A083860 First subdiagonal of generalized Fibonacci array A083856.

0, 1, 1, 5, 11, 55, 176, 937, 3781, 21571, 102455, 624493, 3356640, 21752431, 129055681, 884773585, 5696734715, 41129090011, 283908657880, 2149818248341, 15765656131765, 124759995175751, 965186517474191, 7956847444317049, 64577172850366176, 553048437381116275

Offset: 0

Author

Paul Barry, May 06 2003

Keywords

• easy
• nonn

Crossrefs

Cf. A083856, A083859.

Programs

Maple

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;
seq(T(n + 1, n), n = 0 .. 40); # Petros Hadjicostas, Dec 25 2019

Mathematica

T[, 0] = 0; T[, 1|2] = 1;
T[n_, k_] := T[n, k] = T[n, k-1] + n T[n, k-2];
a[n_] := T[n+1, n];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Sep 26 2022 *)

Formula

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

a(n) = A193376(n-1, n+1) for n >= 2. - R. J. Mathar, Aug 23 2011

a(n) = Sum_{s = 0..floor((n-1)/2)} binomial(n-1-s, s) * (n+1)^s. - Petros Hadjicostas, Dec 25 2019