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%I #19 Mar 12 2016 19:50:34
%S 1,4,12,34,85,203,454,985,2060,4205,8363,16298,31103,58319,107471,
%T 195037,348795,615550,1072706,1847867,3148444,5309948,8869172,
%U 14680261,24090035,39210436,63327665,101527253,161626560,255579456,401556210,627039569,973374176
%N Number of partitions of 5n such that cn(0,5) <= cn(1,5) = cn(4,5) <= cn(2,5) = cn(3,5).
%C Alternatively, number of partitions of 5n such that cn(0,5) <= cn(2,5) = cn(3,5) <= 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) = A202087(n) + A036883(n)
%F a(n) = A036884(n) + A036888(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[1]<=t[2] and t[2]=t[5] and t[5]<=t[3] and t[3]=t[4], 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_] := Module[{ll, mn, x}, ll = MapAt[#+1&, l, Mod[i, 5]+1]; mn = Min[ll]; If[mn==0, ll, Map[#-mn&, ll]]]; g[n_, i_, t_List] := g[n, i, t] = Which[n<0, 0, n == 0 , If[t[[1]] <= t[[2]] && t[[2]] == t[[5]] && t[[5]] <= t[[3]] && t[[3]] == t[[4]], 1, 0], 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]], t[[2]], t[[4]], t[[4]], t[[5]]}]], True, g[n, i-1, t] + g[n-i, i, mkl[i, t]]]; a[n_] := g[5*n, 5*n, {0, 0, 0, 0, 0}]; Table[a[n], {n, 1, 15}] (* _Jean-François Alcover_, Jul 29 2015, after _Alois P. Heinz_ *)
%K nonn
%O 1,2
%A _Olivier Gérard_
%E a(10)-a(31) from _Alois P. Heinz_, Jul 02 2009
%E Edited by _Max Alekseyev_, Dec 11 2011
%E a(32)-a(33) from _Alois P. Heinz_, Mar 12 2016
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