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Partial sums of A062074.
2

%I #28 Apr 16 2023 20:28:46

%S 0,3,75,804,5988,36363,193827,943968,4303200,18652107,77701107,

%T 313483764,1231813812,4734541443,17859008379,66286569504,242605938720,

%U 877071559539,3136507851387,11108459253540,39002734461540,135876065474523,470021588191155,1615461644979264

%N Partial sums of A062074.

%H G. C. Greubel, <a href="/A343808/b343808.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 (13,-66,162,-189,81).

%F a(n) = Sum_{k=0..n} 3^k*k^3.

%F a(n) = Sum_{k=0..n} A062074(k).

%F G.f.: 3*x*(1 + 12*x + 9*x^2)/((1 - 3*x)^4*(1 - x)). - _Stefano Spezia_, May 01 2021

%F a(n) = ((4*n^3-6*n^2+12*n-11)*3^(n+1) + 33) / 8. - _Kevin Ryde_, May 01 2021

%F E.g.f.: (3/8)*(11*exp(x) + (-11 + 30*x + 54*x^2 + 108*x^3)*exp(3*x)). - _G. C. Greubel_, May 18 2022

%F a(n) = 13*a(n-1) - 66*a(n-2) + 162*a(n-3) - 189*a(n-4) + 81*a(n-5). - _Wesley Ivan Hurt_, Apr 16 2023

%e a(3) = 1^3*3^1+2^3*3^2+3^3*3^3 = 3+8*9+27*27 = 804.

%t CoefficientList[Series[3x(1 +12x +9x^2)/((1-3x)^4*(1-x)), {x, 0, 23}], x] (* _Michael De Vlieger_, May 01 2021 *)

%o (PARI) a(n) = sum(k=0, n, 3^k*k^3); \\ _Michel Marcus_, Apr 30 2021

%o (PARI) a(n) = ((((n<<2 - 6)*n + 12)*n - 11)*3^(n+1) + 33) >> 3; \\ _Kevin Ryde_, May 01 2021

%o (SageMath) [((4*n^3-6*n^2+12*n-11)*3^(n+1) +33)/8 for n in (0..30)] # _G. C. Greubel_, May 18 2022

%Y Cf. A036827, A062074.

%K nonn,easy

%O 0,2

%A _Sebastian Krüger_, Apr 30 2021