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