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A341033 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)).

%I A341033 #27 Feb 04 2021 11:23:17
%S A341033 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 A341033 11,121,1009,4361,4051,1,1,1,13,181,2161,17341,62701,37633,1,1,1,15,
%U A341033 253,3961,48081,355951,1044205,394353,1
%N A341033 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 A341033 Seiichi Manyama, <a href="/A341033/b341033.txt">Antidiagonals n = 0..139, flattened</a>
%F A341033 T(n,k) = Sum_{j=1..n} k^(n-j) * (n!/j!) * binomial(n-1,j-1) for n > 0.
%F A341033 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 A341033 Square array begins:
%e A341033   1,   1,    1,     1,     1,      1, ...
%e A341033   1,   1,    1,     1,     1,      1, ...
%e A341033   1,   3,    5,     7,     9,     11, ...
%e A341033   1,  13,   37,    73,   121,    181, ...
%e A341033   1,  73,  361,  1009,  2161,   3961, ...
%e A341033   1, 501, 4361, 17341, 48081, 108101, ...
%t A341033 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 A341033 (PARI) {T(n, k) = if(n==0, 1, n!*sum(j=1, n, k^(n-j)*binomial(n-1, j-1)/j!))}
%o A341033 (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 A341033 Columns 0..7 give A000012, A000262, A025168, A321837, A321847, A321848, A321849, A321850.
%Y A341033 Main diagonal gives A293146.
%Y A341033 Cf. A253286, A341014.
%K A341033 nonn,tabl
%O A341033 0,9
%A A341033 _Seiichi Manyama_, Feb 03 2021