%I #61 Feb 22 2024 09:05:33

%S 1,0,0,5,21,56,120,225,385,616,936,1365,1925,2640,3536,4641,5985,7600,

%T 9520,11781,14421,17480,21000,25025,29601,34776,40600,47125,54405,

%U 62496,71456,81345,92225,104160,117216,131461,146965,163800,182040,201761,223041

%N a(n) = (n-1)*(n-2)*(n+3)*(n+2)/12.

%H Colin Barker, <a href="/A299120/b299120.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F a(n) = n^4/12 + n^3/6 - 7*n^2/12 - 2*n/3 + 1 = (n-1)*(n-2)*(n+3)*(n+2)/12.

%F From _Colin Barker_, Feb 05 2018: (Start)

%F G.f.: (1 - 5*x + 10*x^2 - 5*x^3 + x^4) / (1 - x)^5.

%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5. (End)

%F a(n) = A033275(n+2) for n > 1. - _Georg Fischer_, Oct 09 2018

%F From _Amiram Eldar_, Jan 12 2021: (Start)

%F Sum_{n>=3} 1/a(n) = 43/150.

%F Sum_{n>=3} (-1)^(n+1)/a(n) = 16*log(2)/5 - 154/75. (End)

%F E.g.f.: exp(x)*(12 - 12*x + 6*x^2 + 8*x^3 + x^4)/12. - _Stefano Spezia_, Feb 21 2024

%p seq(n^4/12+n^3/6-7*n^2/12-2*n/3+1, n=0..10^3); # _Muniru A Asiru_, Feb 04 2018

%t Rest@ CoefficientList[Series[(1 - 5 x + 10 x^2 - 5 x^3 + x^4)/(1 - x)^5, {x, 0, 41}], x] (* _Michael De Vlieger_, Feb 10 2018 *)

%t f[n_] := n^4/12 + n^3/6 - 7*n^2/12 - 2*n/3 + 1; Array[f, 40, 0] (* or *)

%t LinearRecurrence[{5, -10, 10, -5, 1}, {1, 0, 0, 5, 21}, 40] (* _Robert G. Wilson v_, Mar 12 2018 *)

%o (Magma) [n^4/12 + n^3/6 - 7*n^2/12 - 2*n/3 + 1: n in [0..40]];

%o (GAP) List([0..10^3], n->n^4/12+n^3/6-7*n^2/12-2*n/3+1); # _Muniru A Asiru_, Feb 04 2018

%o (PARI) Vec((1 - 5*x + 10*x^2 - 5*x^3 + x^4) / (1 - x)^5 + O(x^50)) \\ _Colin Barker_, Feb 05 2018

%Y Cf. A001263, A033275, A077415, A299146, A299198.

%K nonn,easy

%O 0,4

%A _Juri-Stepan Gerasimov_, Feb 03 2018

%E Edited by _Wolfdieter Lang_, Apr 06 2018