|
| |
|
|
A047520
|
|
a(n) = 2*a(n-1) + n^2, a(0) = 0.
|
|
9
|
|
|
|
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] (* From 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
|
| |
|
|
