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A379587 Array read by ascending antidiagonals: A(n, k) = (k^n - 1)^2/(k - 1), with k >= 2.

%I A379587 #8 Dec 26 2024 20:01:43
%S A379587 0,1,0,9,2,0,49,32,3,0,225,338,75,4,0,961,3200,1323,144,5,0,3969,
%T A379587 29282,21675,3844,245,6,0,16129,264992,348843,97344,9245,384,7,0,
%U A379587 65025,2389298,5589675,2439844,335405,19494,567,8,0,261121,21516800,89467563,61027344,12090125,960000,37303,800,9,0
%N A379587 Array read by ascending antidiagonals: A(n, k) = (k^n - 1)^2/(k - 1), with k >= 2.
%F A379587 G.f. of column k: (1 - k)*x*(1 + k*x)/((1 - x)*(1 - k*x)*(1 - k^2*x)).
%F A379587 E.g.f. of column k: exp(x)*(1 - 2*exp((k-1)*x) + exp((k^2-1)*x))/(k - 1).
%F A379587 A(2, n) = A027620(n-2) for n > 1.
%e A379587 The array begins as:
%e A379587     0,     0,      0,       0,        0,        0, ...
%e A379587     1,     2,      3,       4,        5,        6, ...
%e A379587     9,    32,     75,     144,      245,      384, ...
%e A379587    49,   338,   1323,    3844,     9245,    19494, ...
%e A379587   225,  3200,  21675,   97344,   335405,   960000, ...
%e A379587   961, 29282, 348843, 2439844, 12090125, 47073606, ...
%e A379587   ...
%t A379587 A[n_,k_]:=(k^n-1)^2/(k-1); Table[A[n-k+2,k],{n,0,9},{k,2,n+2}]//Flatten
%Y A379587 Cf. A027620, A060867 (k=2), A060868 (k=3), A060869 (k=4), A060870 (k=5), A060871 (k=7), A361475, A379588 (antidiagonal sums).
%K A379587 nonn,easy,tabl
%O A379587 0,4
%A A379587 _Stefano Spezia_, Dec 26 2024