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A046787 Number of partitions of 5n with equal nonzero number of parts congruent to each of 1, 2, 3 and 4 modulo 5. 11
0, 0, 1, 5, 17, 46, 113, 254, 546, 1122, 2242, 4354, 8286, 15441, 28303, 51025, 90699, 159003, 275355, 471216, 797761, 1336686, 2218393, 3648177, 5948503, 9620406, 15439833, 24597942, 38916192, 61159549, 95508014, 148241050, 228753319, 351022425, 535760584 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
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
Andrew Howroyd, Table of n, a(n) for n = 0..1000 (terms n=0..100 from Alois P. Heinz)
FORMULA
a(n) = A046776(n) + A202086(n) + A202088(n) - A000041(n) = A202192(n) - A000041(n). - Max Alekseyev
G.f.: (Sum_{k>0} x^(2*k)/(Product_{j=1..k} 1 - x^j)^4)/(Product_{j>0} 1 - x^j). - Andrew Howroyd, Sep 16 2019
MAPLE
mkl:= proc(i, l) local ll, mn, x; ll:= `if`(irem(i, 5)=0, l, applyop(x->x+1, irem(i, 5), l)); mn:= min(l[])-1; `if`(mn<=0, ll, map(x->x-mn, ll)) end:
g:= proc(n, i, t) local m, mx; if n<0 then 0 elif n=0 then `if`(t[1]>0 and t[1]=t[2] and t[2]=t[3] and t[3]=t[4], 1, 0) elif i=0 then 0 elif i<5 then mx:= max(t[]); m:= n-10*mx +t[1] +t[2]*2 +t[3]*3 +t[4]*4; `if`(m>=0 and irem(m, 10)=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, 0, 0, 0]):
seq(a(n), n=0..20); # Alois P. Heinz, Jul 04 2009
MATHEMATICA
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]]];
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]]]]]]];
a[n_] := g[5n, 5n, {0, 0, 0, 0}];
Table[a[n], {n, 0, 34}] (* Jean-François Alcover, May 25 2019, after Alois P. Heinz *)
PROG
(PARI) seq(n)={Vec(sum(k=1, n\2, x^(2*k)/prod(j=1, k, 1 - x^j + O(x*x^(n-2*k)))^4)/prod(j=1, n, 1 - x^j + O(x*x^n)), -(n+1))} \\ Andrew Howroyd, Sep 16 2019
CROSSREFS
Other similar sequences include:
Sequence in context: A147043 A146264 A146216 * A003295 A228857 A253427
KEYWORD
nonn
AUTHOR
EXTENSIONS
a(17)-a(32) from Alois P. Heinz, Jul 04 2009
a(33)-a(34) from Alois P. Heinz, Aug 13 2013
STATUS
approved

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Last modified June 14 07:34 EDT 2024. Contains 373393 sequences. (Running on oeis4.)