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A077044 Largest coefficient in expansion of (1 + x + x^2 + ... + x^(n-1))^5 = ((1-x^n)/(1-x))^5, i.e., the coefficient of x^floor(5*(n-1)/2) and of x^ceiling(5*(n-1)/2); also number of compositions of floor(5*(n+1)/2) into exactly 5 positive integers each no more than n.

%I A077044 #29 Jan 14 2023 10:53:47
%S A077044 0,1,10,51,155,381,780,1451,2460,3951,6000,8801,12435,17151,23030,
%T A077044 30381,39280,50101,62910,78151,95875,116601,140360,167751,198780,
%U A077044 234131,273780,318501,368235,423851,485250,553401,628160,710601,800530
%N A077044 Largest coefficient in expansion of (1 + x + x^2 + ... + x^(n-1))^5 = ((1-x^n)/(1-x))^5, i.e., the coefficient of x^floor(5*(n-1)/2) and of x^ceiling(5*(n-1)/2); also number of compositions of floor(5*(n+1)/2) into exactly 5 positive integers each no more than n.
%H A077044 Vincenzo Librandi, <a href="/A077044/b077044.txt">Table of n, a(n) for n = 0..10000</a>
%H A077044 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%H A077044 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-6,0,6,-2,-2,1).
%F A077044 a(n) = (230*n^4 + 70*n^2 + 27 - (30*n^2 + 27)*(-1)^n)/384 = A077042(n, 5).
%F A077044 a(n) = 2*a(n-1) + 2*a(n-2) - 6*a(n-3) + 6*a(n-5) - 2*a(n-6) - 2*a(n-7) + a(n-8).
%F A077044 G.f.: -x*(1 + 8*x + 29*x^2 + 39*x^3 + 29*x^4 + 8*x^5 + x^6) / ( (1+x)^3*(x-1)^5 ). - _R. J. Mathar_, Sep 04 2011
%e A077044 a(2)=10 since the compositions of floor(5*(2+1)/2) = 7 into exactly 5 positive integers each no more than 2 are: 1+1+1+2+2, 1+1+2+1+2, 1+1+2+2+1, 1+2+1+1+2, 1+2+1+2+1, 1+2+2+1+1, 2+1+1+1+2, 2+1+1+2+1, 2+1+2+1+1, 2+2+1+1+1.
%t A077044 LinearRecurrence[{2,2,-6,0,6,-2,-2,1},{0,1,10,51,155,381,780,1451},40] (* _Harvey P. Dale_, Mar 05 2015 *)
%o A077044 (Magma) [(230*n^4+70*n^2+27-(30*n^2+27)*(-1)^n)/384: n in [0..40]]; // _Vincenzo Librandi_, Sep 05 2011
%o A077044 (PARI) a(n)=(230*n^4+70*n^2-30*n^2*(-1)^n)\/384 \\ _Charles R Greathouse IV_, Sep 25 2012
%K A077044 nonn,easy
%O A077044 0,3
%A A077044 _Henry Bottomley_, Oct 22 2002