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A202192 Number of partitions of 5n with equal number of parts congruent to each of 1, 2, 3 and 4 modulo 5.

%I A202192 #14 May 25 2019 10:20:49
%S A202192 1,1,3,8,22,53,124,269,568,1152,2284,4410,8363,15542,28438,51201,
%T A202192 90930,159300,275740,471706,798388,1337478,2219395,3649432,5950078,
%U A202192 9622364,15442269,24600952,38919910,61164114,95513618,148247892,228761668,351032568,535772894
%N A202192 Number of partitions of 5n with equal number of parts congruent to each of 1, 2, 3 and 4 modulo 5.
%H A202192 Alois P. Heinz, <a href="/A202192/b202192.txt">Table of n, a(n) for n = 0..100</a>
%H A202192 <a href="/wiki/Partitions_of_5n">Index and properties of sequences related to partitions of 5n</a>
%F A202192 a(n) = A046776(n) + A202086(n) + A202088(n).
%F A202192 a(n) = A046787(n) + A000041(n).
%t A202192 mkl[i_, l_] := Module[{ll, mn, x}, ll = If[Mod[i, 5] == 0, l, MapAt[#+1&, l, Mod[i, 5]]]; mn = Min[l] - 1; If[mn <= 0, ll, Map[# - mn&, ll]]];
%t A202192 g[n_, i_, t_] := g[n, i, t] = Module[{m, mx}, If[n < 0, 0, If[n == 0, If[ t[[1]] > 0 && Equal @@ t[[1 ;; 4]], 1, 0] , If[i == 0, 0, If[i < 5, mx = Max[t]; m = n - 10 mx + t[[1]] + 2 t[[2]] + 3 t[[3]] + 4 t[[4]]; If[m >= 0 && Mod[m, 10] == 0, 1, 0], g[n, i-1, t] + g[n-i, i, mkl[i, t]]]]]]];
%t A202192 a[n_] :=  g[5n, 5n, {0, 0, 0, 0}] + PartitionsP[n];
%t A202192 Table[a[n], {n, 0, 34}] (* _Jean-François Alcover_, May 25 2019, after _Alois P. Heinz_ in A046787 *)
%Y A202192 Cf. A046776.
%K A202192 nonn
%O A202192 0,3
%A A202192 _Max Alekseyev_, Dec 14 2011
%E A202192 a(33)-a(34) from _Alois P. Heinz_, May 24 2019