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%I #19 Mar 19 2023 11:53:57
%S 1,1,1,1,121,721,2521,6721,128521,1277641,7539841,32527441,281835841,
%T 3031468441,23779315561,139431015361,962322302761,9034098300361,
%U 79726215362761,569831799431881,3952559737085401,32660742079719601,289694072383115401
%N Diagonal of the rational function 1/(1 - v*w*x*y*z * (1 + 1/v + 1/w + 1/x + 1/y + 1/z)).
%H Winston de Greef, <a href="/A361636/b361636.txt">Table of n, a(n) for n = 0..1064</a>
%F a(n) = Sum_{k=0..floor(n/4)} (n+k)!/(k!^5 * (n-4*k)!).
%F G.f.: Sum_{k>=0} (5*k)!/k!^5 * x^(4*k)/(1-x)^(5*k+1).
%F Recurrence: n^4*(5*n - 19)*(5*n - 18)*(5*n - 17)*(5*n - 14)*(5*n - 13)*(5*n - 9)*a(n) = (5*n - 19)*(5*n - 18)*(5*n - 14)*(625*n^7 - 6125*n^6 + 23025*n^5 - 43195*n^4 + 45394*n^3 - 28716*n^2 + 10144*n - 1536)*a(n-1) - (5*n - 19)*(31250*n^9 - 568750*n^8 + 4441875*n^7 - 19516000*n^6 + 53172025*n^5 - 93366740*n^4 + 106140132*n^3 - 75781664*n^2 + 30987264*n - 5529600)*a(n-2) + (5*n - 4)*(31250*n^9 - 725000*n^8 + 7354375*n^7 - 42784750*n^6 + 157237100*n^5 - 378480620*n^4 + 596812963*n^3 - 594970390*n^2 + 340845072*n - 85743360)*a(n-3) + (5*n - 16)*(5*n - 9)*(5*n - 8)*(5*n - 4)*(78000*n^6 - 1450800*n^5 + 11179085*n^4 - 45672814*n^3 + 104341702*n^2 - 126378083*n + 63400710)*a(n-4) + (n-4)^4*(5*n - 14)*(5*n - 13)*(5*n - 12)*(5*n - 9)*(5*n - 8)*(5*n - 4)*a(n-5). - _Vaclav Kotesovec_, Mar 19 2023
%t Table[Sum[(n + k)!/(k!^5*(n - 4*k)!), {k, 0, n/4}], {n, 0, 25}] (* _Vaclav Kotesovec_, Mar 19 2023 *)
%o (PARI) a(n) = sum(k=0, n\4, (n+k)!/(k!^5*(n-4*k)!));
%Y Cf. A001850, A208425, A274783.
%Y Cf. A008978.
%K nonn
%O 0,5
%A _Seiichi Manyama_, Mar 19 2023
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