cp's OEIS Frontend

A215570 Number of permutations of n indistinguishable copies of 1..5 with every partial sum <= the same partial sum averaged over all permutations.

%I A215570 #24 Mar 07 2023 06:57:35
%S A215570 1,35,18720,19369350,27032968200,44776592395920,82881380383401600,
%T A215570 165850226337286576800,351597937025844947295000,
%U A215570 779279938350147159519336600,1789294251011628021153241548800,4228135363283244543270651711564000,10232120200642411474243152429724152000
%N A215570 Number of permutations of n indistinguishable copies of 1..5 with every partial sum <= the same partial sum averaged over all permutations.
%H A215570 Manuel Kauers and Christoph Koutschan, <a href="/A215570/b215570.txt">Table of n, a(n) for n = 0..50</a> (terms 0..47 from Vaclav Kotesovec).
%H A215570 M. Kauers and C. Koutschan, <a href="https://arxiv.org/abs/2303.02793">Some D-finite and some possibly D-finite sequences in the OEIS</a>, arXiv:2303.02793 [cs.SC], 2023.
%F A215570 a(n) ~ (3*sqrt(5)-5) * 5^(5*n) / (8*Pi^2*n^3). - _Vaclav Kotesovec_, Sep 06 2016
%F A215570 Conjectured recurrence of order 3 and degree 15: 3*(n + 1)*(n + 2)^3*(n + 4)^2*(3*n + 8)*(3*n + 10)*(65*n^3 + 398*n^2 + 781*n + 496)*(n + 3)^4*a(n + 3) - 20*(n + 1)*(n + 2)^3*(5*n + 11)*(5*n + 12)*(5*n + 13)*(5*n + 14)*(910*n^5 + 11032*n^4 + 52047*n^3 + 119686*n^2 + 134365*n + 58980)*(n + 3)^2*a(n + 2) + 25*(n + 1)*(n + 2)*(5*n + 6)*(5*n + 7)*(5*n + 8)*(5*n + 9)*(5*n + 11)*(5*n + 12)*(5*n + 13)*(5*n + 14)*(2015*n^5 + 24428*n^4 + 114387*n^3 + 258294*n^2 + 281088*n + 118368)*a(n + 1) - 250*(5*n + 1)*(5*n + 2)*(5*n + 3)*(5*n + 4)*(5*n + 6)*(5*n + 7)*(5*n + 8)*(5*n + 9)*(5*n + 11)*(5*n + 12)*(5*n + 13)*(5*n + 14)*(65*n^3 + 593*n^2 + 1772*n + 1740)*a(n) = 0. - _Manuel Kauers_ and _Christoph Koutschan_, Mar 02 2023
%e A215570 a(1) = 35: (1,2,3,4,5), (1,2,3,5,4), ..., (3,2,1,5,4), (3,2,4,1,5).
%p A215570 b:= proc(l) option remember; local m, n, g;
%p A215570       m, n:= nops(l), add(i, i=l);
%p A215570       g:= add(i*l[i], i=1..m)-(m+1)/2*(n-1);
%p A215570      `if`(n<2, 1, add(`if`(l[i]>0 and i<=g,
%p A215570         b(subsop(i=l[i]-1, l)), 0), i=1..m))
%p A215570     end:
%p A215570 a:= n-> b([n$5]):
%p A215570 seq(a(n), n=0..15);
%t A215570 b[l_] := b[l] = Module[{m, n, g}, {m, n} = {Length[l], Total[l]}; g = Sum[i*l[[i]], {i, 1, m}] - (m + 1)/2*(n - 1); If[n < 2, 1, Sum[If[l[[i]] > 0 && i <= g, b[ReplacePart[l, i -> l[[i]] - 1]], 0], {i, 1, m}]]];
%t A215570 a[k_] := b[Array[k&, 5]];
%t A215570 a /@ Range[0, 15] (* _Jean-François Alcover_, Dec 18 2020, after _Alois P. Heinz_ *)
%Y A215570 Row n=5 of A215561.
%K A215570 nonn
%O A215570 0,2
%A A215570 _Alois P. Heinz_, Aug 16 2012