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A176027 Binomial transform of A005563.

%I A176027 #38 Aug 13 2022 06:22:39
%S A176027 0,3,14,48,144,400,1056,2688,6656,16128,38400,90112,208896,479232,
%T A176027 1089536,2457600,5505024,12255232,27131904,59768832,131072000,
%U A176027 286261248,622854144,1350565888,2919235584,6291456000,13522436096
%N A176027 Binomial transform of A005563.
%C A176027 The numbers appear on the diagonal of a table T(n,k), where the left column contains the elements of A005563, and further columns are recursively T(n,k) = T(n,k-1)+T(n-1,k-1):
%C A176027 ....0....-1.....0.....0.....0.....0.....0.....0.....0.....0.
%C A176027 ....3.....3.....2.....2.....2.....2.....2.....2.....2.....2.
%C A176027 ....8....11....14....16....18....20....22....24....26....28.
%C A176027 ...15....23....34....48....64....82...102...124...148...174.
%C A176027 ...24....39....62....96...144...208...290...392...516...664.
%C A176027 ...35....59....98...160...256...400...608...898..1290..1806.
%C A176027 ...48....83...142...240...400...656..1056..1664..2562..3852.
%C A176027 ...63...111...194...336...576...976..1632..2688..4352..6914.
%C A176027 ...80...143...254...448...784..1360..2336..3968..6656.11008.
%C A176027 ...99...179...322...576..1024..1808..3168..5504..9472.16128.
%C A176027 ..120...219...398...720..1296..2320..4128..7296.12800.22272.
%C A176027 The second column is A142463, the third A060626, the fourth essentially A035008 and the fifth essentially A016802. Transposing the array gives A005563 and its higher order differences in the individual rows.
%H A176027 Vincenzo Librandi, <a href="/A176027/b176027.txt">Table of n, a(n) for n = 0..3000</a>
%H A176027 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-12,8).
%F A176027 G.f.: x*(-3+4*x)/(2*x-1)^3. - _R. J. Mathar_, Dec 11 2010
%F A176027 a(n) = 2^(n-2)*n*(5+n). - _R. J. Mathar_, Dec 11 2010
%F A176027 a(n) = A127276(n) - A127276(n+1).
%F A176027 a(n+1)-a(n) = A084266(n+1).
%F A176027 a(n+2) = 16*A058396(n) for n > 0.
%F A176027 a(n) = 2*a(n-1) + A001792(n).
%F A176027 a(n) = A001793(n) - 2^(n-1) for n > 0. - _Brad Clardy_, Mar 02 2012
%F A176027 a(n) = Sum_{k=0..n-1} Sum_{i=0..n-1} (k+3) * C(n-1,i). - _Wesley Ivan Hurt_, Sep 20 2017
%F A176027 From _Amiram Eldar_, Aug 13 2022: (Start)
%F A176027 Sum_{n>=1} 1/a(n) = 1322/75 - 124*log(2)/5.
%F A176027 Sum_{n>=1} (-1)^(n+1)/a(n) = 132*log(3/2)/5 - 782/75. (End)
%t A176027 LinearRecurrence[{6,-12,8},{0,3,14},30] (* _Harvey P. Dale_, Oct 19 2015 *)
%o A176027 (Magma) [2^(n-2)*n*(5+n) : n in [0..30]]; // _Vincenzo Librandi_, Oct 08 2011
%o A176027 (PARI) a(n)=n*(n+5)<<(n-2) \\ _Charles R Greathouse IV_, Sep 21 2017
%Y A176027 Cf. A001792, A001793, A005563, A058396, A084266, A127276.
%K A176027 nonn,easy
%O A176027 0,2
%A A176027 _Paul Curtz_, Dec 06 2010