A379703 - OEIS

%I #13 Jan 02 2025 12:47:40

%S 0,0,0,6,18,31,57,80,125,161,230,282,380,451,583,676,847,965,1180,

%T 1326,1590,1767,2085,2296,2673,2921,3362,3650,4160,4491,5075,5452,

%U 6115,6541,7288,7766,8602,9135,10065,10656,11685,12337,13470,14186,15428,16211,17567,18420,19895,20821

%N Antidiagonal sums of A343052.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-3,-3,3,1,-1).

%F a(n) = (n - 2)*(108 + 43*n + 8*n^2 + 3*(-1)^n*(4 + n))/48 for n > 1.

%F a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7) for n > 8.

%F G.f.: x^3*(6 + 12*x- 5*x^2 - 10*x^3 + 2*x^4 + 3*x^5)/((1 - x)^4*(1 + x)^3).

%F E.g.f.: (x^3/6 + 9*x^2/8 + x - 5)*cosh(x) + (4*x^3 + 24*x^2 + 33*x - 96)*sinh(x)/24.

%F Conjecture: a(n) = A273790((n-1)/2) for n odd and > 1.

%t LinearRecurrence[{1,3,-3,-3,3,1,-1},{0,0,0,6,18,31,57,80,125},50]

%t CoefficientList[Series[x^3*(6+12*x-5*x^2-10*x^3+2*x^4+3*x^5)/((1-x)^4*(1+x)^3),{x,0,60}],x] (* _Vincenzo Librandi_, Jan 02 2025 *)

%o (Magma) [0,0] cat [(n - 2)*(108 + 43*n + 8*n^2 + 3*(-1)^n*(4 + n))/48: n in [2..60]]; // _Vincenzo Librandi_, Jan 02 2025

%Y Cf. A273790, A343052.

%K nonn,easy

%O 0,4

%A _Stefano Spezia_, Dec 30 2024