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Diagonal sums of number array A082046.
6

%I #15 Jan 25 2024 15:22:02

%S 1,2,7,24,69,170,371,736,1353,2338,3839,6040,9165,13482,19307,27008,

%T 37009,49794,65911,85976,110677,140778,177123,220640,272345,333346,

%U 404847,488152,584669,695914,823515,969216,1134881,1322498,1534183

%N Diagonal sums of number array A082046.

%H G. C. Greubel, <a href="/A082047/b082047.txt">Table of n, a(n) for n = 0..1000</a>

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

%F a(n) = (n^5+15*n^3+14*n+30)/30 = (n+1)*(n^4-n^3+16*n^2-16*n+30)/30.

%F From _G. C. Greubel_, Dec 24 2022: (Start)

%F G.f.: (1 - 4*x + 10*x^2 - 8*x^3 + 5*x^4)/(1-x)^6.

%F E.g.f.: (1/30)*(30 +30*x +60*x^2 +40*x^3 +10*x^4 +x^5)*exp(x). (End)

%t Table[(n+1)*(n*(n-1)*(n^2+16)+30)/30, {n,0,40}] (* _G. C. Greubel_, Dec 24 2022 *)

%t LinearRecurrence[{6,-15,20,-15,6,-1},{1,2,7,24,69,170},40] (* _Harvey P. Dale_, Jan 25 2024 *)

%o (PARI) a(n) = (n^5+15*n^3+14*n+30)/30; \\ _Michel Marcus_, Jan 22 2016

%o (Magma) [(n+1)*(n*(n-1)*(n^2+16)+30)/30: n in [0..40]]; // _G. C. Greubel_, Dec 24 2022

%o (SageMath) [(n+1)*(n*(n-1)*(n^2+16)+30)/30 for n in range(41)] # _G. C. Greubel_, Dec 24 2022

%Y Cf. A082045, A082046, A082107.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Apr 03 2003