OFFSET
0,5
COMMENTS
Number of partitions of m with equal numbers of parts congruent to each of 1, 2, 3 and 4 (mod 5) is 0 unless m == 0 mod 5.
LINKS
FORMULA
MAPLE
mkl:= proc(i, l) local ll, mn, ii, x; ii:= irem(i, 5); ii:= `if`(ii=0, 5, ii); ll:= applyop(x->x+1, ii, l); mn:= min(l[]); `if`(mn=0, ll, map (x->x-mn, ll)) end:
g:= proc(n, i, t) local m, mx, j; if n<0 then 0 elif n=0 then `if`(nops ({t[]})=1, 1, 0) elif i=0 then 0 elif i<6 then mx:= max (t[]); m:= n-15*mx +add(t[j]*j, j=1..5); g(n, i, t):= `if`(m>=0 and irem(m, 15)=0, 1, 0) else g(n, i, t):= g(n, i-1, t) + g(n-i, i, mkl(i, t)) fi end:
a:= n-> g(5*n, 5*n, [0$5]):
seq(a(n), n=0..20); # Alois P. Heinz, Jul 04 2009
MATHEMATICA
$RecursionLimit = 1000; mkl[i_, l_List] := Module[{ ll, mn, ii, x}, ii = Mod[i, 5]; ii = If[ii == 0, 5, ii]; ll = MapAt[#+1&, l, ii]; mn = Min[l]; If[mn == 0, ll, Map [#-mn&, ll]]]; g[n_, i_, t_List] := g[n, i, t] = Module[{ m, mx, j}, Which[n<0, 0 , n == 0, If[Length[t // Union] == 1, 1, 0], i==0, 0, i<6, mx = Max[t]; m = n-15*mx + Sum[t[[j]]*j, {j, 1, 5}]; If[m >= 0 && Mod[m, 15] == 0, 1, 0], 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, 0, 20}] (* Jean-François Alcover, Jul 21 2015, after Alois P. Heinz *)
PROG
(PARI) seq(n)={Vec(sum(k=0, n\3, x^(3*k)/prod(j=1, k, 1 - x^j + O(x*x^n))^5) + O(x*x^n))} \\ Andrew Howroyd, Sep 16 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
a(18)-a(35) from Alois P. Heinz, Jul 04 2009
Edited by Max Alekseyev, Dec 11 2011
a(36) from Alois P. Heinz, Feb 03 2013
STATUS
approved