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A267691 a(n) = (n + 1)*(6*n^4 - 21*n^3 + 31*n^2 - 31*n + 30)/30.

%I A267691 #43 Sep 08 2022 08:46:15
%S A267691 1,1,2,18,99,355,980,2276,4677,8773,15334,25334,39975,60711,89272,
%T A267691 127688,178313,243849,327370,432346,562667,722667,917148,1151404,
%U A267691 1431245,1763021,2153646,2610622,3142063,3756719,4464000,5274000,6197521,7246097,8432018,9768354
%N A267691 a(n) = (n + 1)*(6*n^4 - 21*n^3 + 31*n^2 - 31*n + 30)/30.
%H A267691 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1)
%F A267691 G.f.: (1 - 5*x + 11*x^2 + x^3 + 16*x^4)/(1 - x)^6.
%F A267691 a(n + 1) = a(n) + n^4.
%F A267691 a(n + 1) = A000538(n) + 1.
%F A267691 a(n + 2) - a(n) = A008514(n).
%F A267691 Sum_{n>=0} 1/a(n) = 2.570450909491318975...
%F A267691 Sum_{n>=1} 1/(a(n + 1) - a(n)) = zeta(4) = Pi^4/90.
%e A267691 a(0) = 1,
%e A267691 a(1) = 1 + 0^4 = 1,
%e A267691 a(2) = 1 + 1^4 = 2,
%e A267691 a(3) = 2 + 2^4 = 18,
%e A267691 a(4) = 18+ 3^4 = 99, etc.
%t A267691 Table[(n + 1) (6 n^4 - 21 n^3 + 31 n^2 - 31 n + 30)/30, {n, 0, 30}]
%t A267691 LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 1, 2, 18, 99, 355}, 40] (* _Vincenzo Librandi_, Jan 20 2016 *)
%o A267691 (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 A267691 (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 A267691 (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 A267691 Essentially the same as A000538.
%Y A267691 Cf. A000124, A000583, A008514, A056520, A154323, A263689.
%Y A267691 Cf. A013662 (zeta(4)).
%K A267691 nonn,easy
%O A267691 0,3
%A A267691 _Ilya Gutkovskiy_, Jan 19 2016