A391997 - OEIS

%I #12 May 31 2026 21:52:38

%S 0,0,4,27,112,325,792,1666,3200,5670,9500,15125,23184,34307,49392,

%T 69300,95232,128316,170100,222015,286000,363825,457864,570262,703872,

%U 861250,1045772,1260441,1509200,1795535,2124000,2498600,2924544,3406392,3950052,4560675,5244912

%N a(n) = Sum_{k=0..n} floor((k/2)^2)*n^2. Row sums of A391996.

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-8,6,6,-8,0,3,-1).

%F a(n) = [x^n] x^2*(9*x^4 + 21*x^3 + 31*x^2 + 15*x + 4) / ((x - 1)^6*(x + 1)^3).

%F a(n) ~ n^5/12. - _Charles R Greathouse IV_, May 31 2026

%p seq(add(floor((k/2)^2)*n^2, k = 0..n), n = 0..36);

%p gf := (9*x^4 + 21*x^3 + 31*x^2 + 15*x + 4)*x^2/((x - 1)^6*(x + 1)^3):

%p ser := series(gf, x, 40): seq(coeff(ser, x, n), n = 0..36);

%t LinearRecurrence[{3, 0, -8, 6, 6, -8, 0, 3, -1}, {0, 0, 4, 27, 112, 325, 792, 1666, 3200}, 40] (* _Hugo Pfoertner_, Jan 10 2026 *)

%o (PARI) a(n)=(2*n^5+3*n^4-2*n^3-n%2*3*n^2)/24 \\ _Charles R Greathouse IV_, May 31 2026

%Y Cf. A391997, A391996.

%K nonn,easy

%O 0,3

%A _Peter Luschny_, Jan 10 2026