 2016 *)
%Y A077047 Cf. A077042, A077045, A077046, A077048, A270918.
%K A077047 nonn
%O A077047 0,3
%A A077047 _Henry Bottomley_, Oct 22 2002