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A084851 Binomial transform of binomial(n+2,2).

%I A084851 #31 Sep 08 2022 08:45:11
%S A084851 1,4,13,38,104,272,688,1696,4096,9728,22784,52736,120832,274432,
%T A084851 618496,1384448,3080192,6815744,15007744,32899072,71827456,156237824,
%U A084851 338690048,731906048,1577058304,3388997632,7264534528,15535702016,33151778816
%N A084851 Binomial transform of binomial(n+2,2).
%C A084851 Essentially the same as A049611.
%H A084851 Vincenzo Librandi, <a href="/A084851/b084851.txt">Table of n, a(n) for n = 0..1000</a>
%H A084851 Igor Makhlin, <a href="https://arxiv.org/abs/2003.02916">Gröbner fans of Hibi ideals, generalized Hibi ideals and flag varieties</a>, arXiv:2003.02916 [math.CO], 2020.
%H A084851 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-12,8).
%F A084851 G.f.: (1 - x)^2/(1 - 2*x)^3.
%F A084851 a(n) = (n^2 + 7*n + 8)*2^(n - 3).
%F A084851 a(n) = Sum_{k=0..n} C(n, k)*C(k+2, 2).
%F A084851 a(n) = A049611(n+1).
%e A084851 From _Bruno Berselli_, Jul 17 2018: (Start)
%e A084851 Let the triangle:
%e A084851    1
%e A084851    3,  4
%e A084851    6,  9,  13
%e A084851   10, 16,  25,  38
%e A084851   15, 25,  41,  66, 104
%e A084851   21, 36,  61, 102, 168, 272
%e A084851   28, 49,  85, 146, 248, 416,  688
%e A084851   36, 64, 113, 198, 344, 592, 1008, 1696, etc.
%e A084851 where the first column is A000217 (without 0). The other terms are calculated with the recurrence T(r, c) = T(r-1, c-1) + T(r, c-1).
%e A084851 The sequence is the right side of the triangle.
%e A084851 (End)
%p A084851 a := n -> hypergeom([-n, 3], [1], -1);
%p A084851 seq(round(evalf(a(n),32)), n=0..31); # _Peter Luschny_, Aug 02 2014
%t A084851 CoefficientList[ Series[(1 - x)^2/(1 - 2 x)^3, {x, 0, 28}], x] (* _Robert G. Wilson v_, Jun 28 2005 *)
%t A084851 LinearRecurrence[{6,-12,8},{1,4,13},30] (* _Harvey P. Dale_, Aug 05 2019 *)
%o A084851 (Magma) [(n^2+7*n+8)*2^(n-3): n in [0..40]]; // _Vincenzo Librandi_, Aug 03 2014
%Y A084851 Cf. A000217, A049611, A058396 (first differences).
%K A084851 nonn,easy
%O A084851 0,2
%A A084851 _Paul Barry_, Jun 09 2003