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

0,4

Links

Table of n, a(n) for n=0..25.

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

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 *)

Crossrefs

Cf. A083856, A083859.

Sequence in context: A149529 A149530 A149531 * A141496 A192428 A060358

Adjacent sequences: A083857 A083858 A083859 * A083861 A083862 A083863

Keyword

easy,nonn

Author

Paul Barry, May 06 2003

Status

approved