A037237 Expansion of (3 + x^2) / (1 - x)^4.

3, 12, 31, 64, 115, 188, 287, 416, 579, 780, 1023, 1312, 1651, 2044, 2495, 3008, 3587, 4236, 4959, 5760, 6643, 7612, 8671, 9824, 11075, 12428, 13887, 15456, 17139, 18940, 20863, 22912, 25091, 27404

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

This sequence is the partial sums of A058331. - J. M. Bergot, May 31 2012

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1)

Programs

Magma

I:=[3, 12, 31, 64]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)- Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jun 21 2012

Mathematica

CoefficientList[Series[(3+x^2)/(1-x)^4,{x,0,50}],x]  (* Harvey P. Dale, Mar 06 2011 *)
LinearRecurrence[{4,-6,4,-1},{3,12,31,64},40] (* Vincenzo Librandi Jun 21 2012 *)

PARI

x='x+O('x^50); Vec((3+x^2)/(1-x)^4) \\ G. C. Greubel, Jul 22 2017

Formula

a(n) = Sum_{k=0..n} (2*(k+1)^2 + 1). - Mike Warburton, Jul 07 2007, Sep 07 2007

a(n) = (n+1)*(2*n^2 + 7*n + 9)/3. - R. J. Mathar, Mar 29 2010

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 21 2012

E.g.f.: (1/3)*(9 + 27*x + 15*x^2 + 2*x^3)*exp(x). - G. C. Greubel, Jul 22 2017