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A385897 a(n) = 1 - 5*(n + 1)^2 + 5*(n + 1)^4.

%I A385897 #12 Jul 24 2025 10:52:30
%S A385897 1,61,361,1201,3001,6301,11761,20161,32401,49501,72601,102961,141961,
%T A385897 191101,252001,326401,416161,523261,649801,798001,970201,1168861,
%U A385897 1396561,1656001,1950001,2281501,2653561,3069361,3532201,4045501,4612801,5237761,5924161,6675901
%N A385897 a(n) = 1 - 5*(n + 1)^2 + 5*(n + 1)^4.
%H A385897 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A385897 a(n) = [x^n] (-x^4 + 4*x^3 - 66*x^2 - 56*x - 1)/(x - 1)^5.
%F A385897 a(n) = 5! * [x^5] (1 - sin(n*x))^(-1/n) for n > 0.
%F A385897 a(n) = A385896(n + 5, 5).
%F A385897 A000290(n) = (a(n) - 2*a(n-1) + a(n-2)) / 60.
%F A385897 A008512(n) = (a(n) + 2*a(n-1) + a(n-2)) / 2.
%F A385897 A022521(n) = (a(n-1) + a(n)) / 2.
%F A385897 A061317(n) = (a(n) - a(n-2)) / 20.
%F A385897 A063497(n) = a(n) - a(n-1).
%F A385897 gcd(a(n), a(n+1)) = 1.
%p A385897 gf := (-x^4 + 4*x^3 - 66*x^2 - 56*x - 1)/(x - 1)^5:
%p A385897 ser := series(gf, x, 35): seq(coeff(ser, x, n), n = 0..33);
%t A385897 a[n_] := With[{h = (n + 1)^2}, 5 (h^2 - h) + 1]; Table[a[n], {n, 0, 33}]
%Y A385897 Cf. A385896 (column 5), A063497, A000290, A061317, A022521, A008512.
%K A385897 nonn,easy
%O A385897 0,2
%A A385897 _Peter Luschny_, Jul 21 2025