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%I #44 Feb 16 2023 19:24:43
%S 1,2,10,37,101,226,442,785,1297,2026,3026,4357,6085,8282,11026,14401,
%T 18497,23410,29242,36101,44101,53362,64010,76177,90001,105626,123202,
%U 142885,164837,189226,216226,246017,278785,314722,354026,396901,443557,494210,549082,608401,672401,741322,815410,894917,980101
%N Central coefficients of number triangle A113582.
%C a(n) equals n!^3 times the determinant of the n X n matrix whose (i,j)-entry is KroneckerDelta[i, j] (((i^3 + 1)/(i^3)) - 1) + 1. - _John M. Campbell_, May 20 2011
%C Let b(0)=b(1)=1; b(n)=max(b(n-1)+(n-1)^3, b(n-2)+(n-2)^3); then a(n)=b(n+1). - _Yalcin Aktar_, Jul 28 2011
%C a(n-1) is the number of sets of n words of length n over binary alphabet where the first letter occurs n times. a(2) = 10: {aab,abb,bbb}, {aab,bab,bbb}, {aab,bba,bbb}, {aba,abb,bbb}, {aba,bab,bbb}, {aba,bba,bbb}, {abb,baa,bbb}, {abb,bab,bba}, {baa,bab,bbb}, {baa,bba,bbb}. - _Alois P. Heinz_, Feb 16 2023
%H G. C. Greubel, <a href="/A154323/b154323.txt">Table of n, a(n) for n = 0..1000</a>
%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) = (n^4 + 2*n^3 + n^2 + 4)/4.
%F G.f.: (1 - 3*x + 10*x^2 - 3*x^3 + x^4)/(1-x)^5.
%F a(n) = 1 + C(n+1,2)^2 = 1 + A000537(n).
%F From _G. C. Greubel_, Sep 11 2016: (Start)
%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
%F E.g.f.: (1/4)*(4 + 4*x + 14*x^2 + 8*x^3 + x^4)*exp(x). (End)
%F a(n) = a(n-1)+n^3. - _Charles U. Lonappan_, Jun 09 2021
%t s = 1; lst = {s}; Do[s += n^3; AppendTo[lst, s], {n, 1, 42, 1}]; lst (* _Zerinvary Lajos_, Jul 12 2009 *)
%t Table[n!^3*Det[Array[KroneckerDelta[#1,#2](((#1^3+1)/(#1^3))-1)+1&,{n,n}]],{n,1,30}] (* _John M. Campbell_, May 20 2011 *)
%t LinearRecurrence[{5, -10, 10, -5, 1}, {1, 2, 10, 37, 101}, 25] (* or *) Table[(n^4 + 2*n^3 + n^2 + 4)/4, {n,0,25}] (* _G. C. Greubel_, Sep 11 2016 *)
%o (Magma) [(n^4 + 2*n^3 + n^2 + 4)/4: n in [0..40]]; // _Vincenzo Librandi_, Feb 13 2015
%Y Cf. A000537, A113582, A360660.
%Y Main diagonal of A360693.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Jan 07 2009
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