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A231303 Recurrence a(n) = a(n-2) + n^M for M=4, starting with a(0)=0, a(1)=1. 9
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).

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-2k)^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.: (1/30)*x*(30 + 210 x + 185 x^2 + 45 x^3 + 3 x^4)*exp(x). - G. C. Greubel, Apr 24 2016

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, 34}] (* Michael De Vlieger, Dec 22 2015 *)

LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 1, 16, 82, 272, 707}, 35] (* Vincenzo Librandi, Dec 23 2015 *)

PROG

(PARI) nmax=100; 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^60))) \\ Colin Barker, Dec 22 2015

(MAGMA) [1/30*n*(3*n^4+15*n^3+20*n^2-8): n in [0..35]]; // Vincenzo Librandi, Dec 23 2015

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).

Sequence in context: A151502 A030693 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 May 27 22:57 EDT 2017. Contains 287210 sequences.