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
Examples
a(3) = 1^3*3^1+2^3*3^2+3^3*3^3 = 3+8*9+27*27 = 804.
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).
Programs
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Mathematica
CoefficientList[Series[3x(1 +12x +9x^2)/((1-3x)^4*(1-x)), {x, 0, 23}], x] (* Michael De Vlieger, May 01 2021 *) -
PARI
a(n) = sum(k=0, n, 3^k*k^3); \\ Michel Marcus, Apr 30 2021
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PARI
a(n) = ((((n<<2 - 6)*n + 12)*n - 11)*3^(n+1) + 33) >> 3; \\ Kevin Ryde, May 01 2021
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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
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