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%I #37 Feb 21 2023 17:00:53
%S 0,1,11,56,192,517,1183,2408,4488,7809,12859,20240,30680,45045,64351,
%T 89776,122672,164577,217227,282568,362768,460229,577599,717784,883960,
%U 1079585,1308411
%N a(n) = ((n^6 - (n-1)^6) - (n^2 - (n-1)^2))/60.
%C Polynexus numbers of order 6.
%C A polynexus (subtractive) function is composed of two or more subtracted nexus numbers divided by an integer x. The general form of the formula is a(n)=((n^p-(n-1)^p)-(n^q-(n-1)^q))/x, where n, p, q and x are integers.
%C Already known: ((n^5-(n-1)^5) - (n^3-(n-1)^3))/24, giving A006322 for n>1; ((n^4-(n-1)^4) - (n^2-(n-1)^2))/12, giving A000330; ((n^3-(n-1)^3) - (n^1-(n-1)^1))/6, giving A000217; ((n^2-(n-1)^2) - (n^1-(n-1)^1))/2, giving n; ((n^2-(n-1)^2) - (n^0-(n-1)^0))/1, giving 2*n-1. In those examples, x is equal to 1,2,6,12,24, and 3 is also possible.
%C Also number of monotone n-weightings of complete bipartite digraph K(3,2) if offset were 0; cf. A085464-A085465. - Goran Kilibarda, _Vladeta Jovovic_, Jul 01 2003
%C Partial sums of A037270. - _J. M. Bergot_, Jun 07 2012
%H Bruno Berselli, <a href="/A079547/b079547.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) = Sum_{i=1..n} (i^2 + i^4)/2 = n*(2*n+1)*(n+1)*(3*n^2+3*n+4)/60. - _Vladeta Jovovic_, Mar 17 2006
%F G.f.: x^2*(x+1)*(1+4*x+x^2)/(1-x)^6. - _Bruno Berselli_, Feb 13 2012
%F a(n) = Sum_{i=1..n-1} Sum_{j=1..n-1} min(i,j)^3. - _Enrique Pérez Herrero_, Jan 16 2013
%F E.g.f.: x^2*(30 + 80*x + 45*x^2 + 6*x^3)*exp(x)/60. - _G. C. Greubel_, Jun 19 2019
%t Table[((n^6 -(n-1)^6) - (n^2 -(n-1)^2))/60, {n, 30}] (* _Bruno Berselli_, Feb 13 2012 *)
%t LinearRecurrence[{6,-15,20,-15,6,-1},{0,1,11,56,192,517},30] (* _Harvey P. Dale_, Feb 21 2023 *)
%o (PARI) a(n) = n*(6*n^4-15*n^3+20*n^2-15*n+4)/60 \\ _Charles R Greathouse IV_, Jan 16 2013
%o (Magma) [n*(6*n^4-15*n^3+20*n^2-15*n+4)/60: n in [1..30]]; // _G. C. Greubel_, Jun 19 2019
%o (Sage) [n*(6*n^4-15*n^3+20*n^2-15*n+4)/60 for n in (1..30)] # _G. C. Greubel_, Jun 19 2019
%o (GAP) List([1..30], n-> n*(6*n^4-15*n^3+20*n^2-15*n+4)/60) # _G. C. Greubel_, Jun 19 2019
%Y Cf. A006322, A000330, A000217, A047969, A003215, A083200, A088889-A088894.
%K nonn,easy
%O 1,3
%A Xavier Acloque, Jan 22 2003
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