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A379611 Table read by rows: T(n, k) = (n + 1)^(n - 1) - (k - 1)*(n + 1)^(n - 2), by convention T(1, 0) = 1.

%I A379611 #13 Dec 28 2024 10:21:49
%S A379611 2,1,1,4,3,2,20,16,12,8,150,125,100,75,50,1512,1296,1080,864,648,432,
%T A379611 19208,16807,14406,12005,9604,7203,4802,294912,262144,229376,196608,
%U A379611 163840,131072,98304,65536,5314410,4782969,4251528,3720087,3188646,2657205,2125764,1594323,1062882
%N A379611 Table read by rows: T(n, k) = (n + 1)^(n - 1) - (k - 1)*(n + 1)^(n - 2), by convention T(1, 0) = 1.
%H A379611 Steve Butler, Kimberly Hadaway, Victoria Lenius, Preston Martens, and Marshall Moats, <a href="https://arxiv.org/abs/2412.07873">Lucky cars and lucky spots in parking functions</a>, arXiv:2412.07873 [math.CO], 2024. See p. 7, corollary 3.1.
%F A379611 T(n, k) = (n + 1)^(n - 2)*(n - k + 2), if (n, k) != (1, 0).
%F A379611 T(n, k) = (1 - (k - 1)/(n + 1))*(n + 1)^(n - 1), if (n, k) != (1, 0).
%e A379611 Triangle starts:
%e A379611   [0]      2;
%e A379611   [1]      1,      1;
%e A379611   [2]      4,      3,      2;
%e A379611   [3]     20,     16,     12,      8;
%e A379611   [4]    150,    125,    100,     75,     50;
%e A379611   [5]   1512,   1296,   1080,    864,    648,    432;
%e A379611   [6]  19208,  16807,  14406,  12005,   9604,   7203,  4802;
%e A379611   [7] 294912, 262144, 229376, 196608, 163840, 131072, 98304, 65536;
%p A379611 T := (n, k) -> ifelse(n=1 and k=0, 1, (n + 1)^(n - 1) - (k - 1)*(n + 1)^(n - 2)):
%t A379611 T[n_, k_] := T[n, k] = (n + 1)^(n - 1) - (k - 1)*(n + 1)^(n - 2); T[1, 0] := 1;
%t A379611 Flatten@ Table[T[n, k], {n, 0, 8}, {k, 0, n}] (* _Michael De Vlieger_, Dec 27 2024 *)
%Y A379611 Cf. A007334 (main diagonal), A374756, A375616, A379612 (column 0), A379613.
%K A379611 nonn,tabl
%O A379611 0,1
%A A379611 _Peter Luschny_, Dec 27 2024