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A231303 Recurrence a(n) = a(n-2) + n^M for M=4, starting with a(0)=0, a(1)=1. 10
0, 1, 16, 82, 272, 707, 1568, 3108, 5664, 9669, 15664, 24310, 36400, 52871, 74816, 103496, 140352, 187017, 245328, 317338, 405328, 511819, 639584, 791660, 971360, 1182285, 1428336, 1713726, 2042992, 2421007, 2852992, 3344528, 3901568, 4530449, 5237904 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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).

Partial sums of A062392. - Bruce J. Nicholson, Jun 29 2019

LINKS

Stanislav Sykora, Table of n, a(n) for n = 0..9999

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

FORMULA

a(n) = Sum_{k=0..floor(n/2)} (n - 2*k)^4.

From Colin Barker, Dec 22 2015: (Start)

a(n) = (1/30)*n*(3*n^4 + 15*n^3 + 20*n^2 - 8).

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

(End)

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

From Bruce J. Nicholson, Jun 29 2019: (Start)

a(n) = 12*A000389(n+3) + A000292(n);

a(n) = (12*A000579(n+4)+A000332(n+3)) - (12*A000579(n+3)+A000332(n+2));

a(n) - a(n-2) = A000583(n). (End)

EXAMPLE

a(4) = 4^4 + 2^4 = 272; a(5) = 5^4 + 3^4 + 1^4 = 707.

MATHEMATICA

Table[SeriesCoefficient[x*(1+10*x+x^2)/(1-x)^6, {x, 0, n}], {n, 0, 40}] (* Michael De Vlieger, Dec 22 2015 *)

LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 1, 16, 82, 272, 707}, 40] (* Vincenzo Librandi, Dec 23 2015 *)

PROG

(PARI) nmax=40; a = vector(nmax); a[2]=1; for(i=3, #a, a[i]=a[i-2]+(i-1)^4); print(a);

(PARI) concat(0, Vec(x*(1+10*x+x^2)/(1-x)^6 + O(x^40))) \\ Colin Barker, Dec 22 2015

(MAGMA) [1/30*n*(3*n^4+15*n^3+20*n^2-8): n in [0..40]]; // Vincenzo Librandi, Dec 23 2015

(Sage) [n*(3*n^4+15*n^3+20*n^2-8)/30 for n in (0..40)] # G. C. Greubel, Jul 01 2019

(GAP) List([0..40], n-> n*(3*n^4+15*n^3+20*n^2-8)/30) # G. C. Greubel, Jul 01 2019

CROSSREFS

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).

Cf. A000389, A000579, A000332, A000583.

Sequence in context: A030693 A308249 A285989 * A218082 A159501 A172219

Adjacent sequences:  A231300 A231301 A231302 * A231304 A231305 A231306

KEYWORD

nonn,easy

AUTHOR

Stanislav Sykora, Nov 07 2013

STATUS

approved

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Last modified October 19 16:17 EDT 2019. Contains 328223 sequences. (Running on oeis4.)