A115390 Binomial transform of tribonacci sequence A000073.
0, 0, 1, 4, 12, 34, 96, 272, 772, 2192, 6224, 17672, 50176, 142464, 404496, 1148480, 3260864, 9258528, 26287616, 74638080, 211918912, 601698560, 1708394752, 4850622592, 13772308480, 39103533056, 111026143488, 315235058688, 895042726912, 2541282959872
Offset: 0
Examples
1*0 = 0. 1*0 + 1*0 = 0. 1*0 + 2*0 + 1*1 = 1. 1*0 + 3*0 + 3*1 + 1* 1 = 4. 1*0 + 4*0 + 6*1 + 4*1 + 1*2 = 12.
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
- J. Pan, Multiple Binomial Transforms and Families of Integer Sequences, J. Int. Seq. 13 (2010), 10.4.2
- J. Pan, Some Properties of the Multiple Binomial Transform and the Hankel Transform of Shifted Sequences, J. Int. Seq. 14 (2011) # 11.3.4, remark 14.
- Eric Weisstein's World of Mathematics, Binomial Transform.
- Index entries for linear recurrences with constant coefficients, signature (4,-4,2).
Programs
-
Haskell
a115390 n = a115390_list !! n a115390_list = 0 : 0 : 1 : map (* 2) (zipWith (-) a115390_list (tail $ map (* 2) $ zipWith (-) a115390_list (tail a115390_list))) -- Reinhard Zumkeller, Oct 21 2011
-
Mathematica
b[0]=b[1]=0;b[2]=1;b[n_]:=b[n]=b[n-1]+b[n-2]+b[n-3]; a[n_]:=Sum[n!/(k!*(n-k)!)*b[k],{k,0,n}];Table[a[n],{n,0,27}] (* Farideh Firoozbakht, Mar 11 2006 *) -
Maxima
sum(sum(binomial(j-1,k-1)*2^(j-k)*binomial(n-j+k-1,2*k-1),j,k,n-k),k,1,n); /* Vladimir Kruchinin, Aug 18 2010 */
Formula
a(n) = Sum_{k=0..n} C(n,k)*A000073(k).
O.g.f.: -x^2/(-1+4*x-4*x^2+2*x^3). - R. J. Mathar, Apr 02 2008
a(n) = sum(sum(binomial(j-1,k-1)*2^(j-k)*binomial(n-j+k-1,2*k-1),j,k,n-k),k,1,n). - Vladimir Kruchinin, Aug 18 2010
Comments