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

Author

Stefano Spezia, Dec 30 2024

Keywords

• nonn
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Crossrefs

Cf. A273790, A343052.

Programs

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

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.