cp's OEIS Frontend

A231305 Recurrence a(n) = a(n-2) + n^M for M=6, starting with a(0)=0, a(1)=1.

%I A231305 #33 Aug 23 2025 17:39:17
%S A231305 0,1,64,730,4160,16355,50816,134004,312960,665445,1312960,2437006,
%T A231305 4298944,7263815,11828480,18654440,28605696,42792009,62617920,
%U A231305 89837890,126617920,175604011,239997824,323639900,431100800,567780525,740016576,955201014,1221906880
%N A231305 Recurrence a(n) = a(n-2) + n^M for M=6, starting with a(0)=0, a(1)=1.
%H A231305 Stanislav Sykora, <a href="/A231305/b231305.txt">Table of n, a(n) for n = 0..9999</a>
%H A231305 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 A231305 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).
%F A231305 a(n) = Sum_{k=0..floor(n/2)}(n-2k)^6.
%F A231305 From _Colin Barker_, Dec 22 2015: (Start)
%F A231305 a(n) = 1/42*n*(3*n^6+21*n^5+42*n^4-56*n^2+32).
%F A231305 G.f.: x*(1+56*x+246*x^2+56*x^3+x^4) / (1-x)^8.
%F A231305 (End)
%e A231305 a(5) = 5^6 + 3^6 + 1^6 = 16355.
%p A231305 map(op,ListTools:-PartialSums([seq([(2*i)^6,(2*i+1)^6],i=0..50)])); # _Robert Israel_, Dec 22 2015
%t A231305 Table[SeriesCoefficient[x (1 + 56 x + 246 x^2 + 56 x^3 + x^4)/(1 - x)^8, {x, 0, n}], {n, 0, 28}] (* _Michael De Vlieger_, Dec 22 2015 *)
%t A231305 LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{0,1,64,730,4160,16355,50816,134004},30] (* _Harvey P. Dale_, Aug 23 2025 *)
%o A231305 (PARI) nmax=100; a = vector(nmax); a[2]=1; for(i=3, #a, a[i]=a[i-2]+(i-1)^6); print(a);
%o A231305 (PARI) concat(0, Vec(x*(1+56*x+246*x^2+56*x^3+x^4)/(1-x)^8 + O(x^50))) \\ _Colin Barker_, Dec 22 2015
%Y A231305 Cf. A001477 (M=1), A000292 (M=2), A105636 (M=3), A231303 (M=4), A231304 (M=5), A231306 (M=7), A231307 (M=8), A231308 (M=9), A231309 (M=10).
%K A231305 nonn,easy,changed
%O A231305 0,3
%A A231305 _Stanislav Sykora_, Nov 07 2013