A006698 - OEIS

A006698

T(2,2n), where T(k,m) is the number of sequences a_1,...,a_m of integers 0,1,...,n with n=floor(m/k) such that the 'bumped' sequence b_1,...,b_m has exactly k of each of 0,...,n-1, 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 M4813)

%I M4813 #11 Oct 30 2022 18:19:58

%S 1,1,11,378,27213,3378680,645216039,175804806912,64820487788537,

%T 31093204323279744,18824085922156535715,14040767751007803601664,

%U 12652731866917353207799557,13553071929305974778937888768

%N T(2,2n), where T(k,m) is the number of sequences a_1,...,a_m of integers 0,1,...,n with n=floor(m/k) such that the 'bumped' sequence b_1,...,b_m has exactly k of each of 0,...,n-1, 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.

%F Reference gives recurrences (see Mathematica code).

%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. A006699, A006700.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_.

%E More terms and better description from _Reiner Martin_, Feb 08 2002