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A363365 Array read by ascending antidiagonals: A(1, k) = k; for n > 1, A(n, k) = (k + 1)*A(n-1, k) + k + 1 - n, with k > 0.

%I A363365 #9 May 29 2023 11:22:50
%S A363365 1,2,2,3,7,3,4,21,14,4,5,62,57,23,5,6,184,228,117,34,6,7,549,911,586,
%T A363365 207,47,7,8,1643,3642,2930,1244,333,62,8,9,4924,14565,14649,7465,2334,
%U A363365 501,79,9,10,14766,58256,73243,44790,16340,4012,717,98,10
%N A363365 Array read by ascending antidiagonals: A(1, k) = k; for n > 1, A(n, k) = (k + 1)*A(n-1, k) + k + 1 - n, with k > 0.
%F A363365 A(n, k) = ((k - 1)*(k + 1)^(n+1) + k*n - k^2 + 1)/k^2.
%F A363365 O.g.f. of k-th column: x*(k - (k + 1)*x)/((1 - x)^2*(1 - (k + 1)*x)).
%F A363365 E.g.f. of k-th column: exp(x)*((k^2 - 1)*(exp(k*x) - 1) + k*x)/k^2.
%F A363365 A(2, n) = A008865(n+1).
%e A363365 The array begins:
%e A363365   1,   2,    3,     4,     5, ...
%e A363365   2,   7,   14,    23,    34, ...
%e A363365   3,  21,   57,   117,   207, ...
%e A363365   4,  62,  228,   586,  1244, ...
%e A363365   5, 184,  911,  2930,  7465, ...
%e A363365   6, 549, 3642, 14649, 44790, ...
%e A363365   ...
%t A363365 A[n_,k_]:=((k-1)*(k+1)^(n+1)+k*n-k^2+1)/k^2; Table[A[n-k+1,k],{n,10},{k,n}]//Flatten (* or *)
%t A363365 A[n_,k_]:=SeriesCoefficient[x*(k-(k+1)*x)/((1-x)^2*(1-(k+1)*x)),{x,0,n}]; Table[A[n-k+1,k],{n,10},{k,n}]//Flatten (* or *)
%t A363365 A[n_,k_]:=n!SeriesCoefficient[Exp[x]((k^2-1)(Exp[k x]-1)+k x)/k^2,{x,0,n}]; Table[A[n-k+1,k],{n,10},{k,n}]//Flatten
%Y A363365 Cf. A000027 (n=1 or k=1), A008865, A051846 (diagonal), A064017 (k=9), A353094 (k=2), A353095 (k=3), A353096 (k=4), A353097 (k=5), A353098 (k=6), A353099 (k=7), A353100 (k=8), A363366 (antidiagonal sums).
%K A363365 nonn,tabl
%O A363365 1,2
%A A363365 _Stefano Spezia_, May 29 2023