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A084851 Binomial transform of binomial(n+2,2). 5

%I

%S 1,4,13,38,104,272,688,1696,4096,9728,22784,52736,120832,274432,

%T 618496,1384448,3080192,6815744,15007744,32899072,71827456,156237824,

%U 338690048,731906048,1577058304,3388997632,7264534528,15535702016,33151778816

%N Binomial transform of binomial(n+2,2).

%C Essentially the same as A049611.

%H Vincenzo Librandi, <a href="/A084851/b084851.txt">Table of n, a(n) for n = 0..1000</a>

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

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

%F a(n) = (n^2 + 7*n + 8)*2^(n - 3).

%F a(n) = Sum_{k=0..n} C(n, k)*C(k+2, 2).

%F a(n) = A049611(n+1).

%e From _Bruno Berselli_, Jul 17 2018: (Start)

%e Let the triangle:

%e 1

%e 3, 4

%e 6, 9, 13

%e 10, 16, 25, 38

%e 15, 25, 41, 66, 104

%e 21, 36, 61, 102, 168, 272

%e 28, 49, 85, 146, 248, 416, 688

%e 36, 64, 113, 198, 344, 592, 1008, 1696, etc.

%e 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 The sequence is the right side of the triangle.

%e (End)

%p a := n -> hypergeom([-n, 3], [1], -1);

%p seq(round(evalf(a(n),32)), n=0..31); # _Peter Luschny_, Aug 02 2014

%t CoefficientList[ Series[(1 - x)^2/(1 - 2 x)^3, {x, 0, 28}], x] (* _Robert G. Wilson v_, Jun 28 2005 *)

%t LinearRecurrence[{6,-12,8},{1,4,13},30] (* _Harvey P. Dale_, Aug 05 2019 *)

%o (MAGMA) [(n^2+7*n+8)*2^(n-3): n in [0..40]]; // _Vincenzo Librandi_, Aug 03 2014

%Y Cf. A000217, A049611, A058396 (first differences).

%K nonn,easy

%O 0,2

%A _Paul Barry_, Jun 09 2003

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Last modified June 14 10:04 EDT 2021. Contains 345025 sequences. (Running on oeis4.)