A343808 Partial sums of A062074.

0, 3, 75, 804, 5988, 36363, 193827, 943968, 4303200, 18652107, 77701107, 313483764, 1231813812, 4734541443, 17859008379, 66286569504, 242605938720, 877071559539, 3136507851387, 11108459253540, 39002734461540, 135876065474523, 470021588191155, 1615461644979264

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (13,-66,162,-189,81).

Formula

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

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

G.f.: 3*x*(1 + 12*x + 9*x^2)/((1 - 3*x)^4*(1 - x)). - Stefano Spezia, May 01 2021

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

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

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

Example

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

Mathematica

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

Prog

(PARI) a(n) = sum(k=0, n, 3^k*k^3); \\ Michel Marcus, Apr 30 2021
(PARI) a(n) = ((((n<<2 - 6)*n + 12)*n - 11)*3^(n+1) + 33) >> 3; \\ Kevin Ryde, May 01 2021
(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

Crossrefs

Cf. A036827, A062074.

Sequence in context: A265956 A189805 A230145 * A125520 A163131 A060869

Adjacent sequences: A343805 A343806 A343807 * A343809 A343810 A343811

Keyword

nonn,easy

Author

Sebastian Krüger, Apr 30 2021

Status

approved