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Binomial transform of decagonal numbers A001107.
2

%I #23 Oct 09 2022 20:28:09

%S 0,1,12,60,224,720,2112,5824,15360,39168,97280,236544,565248,1331200,

%T 3096576,7127040,16252928,36765696,82575360,184287232,408944640,

%U 902823936,1983905792,4341104640,9462349824,20552089600,44493176832,96032784384,206695301120,443723808768

%N Binomial transform of decagonal numbers A001107.

%H Vincenzo Librandi, <a href="/A086950/b086950.txt">Table of n, a(n) for n = 0..1000</a>

%H Milan Janjić, <a href="https://www.emis.de/journals/JIS/VOL21/Janjic2/janjic103.html">Pascal Matrices and Restricted Words</a>, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-12,8).

%F a(n) = n*2^n*(2*n-1)/2.

%F G.f.: x*(1+6*x)/(1-2*x)^3.

%F a(0)=0, a(1)=1, a(2)=12, a(n)=6*a(n-1)-12*a(n-2)+8*a(n-3). - _Harvey P. Dale_, Dec 13 2015

%F E.g.f.: exp(2*x)*x*(1 + 4*x). - _Stefano Spezia_, Oct 09 2022

%t LinearRecurrence[{6,-12,8},{0,1,12},30] (* _Harvey P. Dale_, Dec 13 2015 *)

%o (Magma) [n*2^n*(2*n-1)/2: n in [0..30] ]; // _Vincenzo Librandi_, Aug 22 2011

%Y Binomial transform is A086951.

%K easy,nonn

%O 0,3

%A _Paul Barry_, Jul 24 2003