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A341014 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} k^j * j! * binomial(n,j)^2.

%I A341014 #25 Feb 03 2021 07:38:40
%S A341014 1,1,1,1,2,1,1,3,7,1,1,4,17,34,1,1,5,31,139,209,1,1,6,49,352,1473,
%T A341014 1546,1,1,7,71,709,5233,19091,13327,1,1,8,97,1246,13505,95836,291793,
%U A341014 130922,1,1,9,127,1999,28881,318181,2080999,5129307,1441729,1
%N A341014 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} k^j * j! * binomial(n,j)^2.
%H A341014 Seiichi Manyama, <a href="/A341014/b341014.txt">Antidiagonals n = 0..139, flattened</a>
%F A341014 E.g.f. of column k: exp(x/(1-k*x)) / (1-k*x).
%F A341014 T(n,k) = (2*k*n-k+1) * T(n-1,k) - k^2 * (n-1)^2 * T(n-2,k) for n > 1.
%e A341014 Square array begins:
%e A341014   1,    1,     1,     1,      1,      1, ...
%e A341014   1,    2,     3,     4,      5,      6, ...
%e A341014   1,    7,    17,    31,     49,     71, ...
%e A341014   1,   34,   139,   352,    709,   1246, ...
%e A341014   1,  209,  1473,  5233,  13505,  28881, ...
%e A341014   1, 1546, 19091, 95836, 318181, 830126, ...
%t A341014 T[n_, k_] := Sum[If[j == k == 0, 1, k^j]*j!*Binomial[n, j]^2, {j, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Feb 02 2021 *)
%o A341014 (PARI) {T(n,k) = sum(j=0, n, k^j*j!*binomial(n, j)^2)}
%Y A341014 Columns 0..4 give A000012, A002720, A025167, A102757, A102773.
%Y A341014 Rows 0..2 give A000012, A000027(n+1), A056220(n+1).
%Y A341014 Main diagonal gives A330260.
%Y A341014 Cf. A307883.
%K A341014 nonn,tabl
%O A341014 0,5
%A A341014 _Seiichi Manyama_, Feb 02 2021