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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) Index and properties of sequences related to partitions of 5n 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: Mod 4: A046778, A046779, A046780, A046781, A046782. Mod 5: A046783, A046784, A046785, A046786. Cf. A046765, A046776, A202192. Sequence in context: A147043 A146264 A146216 * A003295 A228857 A253427 Adjacent sequences: A046784 A046785 A046786 * A046788 A046789 A046790 KEYWORD nonn AUTHOR David W. Wilson 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.)