A379703 Antidiagonal sums of A343052.
0, 0, 0, 6, 18, 31, 57, 80, 125, 161, 230, 282, 380, 451, 583, 676, 847, 965, 1180, 1326, 1590, 1767, 2085, 2296, 2673, 2921, 3362, 3650, 4160, 4491, 5075, 5452, 6115, 6541, 7288, 7766, 8602, 9135, 10065, 10656, 11685, 12337, 13470, 14186, 15428, 16211, 17567, 18420, 19895, 20821
Offset: 0
Links
- Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).
Programs
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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
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Mathematica
LinearRecurrence[{1,3,-3,-3,3,1,-1},{0,0,0,6,18,31,57,80,125},50] 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 *)
Formula
a(n) = (n - 2)*(108 + 43*n + 8*n^2 + 3*(-1)^n*(4 + n))/48 for n > 1.
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.
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).
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.
Conjecture: a(n) = A273790((n-1)/2) for n odd and > 1.