A179441 - OEIS

%I #19 Oct 21 2022 21:55:10

%S 1,21,121,432,1182,2723,5558,10368,18039,29689,46695,70720,103740,

%T 148071,206396,281792,377757,498237,647653,830928,1053514,1321419,

%U 1641234,2020160,2466035,2987361,3593331,4293856,5099592,6021967,7073208,8266368,9615353,11134949,12840849

%N Number of solutions to a+b+c < d+e with each of a,b,c,d,e in {1..n+1}.

%D Mathematics and Computer Education 1988 - 89 #261 Unsolved.

%H Andrew Howroyd, <a href="/A179441/b179441.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).

%F a(n) = (1/120)*(27*n^5 + 80*n^4 + 65*n^3 - 20*n^2 - 32*n).

%F a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n > 6.

%F G.f.: x*(1 + 15*x + 10*x^2 + x^3)/(1 - x)^6.

%e a(1) = 1 since from {1,2} there is only one solution: {1,1,1} for a,b,c and {2,2} for d,e.

%e a(2) = 21 since from {1,2,3} there are 21 ways to select a,b,c,d,e such that a+b+c < d+e.

%t k=10;

%t Table[p=Expand[Sum[x^k,{k,1,n}]^2 Sum[1/x^k,{k,1,n}]^3];

%t Twowins=Drop[CoefficientList[p,x],1]//Total,{n,2,k}]

%o (PARI) a(n)=(27*n^5 + 80*n^4 + 65*n^3 - 20*n^2 - 32*n)/120 \\ _Andrew Howroyd_, Apr 15 2021

%Y Cf. A197083.

%K nonn,easy

%O 1,2

%A _Bobby Milazzo_, Jul 14 2010

%E Name edited and terms a(24) and beyond from _Andrew Howroyd_, Apr 15 2021