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%I #41 Sep 08 2022 08:45:32
%S 1,3,8,20,45,91,168,288,465,715,1056,1508,2093,2835,3760,4896,6273,
%T 7923,9880,12180,14861,17963,21528,25600,30225,35451,41328,47908,
%U 55245,63395,72416,82368,93313,105315,118440,132756,148333,165243,183560,203360,224721,247723,272448
%N Row sums of triangle A134392.
%C Binomial transform of [1, 2, 3, 4, 2, 0, 0, 0, ...].
%C The Kn4 triangle sums of A139600 are given by this sequence. For the definitions of the Kn4 and other triangle sums see A180662. - _Johannes W. Meijer_, Apr 29 2011
%H Vincenzo Librandi, <a href="/A134393/b134393.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F From _R. J. Mathar_, Jun 08 2008: (Start)
%F O.g.f.: x*(1-2*x+3*x^2)/(1-x)^5.
%F a(n) = A014628(n+1). (End)
%F a(n) = binomial(n+3,4) - 2*binomial(n+2,4) + 3*binomial(n+1,4). - _Johannes W. Meijer_, Apr 29 2011, corrected by _Eric Rowland_, Aug 16 2017
%F a(n) = n*(n + 1)*(n^2 - 3*n + 8)/12. - _Johannes W. Meijer_, Apr 29 2011, corrected by _Eric Rowland_, Aug 16 2017
%e a(4) = 20 = (1, 3, 3, 1) dot (1, 2, 3, 4) = (1 + 6 + 9 + 4).
%e a(4) = sum of row 4 terms of triangle A134392: (8 + 7 + 4 + 1).
%t Table[(n^4 - 2*n^3 + 5*n^2 + 8*n)/12, {n, 1, 40}] (* _Vincenzo Librandi_, Feb 04 2013 *)
%t LinearRecurrence[{5,-10,10,-5,1},{1,3,8,20,45},50] (* _Harvey P. Dale_, May 28 2018 *)
%o (Magma) [Binomial(n+3, 4)-2*Binomial(n+2, 4)+ 3*Binomial(n+1, 4): n in [1..40]]; // _Vincenzo Librandi_, Feb 04 2013
%o (PARI) x='x+O('x^99); Vec(x*(1-2*x+3*x^2)/(1-x)^5) \\ _Altug Alkan_, Aug 16 2017
%Y Cf. A000332, A014628, A134392.
%K nonn,easy
%O 1,2
%A _Gary W. Adamson_, Oct 23 2007
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