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A sequence related to binomial(n+2, 2).
2

%I #18 Sep 08 2022 08:45:09

%S 1,8,61,450,3240,22896,159408,1096416,7464960,50388480,337602816,

%T 2247326208,14874679296,97955205120,642150789120,4192482779136,

%U 27270729105408,176789554200576,1142549512519680,7363096858460160

%N A sequence related to binomial(n+2, 2).

%C Binomial transform of A081894.

%C 5th binomial transform of binomial(n+2, 2), A000217.

%C 6th binomial transform of (1,2,1,0,0,0,...).

%H G. C. Greubel, <a href="/A081907/b081907.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (18,-108,216).

%F a(n) = 6^n*(n^2 + 23*n + 72)/72.

%F G.f.: (1-5*x)^2/(1-6*x)^3.

%F E.g.f.: (2 + 4*x + x^2)*exp(6*x)/2. - _G. C. Greubel_, Oct 17 2018

%p seq(coeff(series((1-5*x)^2/(1-6*x)^3,x,n+1), x, n), n = 0 .. 20); # _Muniru A Asiru_, Oct 18 2018

%t Table[6^n*(n^2+23*n+72)/72, {n,0,50}] (* or *) LinearRecurrence[{18,-108, 216}, {1, 8, 61}, 50] (* _G. C. Greubel_, Oct 17 2018 *)

%o (PARI) vector(50, n, n--; 6^n*(n^2 +23*n +72)/72) \\ _G. C. Greubel_, Oct 17 2018

%o (Magma) [6^n*(n^2 +23*n +72)/72: n in [0..50]]; // _G. C. Greubel_, Oct 17 2018

%o (GAP) List([1..20],n->6^(n-1)*(n^2+21*n+50))/72; # _Muniru A Asiru_, Oct 18 2018

%K easy,nonn

%O 0,2

%A _Paul Barry_, Mar 30 2003