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%I #49 Sep 08 2022 08:45:09
%S 1,3,5,6,5,1,-7,-20,-39,-65,-99,-142,-195,-259,-335,-424,-527,-645,
%T -779,-930,-1099,-1287,-1495,-1724,-1975,-2249,-2547,-2870,-3219,
%U -3595,-3999,-4432,-4895,-5389,-5915,-6474,-7067,-7695,-8359,-9060,-9799,-10577,-11395,-12254,-13155,-14099,-15087,-16120
%N Consider the triangle in which the n-th row starts with n, contains n terms and the difference of successive terms is 1,2,3,... up to n-1. Sequence gives row sums.
%C The triangle whose row sums are being considered is:
%C 1;
%C 2, 1;
%C 3, 2, 0;
%C 4, 3, 1, -2;
%C 5, 4, 2, -1, -5;
%C 6, 5, 3, 0, -4, -9;
%C 7, 6, 4, 1, -3, -8, -14;
%C The leading diagonal is given by A080956(n-1) = n*(3-n)/2.
%H Robert Israel, <a href="/A081498/b081498.txt">Table of n, a(n) for n = 1..10000</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^2 - binomial(n+1, n-2). - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 20 2004
%F a(n) = binomial(n,2)+binomial(n,1)-binomial(n,3). - _Zerinvary Lajos_, Jul 23 2006
%F a(n) = n*(1+6*n-n^2)/6. - _Karen A. Yeats_, Nov 20 2006
%F From _Michael Somos_, Jul 04 2012: (Start)
%F G.f.: x * (1 - x - x^2) / (1 - x)^4.
%F a(-1 - n) = A008778(n). (End)
%F E.g.f.: x*(6 +3*x -x^2)*exp(x)/6. - _G. C. Greubel_, Mar 06 2019
%e G.f. = x * (1 + 3*x + 5*x^2 + 6*x^3 + 5*x^4 + x^5 - 7*x^6 - 20*x^7 - 39*x^8 - 65*x^9 + ...).
%p seq(n^2-binomial(n+1,n-2),n=1..50); # C. Ronaldo
%p [seq(binomial(n,2)+binomial(n,1)-binomial(n,3), n=1..49)]; # _Zerinvary Lajos_, Jul 23 2006
%t LinearRecurrence[{4,-6,4,-1}, {1,3,5,6}, 50] (* _G. C. Greubel_, Mar 06 2019 *)
%o (PARI) {a(n) = if( n< 0, n = -2 - n; polcoeff( (1 + x - x^2) / (1 - x)^4 + x * O(x^n), n), polcoeff( (1 - x - x^2) / (1 - x)^4 + x * O(x^n), n))} /* _Michael Somos_, Jul 04 2012 */
%o (PARI) vector(50, n, n*(1+6*n-n^2)/6) \\ _G. C. Greubel_, Mar 06 2019
%o (GAP) List([1..50],n->n^2-Binomial(n+1,n-2)); # _Muniru A Asiru_, Mar 05 2019
%o (Magma) [n*(1+6*n-n^2)/6: n in [1..50]]; // _G. C. Greubel_, Mar 06 2019
%o (Sage) [n*(1+6*n-n^2)/6 for n in (1..50)] # _G. C. Greubel_, Mar 06 2019
%Y Cf. A008778, A080956, A081499.
%K sign,easy
%O 1,2
%A _Amarnath Murthy_, Mar 25 2003
%E More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 20 2004
%E Offset changed to 1 at the suggestion of _Michel Marcus_, Mar 05 2019
%E Formulas and programs addapted for offset 1 by _Michel Marcus_, Mar 05 2019