A081912 - OEIS

%I #16 Nov 12 2025 23:21:35

%S 1,6,37,234,1512,9936,66096,443232,2985984,20155392,136048896,

%T 917070336,6167549952,41358864384,276451356672,1841557856256,

%U 12224809598976,80871817347072,533189772509184,3503818505060352,22952550207062016,149902496042582016,976194303496814592

%N a(n) = 6^n*(n^2 - n + 72)/72.

%C Binomial transform of A081911 6th binomial transform of (1,0,1,0,0,0,...). Case k=6 where a(n,k) = k^n*(n^2 - n + 2*k^2)/(2*k^2) with g.f. (1 - 2*k*x + (k^2+1)*x^2)/(1-k*x)^3.

%H Vincenzo Librandi, <a href="/A081912/b081912.txt">Table of n, a(n) for n = 0..150</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 - n + 72)/72.

%F G.f.: (1 - 12*x + 37*x^2)/(1-6*x)^3.

%F From _Elmo R. Oliveira_, Nov 12 2025: (Start)

%F E.g.f.: (1 + x^2/2)*exp(6*x).

%F a(n) = 18*a(n-1) - 108*a(n-2) + 216*a(n-3). (End)

%o (Magma) [6^n*(n^2-n+72)/72: n in [0..40]]; // _Vincenzo Librandi_, Apr 27 2011

%Y Cf. A081911.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Mar 31 2003