A103424 - OEIS

%I #22 Apr 11 2022 18:01:57

%S 1,2,0,8,0,32,0,128,0,512,0,2048,0,8192,0,32768,0,131072,0,524288,0,

%T 2097152,0,8388608,0,33554432,0,134217728,0,536870912,0,2147483648,0,

%U 8589934592,0,34359738368,0,137438953472,0,549755813888,0,2199023255552

%N Expansion of e.g.f.: 1 + sinh(2*x).

%C Binomial transform is A103425.

%H Vincenzo Librandi, <a href="/A103424/b103424.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,4).

%F G.f.: (1+2*x-4*x^2)/(1-4*x^2).

%F E.g.f.: 1 + sinh(2*x).

%F a(n) = 0^n+(2^n-(-2)^n)/2.

%F a(n) = Sum_{k=0..n} binomial(n, k)*(-1)^(k(n-k)).

%F a(n+1) = 2*A199572(n) = 2*A077957(n)^2. [_Ralf Stephan_, Jul 17 2013]

%t With[{nn=50},CoefficientList[Series[1+Sinh[2x],{x,0,nn}],x] Range[ 0,nn-1]!] (* _Harvey P. Dale_, Jun 29 2014 *)

%t CoefficientList[Series[(1 + 2 x - 4 x^2)/(1 - 4 x^2), {x, 0, 50}], x] (* _Vincenzo Librandi_, Jun 30 2014 *)

%Y Cf. A077957, A103425, A199572.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Feb 05 2005