A047520 - OEIS

%I #55 Sep 08 2022 08:44:57

%S 0,1,6,21,58,141,318,685,1434,2949,5998,12117,24378,48925,98046,

%T 196317,392890,786069,1572462,3145285,6290970,12582381,25165246,

%U 50331021,100662618,201325861,402652398,805305525,1610611834,3221224509

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

%C Convolution of squares (A000290) and powers of 2 (A000079). - _Graeme McRae_, Jun 07 2006

%C Antidiagonal sums of the convolution array A213568. - _Clark Kimberling_, Jun 18 2012

%C This is the partial sums of A050488. - _J. M. Bergot_, Oct 01 2012

%C From _Peter Bala_, Nov 29 2012: (Start)

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

%C 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.

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

%C (End)

%C Convolution of A000225 with A005408. - _J. M. Bergot_, Sep 19 2017

%H Vincenzo Librandi, <a href="/A047520/b047520.txt">Table of n, a(n) for n = 0..3000</a>

%H Filippo Disanto, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Disanto/disanto5.html">Some Statistics on the Hypercubes of Catalan Permutations</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.2.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5,-9,7,-2).

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

%F a(n) = 2^n*Sum_{i=1..n} i^2 / 2^i. - _Benoit Cloitre_, Jan 27 2002

%F 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

%F G.f.: x*(1+x)/((1-x)^3*(1-2*x)). - _Harvey P. Dale_, Aug 21 2011

%F a(n) = Sum_{k=0..n-1} A000079(n-k) * A000290(k). - _Reinhard Zumkeller_, Nov 30 2012

%F E.g.f.: 6*exp(2*x) -(6 +5*x +x^2)*exp(x). - _G. C. Greubel_, Jul 25 2019

%t 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 *)

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

%o (Magma) [ 6*2^n-n^2-4*n-6: n in [0..30]]; // _Vincenzo Librandi_, Aug 22 2011

%o (Haskell)

%o a047520 n = sum $ zipWith (*)

%o                   (reverse $ take n $ tail a000290_list) a000079_list

%o -- _Reinhard Zumkeller_, Nov 30 2012

%o (PARI) vector(30, n, n--; 6*2^n -(n^2+4*n+6)) \\ _G. C. Greubel_, Jul 25 2019

%o (Sage) [6*2^n -(n^2+4*n+6) for n in (0..30)] # _G. C. Greubel_, Jul 25 2019

%o (GAP) List([0..30], n-> 6*2^n -(n^2+4*n+6)); # _G. C. Greubel_, Jul 25 2019

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

%K nonn,easy

%O 0,3

%A _Henry Bottomley_, Jul 04 2000