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A305492 a(n) = ((1 + y)^n - (1 - y)^n)/y with y = sqrt(8).

%I A305492 #36 Dec 11 2024 07:03:58
%S A305492 0,2,4,22,72,298,1100,4286,16272,62546,238996,915814,3504600,13419898,
%T A305492 51371996,196683278,752970528,2882724002,11036241700,42251551414,
%U A305492 161756794728,619274449354,2370846461804,9076614069086,34749153370800
%N A305492 a(n) = ((1 + y)^n - (1 - y)^n)/y with y = sqrt(8).
%H A305492 Colin Barker, <a href="/A305492/b305492.txt">Table of n, a(n) for n = 0..1000</a>
%H A305492 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,7).
%F A305492 E.g.f.: 2*exp(x)*sinh(sqrt(n)*x)/sqrt(n) for n = 8.
%F A305492 From _Colin Barker_, Jun 02 2018: (Start)
%F A305492 G.f.: 2*x / (1 - 2*x - 7*x^2).
%F A305492 a(n) = 2*a(n-1) + 7*a(n-2) for n>1.
%F A305492 (End)
%e A305492 Array ((1+y)^n - (1-y)^n)/y with y = sqrt(k).
%e A305492 [k\n]
%e A305492 [1]   1, 2, 4,  8, 16, 32,   64,  128,    256,   512,   1024, ...
%e A305492 [2]   0, 2, 4, 10, 24, 58,  140,  338,    816,  1970,   4756, ...
%e A305492 [3]   0, 2, 4, 12, 32, 88,  240,  656,   1792,  4896,  13376, ...
%e A305492 [4]   0, 2, 4, 14, 40, 122, 364,  1094,  3280,  9842,  29524, ...
%e A305492 [5]   0, 2, 4, 16, 48, 160, 512,  1664,  5376, 17408,  56320, ...
%e A305492 [6]   0, 2, 4, 18, 56, 202, 684,  2378,  8176, 28242,  97364, ...
%e A305492 [7]   0, 2, 4, 20, 64, 248, 880,  3248, 11776, 43040, 156736, ...
%e A305492 [8]   0, 2, 4, 22, 72, 298, 1100, 4286, 16272, 62546, 238996, ...
%e A305492 [9]   0, 2, 4, 24, 80, 352, 1344, 5504, 21760, 87552, 349184, ...
%p A305492 egf :=  (n,x) -> 2*exp(x)*sinh(sqrt(n)*x)/sqrt(n):
%p A305492 ser := series(egf(8,x), x, 26):
%p A305492 seq(n!*coeff(ser,x, n), n=0..24);
%t A305492 Table[Simplify[((1 + Sqrt[8])^n - (1 - Sqrt[8])^n)/ Sqrt[8]], {n, 0, 24}]
%o A305492 (PARI) concat(0, Vec(2*x / (1 - 2*x - 7*x^2) + O(x^40))) \\ _Colin Barker_, Jun 05 2018
%Y A305492 Let f(n, y) = ((1 + y)^n - (1 - y)^n)/y.
%Y A305492 f(n,      1 ) = A000079(n);
%Y A305492 f(n, sqrt(2)) = A163271(n+1);
%Y A305492 f(n, sqrt(3)) = A028860(n+2);
%Y A305492 f(n,      2 ) = A152011(n) for n>0;
%Y A305492 f(n, sqrt(5)) = A103435(n);
%Y A305492 f(n, sqrt(6)) = A083694(n);
%Y A305492 f(n, sqrt(7)) = A274520(n);
%Y A305492 f(n, sqrt(8)) = a(n);
%Y A305492 f(n,      3 ) = A192382(n+1);
%Y A305492 Cf. A305491.
%Y A305492 Equals 2 * A015519.
%K A305492 nonn,easy
%O A305492 0,2
%A A305492 _Peter Luschny_, Jun 02 2018