OFFSET
1,2
COMMENTS
The first differences are in the third row of the square array of A072590.
The general formula for the partial sums of the sequence 1, 4*m, 9*m^2, 16*m^3, 25*m^4,...,n^2*m^(n-1),... is (n^2*m^(n+2)-(2*n*(n+1)-1)*m^(n+1)+(n+1)^2*m^n-m-1)/(m-1)^3 with m>1 (see also References).
REFERENCES
"Supplemento al Periodico di Matematica", Raffaello Giusti Editore (Livorno) - Apr / May, 1913 - p. 99 (Problem 1277, case x=3).
LINKS
Bruno Berselli, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (10,-36,54,-27).
FORMULA
a(n) = (3^n*(n^2 - n + 1) - 1)/2.
G.f.: x*(1+3*x)/((1-x)*(1-3*x)^3).
a(n) = 10*a(n-1) - 36*a(n-2) + 54*a(n-3) - 27a(n-4) for n>4.
E.g.f.: (1/2)*((1 + 9*x^2)*exp(x) - exp(-x))*exp(2*x). - G. C. Greubel, Aug 24 2016
MATHEMATICA
CoefficientList[Series[(1 + 3 x) / ((1 - x) (1 - 3 x)^3), {x, 0, 25}], x] (* Vincenzo Librandi, Aug 19 2013 *)
PROG
(PARI) a(n) = (3^n*(n^2-n+1)-1)/2 \\ Michel Marcus, Jun 07 2013
(Magma) [(3^n*(n^2-n+1)-1)/2: n in [1..25]]; // Vincenzo Librandi, Aug 19 2013
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Bruno Berselli, Dec 12 2010
STATUS
approved