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A136680 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.

%I A136680 #8 Dec 02 2022 07:05:33
%S A136680 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 A136680 26440,8766080,1,1,4,20,520,26440,8766080,6939853440,1,1,4,20,520,
%U A136680 26440,8766080,6939853440,41934828744960,1,1,4,20,520,26440,8766080,6939853440,41934828744960,694027278828744960
%N A136680 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 A136680 G. C. Greubel, <a href="/A136680/b136680.txt">Rows n = 1..30 of the triangle, flattened</a>
%F A136680 T(k) = T(k-1) + n^(k-2)*T(k-2), with T(0) = 0, T(1) = 1.
%F A136680 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 A136680 Triangle begins as:
%e A136680   1;
%e A136680   1, 1;
%e A136680   1, 1, 4;
%e A136680   1, 1, 4, 20;
%e A136680   1, 1, 4, 20, 520;
%e A136680   1, 1, 4, 20, 520, 26440;
%e A136680   1, 1, 4, 20, 520, 26440, 8766080;
%e A136680   1, 1, 4, 20, 520, 26440, 8766080, 6939853440;
%e A136680   1, 1, 4, 20, 520, 26440, 8766080, 6939853440, 41934828744960;
%t A136680 T[k_]:= T[k]= If[k<2, k, T[k-1] + n^(k-2)*T[k-2]];
%t A136680 Table[T[k], {n,10}, {k,n}]//Flatten
%o A136680 (Magma)
%o A136680 function f(k)
%o A136680   if k lt 2 then return k;
%o A136680   else return f(k-1) + k^(k-2)*f(k-2);
%o A136680   end if; return f;
%o A136680 end function;
%o A136680 A136680:= func< n,k | k le n select f(k) else 0 >;
%o A136680 [A136680(n,k): k in [1..n], n in [1..14]]; // _G. C. Greubel_, Dec 01 2022
%o A136680 (SageMath)
%o A136680 @CachedFunction
%o A136680 def f(k):
%o A136680     if (k<2): return k
%o A136680     else: return f(k-1) + k^(k-2)*f(k-2)
%o A136680 def A136680(n,k): return f(k) if (k < n+1) else 0
%o A136680 flatten([[A136680(n,k) for k in range(1,n+1)] for n in range(1,15)]) # _G. C. Greubel_, Dec 01 2022
%Y A136680 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 A136680 nonn,tabl
%O A136680 1,6
%A A136680 _Roger L. Bagula_, Apr 06 2008
%E A136680 Edited by _G. C. Greubel_, Dec 01 2022