login
Binomial transform of Thue-Morse sequence A010060.
2

%I #30 Oct 01 2024 23:18:48

%S 0,1,3,6,11,20,36,64,115,216,430,892,1872,3888,7920,15840,31315,61744,

%T 122418,245348,497650,1019032,2096680,4312224,8826320,17925376,

%U 36070128,71915616,142239056,279671360,548106816,1073741824,2108053075

%N Binomial transform of Thue-Morse sequence A010060.

%H Reinhard Zumkeller, <a href="/A019302/b019302.txt">Table of n, a(n) for n = 0..1000</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F a(n) = Sum_{k=0..n} A010060(k) * A007318(n,k). - _Reinhard Zumkeller_, May 07 2014

%F G.f.: (1/2)*(1/(1 - 2*x) - (1/(1 - x))*Product_{k>=0} (1 - x^(2^k)/(1 - x)^(2^k))). - _Ilya Gutkovskiy_, Aug 20 2018

%t tm[0] = 0;

%t tm[n_?EvenQ] := tm[n] = tm[n/2]; tm[n_] := tm[n] = 1-tm[(n-1)/2];

%t a[n_] := Sum[tm[k]*Binomial[n, k], {k, 0, n}];

%t Table[a[n], {n, 0, 40}]

%t (* or (since 2015): *)

%t a[n_] := Sum[ThueMorse[k]*Binomial[n, k], {k, 0, n}];

%t Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Jun 30 2015, updated Jul 19 2022 *)

%o (Haskell)

%o a019302 = sum . zipWith (*) a010060_list . a007318_row

%o -- _Reinhard Zumkeller_, May 07 2014

%Y Cf. A007318, A010060, A019301.

%K nonn,nice

%O 0,3

%A _Jonas Wallgren_

%E More terms from _Carl Najafi_, Sep 08 2011