login
A047520
a(n) = 2*a(n-1) + n^2, a(0) = 0.
16
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
0,3
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)
Convolution of A000225 with A005408. - J. M. Bergot, Sep 19 2017
LINKS
Filippo Disanto, Some Statistics on the Hypercubes of Catalan Permutations, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.2.
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
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 *)
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
(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
(GAP) List([0..30], n-> 6*2^n -(n^2+4*n+6)); # G. C. Greubel, Jul 25 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Henry Bottomley, Jul 04 2000
STATUS
approved