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A115390 Binomial transform of tribonacci sequence A000073. 6

%I

%S 0,0,1,4,12,34,96,272,772,2192,6224,17672,50176,142464,404496,1148480,

%T 3260864,9258528,26287616,74638080,211918912,601698560,1708394752,

%U 4850622592,13772308480,39103533056,111026143488,315235058688

%N Binomial transform of tribonacci sequence A000073.

%C See also A117189 Binomial transform of the tribonacci sequence A000073.

%H Reinhard Zumkeller, <a href="/A115390/b115390.txt">Table of n, a(n) for n = 0..1000</a>

%H J. Pan, <a href="https://cs.uwaterloo.ca/journals/JIS/OL13/Pan/pan8.html">Multiple Binomial Transforms and Families of Integer Sequences </a>, J. Int. Seq. 13 (2010), 10.4.2

%H J. Pan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Pan/pan12.html"> Some Properties of the Multiple Binomial Transform and the Hankel Transform of Shifted Sequences </a>, J. Int. Seq. 14 (2011) # 11.3.4, remark 14.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BinomialTransform.html">Binomial Transform</a>.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4, -4, 2).

%F a(n) = SUM[k=0..n] C(n,k)*A000073(k).

%F O.g.f.: -x^2/(-1+4*x-4*x^2+2*x^3). - _R. J. Mathar_, Apr 02 2008

%F 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

%e 1*0 = 0.

%e 1*0 + 1*0 = 0.

%e 1*0 + 2*0 + 1*1 = 1.

%e 1*0 + 3*0 + 3*1 + 1* 1 = 4.

%e 1*0 + 4*0 + 6*1 + 4*1 + 1*2 = 12.

%t 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 *)

%o (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

%o (Haskell)

%o a115390 n = a115390_list !! n

%o a115390_list = 0 : 0 : 1 : map (* 2) (zipWith (-) a115390_list

%o (tail $ map (* 2) $ zipWith (-) a115390_list (tail a115390_list)))

%o -- _Reinhard Zumkeller_, Oct 21 2011

%Y Cf. A000073, A117189. Trisection of A103685.

%K easy,nonn

%O 0,4

%A _Jonathan Vos Post_, Mar 08 2006

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Last modified September 25 00:48 EDT 2020. Contains 337333 sequences. (Running on oeis4.)