A387263 - OEIS

%I #21 Dec 29 2025 16:42:28

%S 0,0,1,5,18,56,164,468,1325,3751,10653,30381,87003,250095,721300,

%T 2086308,6049629,17580415,51187605,149293221,436088521,1275559185,

%U 3735597612,10952250100,32143070778,94422309606,277608161509,816829091513,2405170198470,7086849120836

%N Partial sums of A247287.

%C Second partial sums of A097861.

%C For p Pythagorean prime (A002144), a(p) - 1 == 0 (mod p).

%C For p prime of the form 4*k + 3 (A002145), a(p) + 1 == 0 (mod p).

%H Paolo Xausa, <a href="/A387263/b387263.txt">Table of n, a(n) for n = 0..2000</a>

%F G.f.: ((1-x)/sqrt(1-2*x-3*x^2)-1) / (2*(1-x)^3).

%F E.g.f.: exp(x)*(g(x) + 2*Integral_{x=-oo..oo} g(x) dx + Integral_{x=-oo..oo} (Integral_{x=-oo..oo} g(x) dx) dx) where g(x) = (-1+BesselI(0, 2*x))/2.

%F D-finite with recurrence n*a(n) = (6*n-3)*a(n-1) - (11*n-11)*a(n-2) + (4*n-6)*a(n-3) + (9*n-18)*a(n-4) - (10*n-25)*a(n-5) + (3*n-9)*a(n-6).

%F a(n) = Sum_{j=0..n}(Sum_{k=0..j} A390369(j, k)).

%F a(n) = a(n-2) - (n+1) + A383527(n) for n > 1.

%F a(n) = (A385641(n) - binomial(n+2, 2)) / 2.

%F From _Mélika Tebni_, Dec 29 2025: (Start)

%F a(n) = Sum_{k=0..n} A391467(n, k).

%F a(n) = (Sum_{k=0..n} ((n-k+1)*A002426(k) - k) - (n+1)) / 2.

%F For n >= 2, 2*a(n) - 4*a(n-1) + 2*a(n-2) = A180282(n) = A002426(n) - 1. (End)

%p a := series(exp(x)*((-1+BesselI(0, 2*x))/2 + 2*int((-1+BesselI(0, 2*x))/2, x) + int(int((-1+BesselI(0, 2*x))/2, x), x)), x = 0, 30):

%p seq(n!*coeff(a, x, n), n = 0 .. 29);

%p # Recurrence:

%p a := proc (n) option remember; `if`(n < 7, [0, 0, 1, 5, 18, 56, 164][n+1], 1/n*((6*n-3)*a(n-1) - (11*n-11)*a(n-2) + (4*n-6)*a(n-3) + (9*n-18)*a(n-4) - (10*n-25)*a(n-5) + (3*n-9)*a(n-6))) end: seq(a(n), n = 0 .. 29);

%t Module[{x}, CoefficientList[Series[((1 - x)/Sqrt[1 - 2*x - 3*x^2] - 1)/(2*(1 - x)^3), {x, 0, 30}], x]] (* _Paolo Xausa_, Dec 29 2025 *)

%o (Python)

%o from math import comb

%o def a(n):

%o     return (sum(comb(n+1, k+1)*comb(2*(k//2), k//2) for k in range(n + 1)) - comb(n+2, 2))//2

%o print([a(n) for n in range(30)])

%Y Cf. A002144, A002145, A002426, A097861, A180282, A247287, A383527, A385641, A390369, A391467.

%K nonn,easy

%O 0,4

%A _Mélika Tebni_, Nov 25 2025