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A047520 a(n) = 2*a(n-1) + n^2, a(0) = 0. 11
0, 1, 6, 21, 58, 141, 318, 685, 1434, 2949, 5998, 12117, 24378, 48925, 98046, 196317, 392890, 786069, 1572462, 3145285, 6290970, 12582381, 25165246, 50331021, 100662618, 201325861, 402652398, 805305525, 1610611834, 3221224509 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Convolution of squares (A000290) and powers of 2 (A000079) - Graeme McRae, Jun 07 2006

Antidiagonal sums of the convolution array A213568.  [Clark Kimberling, Jun 18 2012]

This is the partial sums of A050488. - J. M. Bergot, Oct 01 2012

From Peter Bala, Nov 29 2012: (Start)

This is the case m = 2 of the recurrence a(n) = m*a(n-1) + n^m, m = 1,2,..., with a(0) = 0.

The recurrence has the solution a(n) = m^n*sum {i = 1..n} i^m/m^i and has the o.g.f. A(m,x)/((1-m*x)*(1-x)^(m+1)), where A(m,x) denotes the m-th Eulerian polynomial of A008292.

For other cases see A000217 (m = 1), A066999 (m = 3) and A067534 (m = 4).

(End)

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..3000

Index to sequences with linear recurrences with constant coefficients, signature (5,-9,7,-2).

FORMULA

a(n) = 6*2^n-n^2-4n-6 = 6*A000225(n)-A028347(n+2)

a(n) = 2^n*sum(i = 1, n, i^2/2^i) - Benoit Cloitre, Jan 27 2002

a(0)=0, a(1)=1, a(2)=6, a(3)=21, a(n)=5*a(n-1)-9*a(n-2)+7*a(n-3)- 2*a(n-4) [From Harvey P. Dale, Aug 21 2011]

G.f.: (x*(x+1))/((x-1)^3*(2*x-1)) [From Harvey P. Dale, Aug 21 2011]

a(n) = sum(A000079(n-k) * A000290(k): k=0..n-1). - Reinhard Zumkeller, Nov 30 2012

MATHEMATICA

k=0; lst={}; Do[k=2*k+n^2; AppendTo[lst, k], {n, 0, 5!}]; lst [From Vladimir Joseph Stephan Orlovsky, Dec 05 2009]

RecurrenceTable[{a[0]==0, a[n]==2a[n-1]+n^2}, a[n], {n, 30}] (* or *) LinearRecurrence[{5, -9, 7, -2}, {0, 1, 6, 21}, 31] (* Harvey P. Dale, Aug 21 2011 *)

f[n_] := 2^n*Sum[i^2/2^i, {i, n}]; Array[f, 30] (* Robert G. Wilson v, Nov 28 2012 *)

PROG

(MAGMA) [ 6*2^n-n^2-4*n-6: n in [0..30]]; // Vincenzo Librandi, Aug 22 2011

(Haskell)

a047520 n = sum $ zipWith (*)

                  (reverse $ take n $ tail a000290_list) a000079_list

-- Reinhard Zumkeller, Nov 30 2012

CROSSREFS

Cf. A000295. A000217, A008292, A066999, A067534.

Sequence in context: A056341 A144899 A053809 * A143115 A066524 A113070

Adjacent sequences:  A047517 A047518 A047519 * A047521 A047522 A047523

KEYWORD

nonn,easy

AUTHOR

Henry Bottomley, Jul 04 2000

STATUS

approved

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Last modified October 24 18:26 EDT 2014. Contains 248516 sequences.