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%I #45 Aug 14 2023 15:51:16
%S 0,1,19,70,170,335,581,924,1380,1965,2695,3586,4654,5915,7385,9080,
%T 11016,13209,15675,18430,21490,24871,28589,32660,37100,41925,47151,
%U 52794,58870,65395,72385,79856,87824,96305,105315,114870,124986,135679
%N a(n) = n*(16*n^2 + 3*n - 13)/6.
%C Generated by the formula n*(n+1)*(2*d*n-(2*d-3))/6 for d=8.
%C In fact, the sequence is related to A001107 by a(n) = n*A001107(n) - Sum_{k=0..n-1} A001107(k), and this is the case d=8 in the identity n*(n*(d*n-d+2)/2) - Sum_{k=0..n-1} k*(d*k-d+2)/2 = n*(n+1)*(2*d*n-2*d+3)/6. - _Bruno Berselli_, Dec 14 2010
%C Inverse binomial transform of this sequence: 0, 1, 17, 16, 0, 0 (0 continued). - _Bruno Berselli_, Dec 14 2010
%C Principal diagonal of the convolution array A213835. - _Clark Kimberling_, Jul 04 2012
%D E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93. - _Bruno Berselli_, Feb 13 2014
%H Vincenzo Librandi, <a href="/A172078/b172078.txt">Table of n, a(n) for n = 0..1000</a>
%H B. Berselli, A description of the recursive method in Comments lines: website <a href="http://www.lanostra-matematica.org/2008/12/sequenze-numeriche-e-procedimenti.html">Matem@ticamente</a> (in Italian), 2008.
%H <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</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) = n*(n+1)*(16*n-13)/6.
%F G.f.: x*(1+15*x)/(1-x)^4. - _Bruno Berselli_, Dec 14 2010
%F a(n) = Sum_{i=0..n-1} (n-i)*(16*i+1), with a(0)=0. - _Bruno Berselli_, Feb 10 2014
%F E.g.f.: x*(6 +51*x +16*x^2)*exp(x)/6. - _G. C. Greubel_, Aug 30 2019
%p A172078:=n->(16*n^3+3*n^2-13*n)/6: seq(A172078(n), n=0..50); # _Wesley Ivan Hurt_, Jan 21 2017
%t LinearRecurrence[{4,-6,4,-1}, {0,1,19,70}, 50] (* _Vincenzo Librandi_, Mar 01 2012 *)
%t Table[n (16n^2+3n-13)/6,{n,0,40}] (* _Harvey P. Dale_, Aug 14 2023 *)
%o (PARI) a(n)=n*(16*n^2+3*n-13)/6 \\ _Charles R Greathouse IV_, Oct 07 2015
%o (Magma) [n*(n+1)*(16*n-13)/6: n in [0..40]]; // _G. C. Greubel_, Aug 30 2019
%o (Sage) [n*(n+1)*(16*n-13)/6 for n in (0..40)] # _G. C. Greubel_, Aug 30 2019
%o (GAP) List([0..40], n-> n*(n+1)*(16*n-13)/6); # _G. C. Greubel_, Aug 30 2019
%Y Cf. A001107.
%Y Cf. similar sequences listed in A237616.
%K nonn,easy
%O 0,3
%A _Vincenzo Librandi_, Jan 25 2010
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