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A372821 Table read by antidiagonals: T(m,n) = number of (m-2)-metered (m,n)-parking functions.

%I A372821 #17 Jul 23 2025 16:05:43
%S A372821 0,1,0,0,4,0,0,4,9,0,0,0,21,16,0,0,0,27,56,25,0,0,0,0,163,115,36,0,0,
%T A372821 0,0,257,483,204,49,0,0,0,0,0,1686,1095,329,64,0,0,0,0,0,3156,5367,
%U A372821 2131,496,81,0,0,0,0,0,0,21858,13076,3747,711,100,0,0,0,0,0,0,47442,73276,27309,6123,980,121,0
%N A372821 Table read by antidiagonals: T(m,n) = number of (m-2)-metered (m,n)-parking functions.
%H A372821 Spencer Daugherty, Pamela E. Harris, Ian Klein, and Matt McClinton, <a href="https://arxiv.org/abs/2406.12941">Metered Parking Functions</a>, arXiv:2406.12941 [math.CO], 2024.
%F A372821 T(m,n) = (n-m+2)^2*(m-1)^(m-3) + Sum_{k=n-m+3...n} binomial(m-2, n-k)*(n-k+1)^(n-k-1)*[binomial(k+1,2)*(n+m+2)*k^(m-n+k-3) + (k*(n-m+1) - binomial(n-m+2,2))*(k-n+m-1)^(k-n+m-3) + Sum_{j=n-m+2} (jk - binomial(j+1,2))*binomial(m-2-n+k, k-1-j)*(n-m+1)*j^(j+m-2-n)*(k-j)^(k-j-2)].
%e A372821 Table begins:
%e A372821   0, 0, 0, 0, 0, 0, 0, ...
%e A372821   1, 4, 9, 16, 25, 36, 49, ...
%e A372821   0, 4, 21, 56, 115, 204, 329, ...
%e A372821   0, 0, 27, 163, 483, 1095, 2131, ...
%e A372821   0, 0, 0, 257, 1686, 5367, 13076, ...
%e A372821   0, 0, 0, 0, 3156, 21858, 73276, ...
%e A372821   0, 0, 0, 0, 0, 47442, 341192, ...
%e A372821   ...
%Y A372821 Main diagonal is A328694.
%Y A372821 Cf. A372817, A372818, A372819, A372820.
%K A372821 nonn,tabl
%O A372821 1,5
%A A372821 _Spencer Daugherty_, May 13 2024