A006700 - OEIS

A006700

T(n,3,1), where T(n,k,s) with 0<=s<n is the number of sequences a_1,...,a_(k*n+s) of integers 0,1,...,n such that the 'bumped' sequence b_1,...,b_(k*n+s) has exactly s n's, where b_i=a_i + j (mod n+1) with minimal j>=0 such that b_0,...,b_i contain at most k elements equal to b_i.
(Formerly M4038)

%I M4038 #11 Oct 30 2022 18:19:59

%S 1,5,393,131473,117316993,219639324573,745567087151089,

%T 4160071323762705545,35561839228620292884321,

%U 441709871992325805279854773,7649447738065395278958097645657

%N T(n,3,1), where T(n,k,s) with 0<=s<n is the number of sequences a_1,...,a_(k*n+s) of integers 0,1,...,n such that the 'bumped' sequence b_1,...,b_(k*n+s) has exactly s n's, where b_i=a_i + j (mod n+1) with minimal j>=0 such that b_0,...,b_i contain at most k elements equal to b_i.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H I. A. Blake and A. G. Konheim, <a href="https://doi.org/10.1145/322033.322038">Big buckets are (are not) better!</a>, J. ACM, 24 (1977), 591-606.

%F Reference gives recurrences.

%t T[k_, m_] := T[k, m] = If[m <= k, 1, Module[{n = Quotient[m, k]}, Sum[Binomial[m - 1, k i - 1] i T[k, k i - 1]T[k, m - k i], {i, 1, n}] + If[n k == m, 0, (n + 1)T[k, m - 1]]]]

%Y Cf. A006698, A006699.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_.

%E More terms and better description from _Reiner Martin_, Feb 07 2002