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%I #18 Dec 23 2015 13:06:17
%S 0,1,4,11,27,63,142,312,665,1382,2795,5524,10674,20228,37634,68886,
%T 124179,220779,387458,671883,1152027,1954614,3283494,5464437,9013558,
%U 14743397,23923577,38526121,61593796,97795238,154251217,241765892,376643803,583370176
%N Number of partitions of 5n such that cn(1,5) = cn(4,5) < cn(0,5) <= cn(2,5) = cn(3,5).
%C Alternatively, number of partitions of 5n such that cn(2,5) = cn(3,5) < cn(0,5) <= cn(1,5) = cn(4,5).
%C For a given partition, cn(i,n) means the number of its parts equal to i modulo n.
%H <a href="/wiki/Partitions_of_5n">Index and properties of sequences related to partitions of 5n</a>
%F a(n) = A036892(n) + A036894(n).
%F a(n) = A036881(n) - A036884(n).
%p mkl:= proc(i,l) local ll, mn, x; ll:= applyop(x->x+1, irem (i,5)+1, l); mn:= min(ll[]); `if`(mn=0, ll, map(x->x-mn, ll)) end:
%p g:= proc (n,i,t) if n<0 then 0 elif n=0 then `if`(t[2]=t[5] and t[3]=t[4] and t[5]<t[1] and t[1]<=t[3], 1, 0) elif i=0 then 0 elif i=1 then g(0, 0, [t[1], t[2]+n, t[3], t[4], t[5]]) elif i=2 then `if`(t[3]>t[4], 0, g(n-2*(t[4]-t[3]), 1, [t[1], t[2], t[4], t[4], t[5]])) else g(n,i,t):= g(n, i-1, t) + g(n-i, i, mkl(i,t)) fi end:
%p a:= n-> g(5*n, 5*n, [0,0,0,0,0]):
%p seq(a(n), n=1..15); # _Alois P. Heinz_, Jul 02 2009
%t mkl[i_, l_List] := Module[{ll, mn, x}, ll = MapAt[#+1&, l, Mod[i, 5]+1]; mn = Min[ll]; If[mn == 0, ll, ll-mn]]; g[n_, i_, t_List] := g[n, i, t] = Which[n<0, 0, n == 0, If[t[[2]] == t[[5]] && t[[3]] == t[[4]] && t[[5]] < t[[1]] && t[[1]] <= t[[3]], 1, 0], True, Which[i == 0, 0, i == 1, g[0, 0, {t[[1]], t[[2]]+n, t[[3]], t[[4]], t[[5]]}], i == 2, If[t[[3]] > t[[4]], 0, g[n-2*(t[[4]] - t[[3]]), 1, t[[{1, 2, 4, 4, 5}]]]], 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, 1, 32}] (* _Jean-François Alcover_, Dec 23 2015, after _Alois P. Heinz_ *)
%K nonn
%O 1,3
%A _Olivier Gérard_
%E a(10)-a(32) from _Alois P. Heinz_, Jul 02 2009
%E a(33)-a(34) from _Max Alekseyev_, Dec 11 2011
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