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a(n) = n*(n+1)*(11*n+1)/6.
8

%I #33 Feb 21 2024 10:23:31

%S 0,4,23,68,150,280,469,728,1068,1500,2035,2684,3458,4368,5425,6640,

%T 8024,9588,11343,13300,15470,17864,20493,23368,26500,29900,33579,

%U 37548,41818,46400,51305,56544,62128,68068,74375,81060,88134,95608,103493,111800

%N a(n) = n*(n+1)*(11*n+1)/6.

%C Sums of rows of the triangle in A132111.

%H Vincenzo Librandi, <a href="/A132112/b132112.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F a(n) = A132121(n,3) for n > 2.

%F G.f.: x*(4+7*x)/(1-x)^4. - _Colin Barker_, Jun 06 2012

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Vincenzo Librandi_, Jun 29 2012

%F a(-n) = -A254407(n-1). - _Bruno Berselli_, Jan 31 2015

%F E.g.f.: exp(x)*x*(24 + 45*x + 11*x^2)/6. - _Stefano Spezia_, Feb 21 2024

%t CoefficientList[Series[x*(4+7*x)/(1-x)^4,{x,0,40}],x] (* _Vincenzo Librandi_, Jun 29 2012 *)

%t LinearRecurrence[{4,-6,4,-1},{0,4,23,68},40] (* _Harvey P. Dale_, Jun 28 2021 *)

%o (Magma) I:=[0, 4, 23, 68]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // _Vincenzo Librandi_, Jun 29 2012

%o (PARI) a(n)=n*(n+1)*(11*n+1)/6 \\ _Charles R Greathouse IV_, Oct 07 2015

%Y Cf. A000330, A033994, A050409, A132111, A132121, A132124, A254407.

%K nonn,easy

%O 0,2

%A _Reinhard Zumkeller_, Aug 10 2007