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A083860 First subdiagonal of generalized Fibonacci array A083856.

%I A083860 #21 Sep 26 2022 08:39:21
%S A083860 0,1,1,5,11,55,176,937,3781,21571,102455,624493,3356640,21752431,
%T A083860 129055681,884773585,5696734715,41129090011,283908657880,
%U A083860 2149818248341,15765656131765,124759995175751,965186517474191,7956847444317049,64577172850366176,553048437381116275
%N A083860 First subdiagonal of generalized Fibonacci array A083856.
%F A083860 a(n) = (((1 + sqrt(4*n + 5))/2)^n - ((1 - sqrt(4*n + 5))/2)^n)/sqrt(4*n + 5).
%F A083860 a(n) = A193376(n-1, n+1) for n >= 2. - _R. J. Mathar_, Aug 23 2011
%F A083860 a(n) = Sum_{s = 0..floor((n-1)/2)} binomial(n-1-s, s) * (n+1)^s. - _Petros Hadjicostas_, Dec 25 2019
%p A083860 T := proc(n, k) local v; option remember; if 0 <= n and k = 0 then v := 0; end if; if 0 <= n and k = 1 then v := 1; end if; if 0 <= n and 2 <= k then v := T(n, k - 1) + n*T(n, k - 2); end if; v; end proc;
%p A083860 seq(T(n + 1, n), n = 0 .. 40); # _Petros Hadjicostas_, Dec 25 2019
%t A083860 T[_, 0] = 0; T[_, 1|2] = 1;
%t A083860 T[n_, k_] := T[n, k] = T[n, k-1] + n T[n, k-2];
%t A083860 a[n_] := T[n+1, n];
%t A083860 Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Sep 26 2022 *)
%Y A083860 Cf. A083856, A083859.
%K A083860 easy,nonn
%O A083860 0,4
%A A083860 _Paul Barry_, May 06 2003