A036800 a(n) = -6 + 2^(n+1)*(3 - 2*n + n^2).

0, 2, 18, 90, 346, 1146, 3450, 9722, 26106, 67578, 169978, 417786, 1007610, 2392058, 5603322, 12976122, 29753338, 67633146, 152567802, 341835770, 761266170, 1686110202, 3716153338, 8153726970, 17817403386, 38788923386

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

This sequence is a part of a class of sequences of the type: a(n) = sum(i=0,n,(C^i)*(i^k)). This sequence has C=2, k=2. Sequence A036799 has C=2, k=1. Suppose C>=2, k>=1 are integers. What is the general closed form for a(n)? - Ctibor O. Zizka, Feb 07 2008

References

• M. Petkovsek et al., A=B, Peters, 1996, p. 97.
• Jolley, Summation of Series, Dover (1961), p. 6.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• S. Sykora, Finite and Infinite Sums of the Power Series (k^p)(x^k), DOI 10.3247/SL1Math06.002, Section V.
• Index entries for linear recurrences with constant coefficients, signature (7,-18,20,-8).

Crossrefs

Cf. A232599, A232600, A232601, A232602.

Programs

Magma

[-6+2^(n+1)*(3-2*n+n^2): n in [0..30]]; // Vincenzo Librandi, Oct 04 2011

Maple

A036800:= n-> 2^(n+1)*(3-2*n+n^2) -6; seq(A036800(n), n=0..30); # G. C. Greubel, Mar 31 2021

Mathematica

Table[ -6+2^(n+1)*(3-2*n+n^2),{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Mar 08 2010 *)
LinearRecurrence[{7,-18,20,-8},{0,2,18,90},30] (* Harvey P. Dale, Jun 13 2015 *)

PARI

a(n)=2^(n+1)*(3 - 2*n + n^2) - 6 \\ Charles R Greathouse IV, Jun 11 2015

Sage

[2^(n+1)*(3-2*n+n^2) -6 for n in (0..30)] # G. C. Greubel, Mar 31 2021

Formula

a(n) = Sum_{k=0..n} 2^k * k^2. - Benoit Cloitre, Jun 11 2003

From R. J. Mathar, Oct 03 2011: (Start)

G.f.: 2*x*(1+2*x) / ( (1-x)*(1-2*x)^3 ).

a(n) = 2*A036826(n). (End)

a(0)=0, a(1)=2, a(2)=18, a(3)=90, a(n)=7*a(n-1)-18*a(n-2)+ 20*a(n-3)- 8*a(n-4). - Harvey P. Dale, Jun 13 2015

a(n) = Sum_{k=0..n} Sum_{i=0..n} k^2 * C(k,i). - Wesley Ivan Hurt, Sep 21 2017

E.g.f.: 2*(3 -2*x +4*x^2)*exp(2*x) -6*exp(x). - G. C. Greubel, Mar 31 2021