A319577 - OEIS

%I #27 Feb 20 2024 09:37:18

%S 0,0,0,24,96,240,544,1288,3136,7392,16320,33528,64416,116688,200928,

%T 331240,525952,808384,1207680,1759704,2508000,3504816,4812192,6503112,

%U 8662720,11389600,14797120,19014840,24189984,30488976,38099040,47229864,58115328,71015296

%N a(n) = (4/45)*n*(n - 2)*(n - 1)*(n^3 - 12*n^2 + 47*n - 15).

%H Seiichi Manyama, <a href="/A319577/b319577.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

%F a(n) = [x^6] JacobiTheta3(x)^n.

%F a(n) = A319574(n,6).

%F From _Colin Barker_, Oct 02 2018: (Start)

%F G.f.: 8*x^3*(3 - 9*x + 9*x^2 + 5*x^3) / (1 - x)^7.

%F a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>6.

%F (End)

%p a := n -> (4/45)*n*(n - 2)*(n - 1)*(n^3 - 12*n^2 + 47*n - 15):

%p seq(a(n), n=0..41);

%t A319577[n_]:=4/45*n*(n-2)*(n-1)*(n^3-12*n^2+47*n-15); Array[A319577, 50, 0] (*or*)

%t LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 0, 0, 24, 96, 240, 544}, 50] (* _Paolo Xausa_, Feb 20 2024 *)

%o (PARI) concat([0,0,0], Vec(8*x^3*(3 - 9*x + 9*x^2 + 5*x^3) / (1 - x)^7 + O(x^40))) \\ _Colin Barker_, Oct 02 2018

%Y Cf. A000012 (m=0), A005843 (m=1), A046092 (m=2), A130809 (m=3), A319575 (m=4), A319576 (m=5), this sequence (m=6).

%Y Column n=6 of A122141.

%Y Cf. A319574.

%K nonn,easy

%O 0,4

%A _Peter Luschny_, Oct 01 2018