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A058294 Successive rows of a triangle, the columns of which are generalized Fibonacci sequences S(j).

%I A058294 #17 Jan 28 2022 01:55:25
%S A058294 1,1,1,1,1,2,3,2,1,1,3,7,10,7,3,1,1,4,13,30,43,30,13,4,1,1,5,21,68,
%T A058294 157,225,157,68,21,5,1,1,6,31,130,421,972,1393,972,421,130,31,6,1,1,7,
%U A058294 43,222,931,3015,6961,9976,6961,3015,931,222,43,7,1
%N A058294 Successive rows of a triangle, the columns of which are generalized Fibonacci sequences S(j).
%C A058294 From _Reinhard Zumkeller_, Sep 14 2014: (Start)
%C A058294 T(n,k) = A102473(n,k), k=1..n;
%C A058294 T(n,k) = A102472(n,k-n+1), k=n..2*n-1;
%C A058294 T(n,n) = A001040(n). (End)
%H A058294 Reinhard Zumkeller, <a href="/A058294/b058294.txt">>Rows n = 1..125 of table, flattened</a>
%H A058294 Russell Walsmith, <a href="/A058294/a058294.pdf">CL-Chemy Transforms Fibonacci-Type Sequences to Arrays</a>
%F A058294 The j-th column S(j) is generated by a(n+1) = (n+j)*a(n) + a(n-1), a(0)=0, a(1)=1.
%e A058294 Triangle begins:
%e A058294   1;
%e A058294   1, 1, 1;
%e A058294   1, 2, 3,  2, 1;
%e A058294   1, 3, 7, 10, 7, 3, 1;
%e A058294   ...
%t A058294 t[n_, n_] = 1; t[n_, k_] := t[n, k] = If[n<k, 0, (n-1)*t[n-1, k] + t[n-2, k]]; row[n_] := With[{ro = Table[t[n, k], {k, 1, n}]}, Join[Reverse[ro], Rest[ro]]]; Array[row, 8] // Flatten (* _Jean-François Alcover_, Oct 05 2016 *)
%o A058294 (Haskell)
%o A058294 a058294 n k = a058294_tabf !! (n-1) !! (k-1)
%o A058294 a058294_row n = a058294_tabf !! (n-1)
%o A058294 a058294_tabf = [1] : zipWith (++) xss (map (tail . reverse) xss)
%o A058294                where xss = tail a102473_tabl
%o A058294 -- _Reinhard Zumkeller_, Sep 14 2014
%Y A058294 A001040, A001053, A058307, A058308, A058309 are columns of this triangle.
%Y A058294 Cf. A102472, A102473.
%K A058294 nonn,tabf,nice,easy
%O A058294 1,6
%A A058294 _Russell Walsmith_, Dec 07 2000