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%I #32 May 01 2023 19:06:53
%S 0,0,2,13,45,115,245,462,798,1290,1980,2915,4147,5733,7735,10220,
%T 13260,16932,21318,26505,32585,39655,47817,57178,67850,79950,93600,
%U 108927,126063,145145,166315,189720,215512,243848,274890,308805
%N Number of (w,x,y,z) with all terms in {1,...,n} and w > x < y >= z.
%C For a guide to related sequences, see A211795.
%C Partial sums of A033994. - _J. M. Bergot_, Jun 14 2013
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 4.
%F G.f.: x^2*(2+3*x)/(1-x)^5. - _Bruno Berselli_, May 31 2012
%F a(n) = (n-1)*n*(n+1)*(5*n-2)/24. - _Bruno Berselli_, May 31 2012
%e a(8)=798 which results from the following: 1*(8+9+10+11+12+13+14) + 2*(8+9+10+11+12+13) + 3*(8+9+10+11+12) + 4*(8+9+10+11) + 5*(8+9+10) + 6*(8+9) + 7*(8) = 798 = 77+126+150+152+135+102+56. - _J. M. Bergot_, Aug 23 2022
%p A212501:=n->(n-1)*n*(n+1)*(5*n-2)/24: seq(A212501(n), n=0..60); # _Wesley Ivan Hurt_, Oct 07 2017
%t t = Compile[{{n, _Integer}}, Module[{s = 0},
%t (Do[If[w > x < y >= z, s = s + 1],
%t {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t Map[t[#] &, Range[0, 40]] (* A212501 *)
%t LinearRecurrence[{5,-10,10,-5,1},{0,0,2,13,45},50] (* _Harvey P. Dale_, May 01 2023 *)
%o (PARI) a(n)=n*(n-1)*(n+1)*(5*n-2)/24 \\ _Charles R Greathouse IV_, Jun 14 2013
%Y Cf. A211795.
%K nonn,easy
%O 0,3
%A _Clark Kimberling_, May 19 2012
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