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A306245 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where A(0,k) = 1 and A(n,k) = Sum_{j=0..n-1} k^j * binomial(n-1,j) * A(j,k) for n > 0.

%I A306245 #79 Jun 18 2022 23:03:43
%S A306245 1,1,1,1,1,1,1,1,2,1,1,1,3,5,1,1,1,4,17,15,1,1,1,5,43,179,52,1,1,1,6,
%T A306245 89,1279,3489,203,1,1,1,7,161,5949,108472,127459,877,1,1,1,8,265,
%U A306245 20591,1546225,26888677,8873137,4140,1
%N A306245 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where A(0,k) = 1 and A(n,k) = Sum_{j=0..n-1} k^j * binomial(n-1,j) * A(j,k) for n > 0.
%H A306245 Seiichi Manyama, <a href="/A306245/b306245.txt">Antidiagonals n = 0..55, flattened</a>
%F A306245 G.f. A_k(x) of column k satisfies A_k(x) = 1 + x * A_k(k * x / (1 - x)) / (1 - x). - _Seiichi Manyama_, Jun 18 2022
%e A306245 Square array begins:
%e A306245    1,  1,    1,      1,       1,        1, ...
%e A306245    1,  1,    1,      1,       1,        1, ...
%e A306245    1,  2,    3,      4,       5,        6, ...
%e A306245    1,  5,   17,     43,      89,      161, ...
%e A306245    1, 15,  179,   1279,    5949,    20591, ...
%e A306245    1, 52, 3489, 108472, 1546225, 12950796, ...
%p A306245 A:= proc(n, k) option remember; `if`(n=0, 1,
%p A306245       add(k^j*binomial(n-1, j)*A(j, k), j=0..n-1))
%p A306245     end:
%p A306245 seq(seq(A(n, d-n), n=0..d), d=0..12);  # _Alois P. Heinz_, Jul 28 2019
%t A306245 A[0, _] = 1;
%t A306245 A[n_, k_] := A[n, k] = Sum[k^j Binomial[n-1, j] A[j, k], {j, 0, n-1}];
%t A306245 Table[A[n-k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, May 29 2020 *)
%Y A306245 Columns k=0..4 give A000012, A000110, A126443, A355081, A355082.
%Y A306245 Rows n=0+1, 2 give A000012, A000027(n+1).
%Y A306245 Main diagonal gives A309401.
%Y A306245 Cf. A309386.
%K A306245 nonn,tabl
%O A306245 0,9
%A A306245 _Seiichi Manyama_, Jul 28 2019