A136680 - OEIS

%I #8 Dec 02 2022 07:05:33

%S 1,1,1,1,1,4,1,1,4,20,1,1,4,20,520,1,1,4,20,520,26440,1,1,4,20,520,

%T 26440,8766080,1,1,4,20,520,26440,8766080,6939853440,1,1,4,20,520,

%U 26440,8766080,6939853440,41934828744960,1,1,4,20,520,26440,8766080,6939853440,41934828744960,694027278828744960

%N Triangle T(n, k) = f(k) for k < n+1, otherwise 0, where f(k) = f(k-1) + k^(k-2)*f(k-2) with f(0) = 0 and f(1) = 1, read by rows.

%H G. C. Greubel, <a href="/A136680/b136680.txt">Rows n = 1..30 of the triangle, flattened</a>

%F T(k) = T(k-1) + n^(k-2)*T(k-2), with T(0) = 0, T(1) = 1.

%F T(n, k) = f(k) for k < n+1, otherwise 0, where f(k) = f(k-1) + k^(k-2)*f(k-2) with f(0) = 0 and f(1) = 1. - _G. C. Greubel_, Dec 01 2022

%e Triangle begins as:

%e   1;

%e   1, 1;

%e   1, 1, 4;

%e   1, 1, 4, 20;

%e   1, 1, 4, 20, 520;

%e   1, 1, 4, 20, 520, 26440;

%e   1, 1, 4, 20, 520, 26440, 8766080;

%e   1, 1, 4, 20, 520, 26440, 8766080, 6939853440;

%e   1, 1, 4, 20, 520, 26440, 8766080, 6939853440, 41934828744960;

%t T[k_]:= T[k]= If[k<2, k, T[k-1] + n^(k-2)*T[k-2]];

%t Table[T[k], {n,10}, {k,n}]//Flatten

%o (Magma)

%o function f(k)

%o   if k lt 2 then return k;

%o   else return f(k-1) + k^(k-2)*f(k-2);

%o   end if; return f;

%o end function;

%o A136680:= func< n,k | k le n select f(k) else 0 >;

%o [A136680(n,k): k in [1..n], n in [1..14]]; // _G. C. Greubel_, Dec 01 2022

%o (SageMath)

%o @CachedFunction

%o def f(k):

%o     if (k<2): return k

%o     else: return f(k-1) + k^(k-2)*f(k-2)

%o def A136680(n,k): return f(k) if (k < n+1) else 0

%o flatten([[A136680(n,k) for k in range(1,n+1)] for n in range(1,15)]) # _G. C. Greubel_, Dec 01 2022

%Y q-Fibonacci numbers include: A015459, A015460, A015461, A015462, A015463, A015464, A015465, A015467, A015468, A015469, A015470, A015473, A015474, A015475, A015476, A015477, A015479, A015480, A015481, A015482, A015484, A015485.

%K nonn,tabl

%O 1,6

%A _Roger L. Bagula_, Apr 06 2008

%E Edited by _G. C. Greubel_, Dec 01 2022