A332698 - OEIS

%I #9 Sep 13 2024 18:04:23

%S 0,6,25,65,134,240,391,595,860,1194,1605,2101,2690,3380,4179,5095,

%T 6136,7310,8625,10089,11710,13496,15455,17595,19924,22450,25181,28125,

%U 31290,34684,38315,42191,46320,50710,55369,60305,65526,71040,76855,82979,89420,96186

%N a(n) = (8*n^3 + 15*n^2 + 13*n)/6.

%C The end values of the partial rows on the main diagonal of A332662 in the representation in the example section.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4, -6, 4, -1).

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

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).

%F a(n) = binomial(n+2, 3) + binomial(n+1, 3) + 2*(n+1)*binomial(n+1, 2) + binomial(n, 1) = A331987(n) + n.

%p a := n -> (8*n^3 + 15*n^2 + 13*n)/6: seq(a(n), n=0..41);

%p gf := (x*(x^2 + x + 6))/(x - 1)^4: ser := series(gf, x, 44):

%p seq(coeff(ser, x, n), n=0..41);

%t LinearRecurrence[{4, -6, 4, -1}, {0, 6, 25, 65}, 42]

%t Table[(8n^3+15n^2+13n)/6,{n,0,50}] (* _Harvey P. Dale_, Sep 13 2024 *)

%Y Cf. A332662, A331987.

%K nonn,easy

%O 0,2

%A _Peter Luschny_, Feb 20 2020