A267691 - OEIS

%I #43 Sep 08 2022 08:46:15

%S 1,1,2,18,99,355,980,2276,4677,8773,15334,25334,39975,60711,89272,

%T 127688,178313,243849,327370,432346,562667,722667,917148,1151404,

%U 1431245,1763021,2153646,2610622,3142063,3756719,4464000,5274000,6197521,7246097,8432018,9768354

%N a(n) = (n + 1)*(6*n^4 - 21*n^3 + 31*n^2 - 31*n + 30)/30.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1)

%F G.f.: (1 - 5*x + 11*x^2 + x^3 + 16*x^4)/(1 - x)^6.

%F a(n + 1) = a(n) + n^4.

%F a(n + 1) = A000538(n) + 1.

%F a(n + 2) - a(n) = A008514(n).

%F Sum_{n>=0} 1/a(n) = 2.570450909491318975...

%F Sum_{n>=1} 1/(a(n + 1) - a(n)) = zeta(4) = Pi^4/90.

%e a(0) = 1,

%e a(1) = 1 + 0^4 = 1,

%e a(2) = 1 + 1^4 = 2,

%e a(3) = 2 + 2^4 = 18,

%e a(4) = 18+ 3^4 = 99, etc.

%t Table[(n + 1) (6 n^4 - 21 n^3 + 31 n^2 - 31 n + 30)/30, {n, 0, 30}]

%t LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 1, 2, 18, 99, 355}, 40] (* _Vincenzo Librandi_, Jan 20 2016 *)

%o (PARI) a(n)=(n+1)*(6*n^4-21*n^3+31*n^2-31*n+30)/30 \\ _Charles R Greathouse IV_, Jan 19 2016

%o (PARI) Vec((1-5*x+11*x^2+x^3+16*x^4)/(x-1)^6 + O(x^100)) \\ _Altug Alkan_, Jan 19 2016

%o (Magma) [(n+1)*(6*n^4-21*n^3+31*n^2-31*n+30)/30: n in [0..35]]; // _Vincenzo Librandi_, Jan 20 2016

%Y Essentially the same as A000538.

%Y Cf. A000124, A000583, A008514, A056520, A154323, A263689.

%Y Cf. A013662 (zeta(4)).

%K nonn,easy

%O 0,3

%A _Ilya Gutkovskiy_, Jan 19 2016