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Triangle read by rows: T(n,k) = (n-1)*T(n-1,k) + T(n-2,k), with T(n,n-1)=1, T(n,n-2)=n-2, for n >= 1, 0 <= k <= n-1.
2

%I #15 Feb 15 2020 11:57:40

%S 1,0,1,1,1,1,3,4,2,1,13,17,9,3,1,68,89,47,16,4,1,421,551,291,99,25,5,

%T 1,3015,3946,2084,709,179,36,6,1,24541,32119,16963,5771,1457,293,49,7,

%U 1,223884,293017,154751,52648,13292,2673,447,64,8,1

%N Triangle read by rows: T(n,k) = (n-1)*T(n-1,k) + T(n-2,k), with T(n,n-1)=1, T(n,n-2)=n-2, for n >= 1, 0 <= k <= n-1.

%H Reinhard Zumkeller, <a href="/A228340/b228340.txt">Rows n = 1..120 of table, flattened</a>

%H C. Cannings, <a href="http://dx.doi.org/10.4236/am.2013.45105">The Stationary Distributions of a Class of Markov Chains</a>, Applied Mathematics, Vol. 4 No. 5, 2013, pp. 769-773.

%e Triangle begins:

%e 1,

%e 0,1,

%e 1,1,1,

%e 3,4,2,1,

%e 13,17,9,3,1,

%e 68,89,47,16,4,1,

%e 421,551,291,99,25,5,1,

%e 3015,3946,2084,709,179,36,6,1,

%e ...

%o (Haskell)

%o a228340 n k = a228340_tabl !! (n-1) !! k

%o a228340_row n = a228340_tabl !! (n-1)

%o a228340_tabl = map (reverse . fst) $ iterate f ([1], [1,0]) where

%o f (us, vs'@(_ : vs@(v : _))) = (vs', ws) where

%o ws = 1 : (v + 1) : zipWith (+) us (map (* (v + 2)) vs)

%o -- _Reinhard Zumkeller_, Aug 31 2013

%Y Diagonals give A058307, A058279, A228341. Row sums give A001040.

%K nonn,tabl,easy

%O 1,7

%A _N. J. A. Sloane_, Aug 29 2013