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%I A231303 #62 Sep 08 2022 08:46:06
%S A231303 0,1,16,82,272,707,1568,3108,5664,9669,15664,24310,36400,52871,74816,
%T A231303 103496,140352,187017,245328,317338,405328,511819,639584,791660,
%U A231303 971360,1182285,1428336,1713726,2042992,2421007,2852992,3344528,3901568,4530449,5237904
%N A231303 Recurrence a(n) = a(n-2) + n^M for M=4, starting with a(0)=0, a(1)=1.
%C A231303 In physics, a(n)/2^(M-1) is the trace of the spin operator |S_z|^M for a particle with spin S=n/2. For example, when S=3/2, the S_z eigenvalues are -3/2, -1/2, +1/2, +3/2 and therefore the sum of their 4th powers is 2*82/16 = a(3)/8 (analogously for other values of M).
%C A231303 Partial sums of A062392. - _Bruce J. Nicholson_, Jun 29 2019
%H A231303 Stanislav Sykora, <a href="/A231303/b231303.txt">Table of n, a(n) for n = 0..9999</a>
%H A231303 Stanislav Sýkora, <a href="http://www.ebyte.it/stan/blog12to14.html#14Dec31">Magnetic Resonance on OEIS</a>, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
%H A231303 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A231303 a(n) = Sum_{k=0..floor(n/2)} (n - 2*k)^4.
%F A231303 From _Colin Barker_, Dec 22 2015: (Start)
%F A231303 a(n) = (1/30)*n*(3*n^4 + 15*n^3 + 20*n^2 - 8).
%F A231303 G.f.: x*(1 + 10*x + x^2) / (1-x)^6.
%F A231303 (End)
%F A231303 E.g.f.: x*(30 + 210*x + 185*x^2 + 45*x^3 + 3*x^4)*exp(x)/30. - _G. C. Greubel_, Apr 24 2016
%F A231303 From _Bruce J. Nicholson_, Jun 29 2019: (Start)
%F A231303 a(n) = 12*A000389(n+3) + A000292(n);
%F A231303 a(n) = (12*A000579(n+4)+A000332(n+3)) - (12*A000579(n+3)+A000332(n+2));
%F A231303 a(n) - a(n-2) = A000583(n). (End)
%e A231303 a(4) = 4^4 + 2^4 = 272; a(5) = 5^4 + 3^4 + 1^4 = 707.
%t A231303 Table[SeriesCoefficient[x*(1+10*x+x^2)/(1-x)^6, {x, 0, n}], {n, 0, 40}] (* _Michael De Vlieger_, Dec 22 2015 *)
%t A231303 LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 1, 16, 82, 272, 707}, 40] (* _Vincenzo Librandi_, Dec 23 2015 *)
%o A231303 (PARI) nmax=40;a = vector(nmax);a[2]=1;for(i=3,#a,a[i]=a[i-2]+(i-1)^4); print(a);
%o A231303 (PARI) concat(0, Vec(x*(1+10*x+x^2)/(1-x)^6 + O(x^40))) \\ _Colin Barker_, Dec 22 2015
%o A231303 (Magma) [1/30*n*(3*n^4+15*n^3+20*n^2-8): n in [0..40]]; // _Vincenzo Librandi_, Dec 23 2015
%o A231303 (Sage) [n*(3*n^4+15*n^3+20*n^2-8)/30 for n in (0..40)] # _G. C. Greubel_, Jul 01 2019
%o A231303 (GAP) List([0..40], n-> n*(3*n^4+15*n^3+20*n^2-8)/30) # _G. C. Greubel_, Jul 01 2019
%Y A231303 Cf. A001477 (M=1), A000292 (M=2), A105636 (M=3), A231304 (M=5), A231305 (M=6), A231306 (M=7), A231307 (M=8), A231308 (M=9), A231309 (M=10).
%Y A231303 Cf. A000389, A000579, A000332, A000583.
%K A231303 nonn,easy
%O A231303 0,3
%A A231303 _Stanislav Sykora_, Nov 07 2013

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