A047520 a(n) = 2*a(n-1) + n^2, a(0) = 0.

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

Offset: 0

Author

Henry Bottomley, Jul 04 2000

Keywords

• nonn
• easy

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)

Convolution of A000225 with A005408. - J. M. Bergot, Sep 19 2017

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..3000
• Filippo Disanto, Some Statistics on the Hypercubes of Catalan Permutations, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.2.
• Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2).

Crossrefs

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

Programs

GAP

List([0..30], n-> 6*2^n -(n^2+4*n+6)); # G. C. Greubel, Jul 25 2019

Haskell

a047520 n = sum $ zipWith (*)
                  (reverse $ take n $ tail a000290_list) a000079_list
-- Reinhard Zumkeller, Nov 30 2012

Magma

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

Mathematica

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 *)

PARI

vector(30, n, n--; 6*2^n -(n^2+4*n+6)) \\ G. C. Greubel, Jul 25 2019

Sage

[6*2^n -(n^2+4*n+6) for n in (0..30)] # G. C. Greubel, Jul 25 2019

Formula

a(n) = 6*2^n - n^2 - 4*n - 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). - Harvey P. Dale, Aug 21 2011

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

a(n) = Sum_{k=0..n-1} A000079(n-k) * A000290(k). - Reinhard Zumkeller, Nov 30 2012

E.g.f.: 6*exp(2*x) -(6 +5*x +x^2)*exp(x). - G. C. Greubel, Jul 25 2019