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A036882 Number of partitions of 5n such that cn(1,5) = cn(4,5) <= cn(2,5) = cn(3,5) <= cn(0,5).

%I A036882 #16 Dec 23 2015 10:15:31
%S A036882 1,1,3,8,22,54,128,282,602,1235,2474,4831,9263,17418,32242,58737,
%T A036882 105519,186976,327238,565896,967910,1638175,2745588,4558864,7503737,
%U A036882 12248234,19835700,31882617,50881290,80648122,126998962,198743334,309163475,478177505,735522058
%N A036882 Number of partitions of 5n such that cn(1,5) = cn(4,5) <= cn(2,5) = cn(3,5) <= cn(0,5).
%C A036882 Alternatively, number of partitions of 5n such that cn(2,5) = cn(3,5) <= cn(1,5) = cn(4,5) <= cn(0,5).
%C A036882 For a given partition cn(i,n) means the number of its parts equal to i modulo n.
%H A036882 <a href="/wiki/Partitions_of_5n">Index and properties of sequences related to partitions of 5n</a>
%F A036882 a(n) = A036889(n) + A036887(n)
%F A036882 a(n) = A202085(n) + A036891(n)
%p A036882 mkl:= proc(i, l) local ll, x, j; j:= irem (i, 5); j:= `if` (j=0, 5, j); ll:= applyop (x->x+1, j, l); map (x-> x-min(ll[]), ll) end:
%p A036882 g:= proc (n, i, t) local x; if n<0 then 0 elif n=0 then `if` (t[1]=t[4] and t[4]<=t[2] and t[2]=t[3] and t[3]<=t[5], 1, 0) elif i=0 then 0 elif i=1 then g (0, 0, applyop (x-> x+n, 1, t)) elif i=2 then `if` (t[2]>t[3], 0, g (n-2*(t[3]-t[2]), 1, subsop(2=t[3], t))) elif (i=3 or i=4) and t[i]>t[5] then 0 else g(n, i, t):= g (n, i-1, t) +g (n-i, i, mkl(i, t)) fi end:
%p A036882 a:= n-> g(5*n, 5*n, [0, 0, 0, 0, 0]):
%p A036882 seq(a(n), n=0..15);  # _Alois P. Heinz_, Jul 07 2009
%t A036882 mkl[i_, l_List] := Module[{ll, x, j}, j = Mod[i, 5]; j = If[j == 0, 5, j]; ll = MapAt [#+1&, l, j]; ll - Min[ll]]; g[n_, i_, t_List] := g[n, i, t] = Which[n<0, 0, n == 0, If[t[[1]] == t[[4]] && t[[4]] <= t[[2]] && t[[2]] == t[[3]] && t[[3]] <= t[[5]], 1, 0], i == 0, 0, i == 1, g[0, 0, MapAt [#+n&, t, 1]], i == 2, If[t[[2]] > t[[3]], 0, g[n - 2*(t[[3]] - t[[2]]), 1, ReplacePart[t, 2 -> t[[3]]]]], (i == 3 || i == 4) && t[[i]] > t[[5]], 0, True, g[n, i, t] = g[n, i-1, t] + g[n-i, i, mkl[i, t]]]; a[n_] := a[n] = g[5*n, 5*n, {0, 0, 0, 0, 0}]; Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 32}] (* _Jean-François Alcover_, Dec 23 2015, after _Alois P. Heinz_ *)
%K A036882 nonn
%O A036882 0,3
%A A036882 _Olivier Gérard_
%E A036882 a(10)-a(32) from _Alois P. Heinz_, Jul 07 2009
%E A036882 Edited by _Max Alekseyev_, Dec 11 2011
%E A036882 More terms from _Alois P. Heinz_, Dec 23 2015