OFFSET
0,3
COMMENTS
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.
(End)
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).
FORMULA
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
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