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Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x/(1-k*x)).
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%I #27 Feb 04 2021 11:23:17

%S 1,1,1,1,1,1,1,1,3,1,1,1,5,13,1,1,1,7,37,73,1,1,1,9,73,361,501,1,1,1,

%T 11,121,1009,4361,4051,1,1,1,13,181,2161,17341,62701,37633,1,1,1,15,

%U 253,3961,48081,355951,1044205,394353,1

%N Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x/(1-k*x)).

%H Seiichi Manyama, <a href="/A341033/b341033.txt">Antidiagonals n = 0..139, flattened</a>

%F T(n,k) = Sum_{j=1..n} k^(n-j) * (n!/j!) * binomial(n-1,j-1) for n > 0.

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

%e Square array begins:

%e 1, 1, 1, 1, 1, 1, ...

%e 1, 1, 1, 1, 1, 1, ...

%e 1, 3, 5, 7, 9, 11, ...

%e 1, 13, 37, 73, 121, 181, ...

%e 1, 73, 361, 1009, 2161, 3961, ...

%e 1, 501, 4361, 17341, 48081, 108101, ...

%t T[0, k_] = 1; T[n_, k_] := n!*Sum[If[k == n - j == 0, 1, k^(n - j)]*Binomial[n - 1, j - 1]/j!, {j, 1, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Feb 03 2021 *)

%o (PARI) {T(n, k) = if(n==0, 1, n!*sum(j=1, n, k^(n-j)*binomial(n-1, j-1)/j!))}

%o (PARI) {T(n, k) = if(n<2, 1, (2*k*n-2*k+1)*T(n-1, k)-k^2*(n-1)*(n-2)*T(n-2, k))}

%Y Columns 0..7 give A000012, A000262, A025168, A321837, A321847, A321848, A321849, A321850.

%Y Main diagonal gives A293146.

%Y Cf. A253286, A341014.

%K nonn,tabl

%O 0,9

%A _Seiichi Manyama_, Feb 03 2021