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%I #18 Sep 16 2017 23:21:36
%S 1,2,7,31,147,801,5028,35757,287703,2594737,25961688,285620919,
%T 3427588046,44559016789,623827340321,9357413642355,149718628050447,
%U 2545216705948425,45813900799435848,870464115447489783,17409282309719616078,365594928506541029523,8043088427150753393871
%N a(n) = n! * Sum_{ 0<=i<=j<=k<=n, i+j+k<=n } 1/(i!*j!*k!).
%H Alois P. Heinz, <a href="/A076177/b076177.txt">Table of n, a(n) for n = 0..449</a>
%F Recurrence: (n-3)*(n-1)*n^2*(63*n^3 - 561*n^2 + 1556*n - 1343)*a(n) = (n-1)*(63*n^7 - 435*n^6 - 1141*n^5 + 17774*n^4 - 59931*n^3 + 89030*n^2 - 60558*n + 15768)*a(n-1) - (315*n^8 - 4821*n^7 + 29287*n^6 - 88832*n^5 + 133626*n^4 - 62593*n^3 - 79108*n^2 + 110766*n - 38016)*a(n-2) + 3*(n-2)*(63*n^7 - 813*n^6 + 3959*n^5 - 9501*n^4 + 13755*n^3 - 16354*n^2 + 14547*n - 4752)*a(n-3) - 9*(n-3)*(n-2)*(126*n^6 - 744*n^5 - 3461*n^4 + 37080*n^3 - 104280*n^2 + 116679*n - 42912)*a(n-4) + 27*(n-4)*(n-3)*(n-2)*(315*n^5 - 3876*n^4 + 17281*n^3 - 34064*n^2 + 29517*n - 9045)*a(n-5) - 81*(n-5)*(n-4)*(n-3)*(n-2)*(63*n^4 - 1002*n^3 + 4508*n^2 - 6960*n + 3096)*a(n-6) - 243*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(63*n^3 - 372*n^2 + 623*n - 285)*a(n-7). - _Vaclav Kotesovec_, Feb 25 2014
%F a(n) ~ c * n!, where c = 7.1557679887402719497137033299521416531568... - _Vaclav Kotesovec_, Feb 25 2014
%p a:=proc(n) option remember; `if`(n<7, [1, 2, 7, 31, 147, 801, 5028][n+1],
%p ((-15309*n^8 +396576*n^7 -4332204*n^6 +25987635*n^5 -93262671*n^4
%p +203936049*n^3 -263303136*n^2 +181302300*n -49863600)*a(n-7)
%p +(-5103*n^8 +152604*n^7 -1863729*n^6 +12224196*n^5 -47180232*n^4
%p +109510056*n^3 -148441896*n^2 +106270704*n -30093120) *a(n-6)
%p +(8505*n^8 -181197*n^7 +1629585*n^6 -8044083*n^5 +23717421*n^4
%p -42527862*n^3 +44992341*n^2 -25476606*n +5861160) *a(n-5) +(-1134*n^8
%p +12366*n^7 -9135*n^6 -449289*n^5 +2794014*n^4 -7745031*n^3 +11267883*n^2
%p -8231706*n +2317248) *a(n-4) +(189*n^8 -2817*n^7 +16755*n^6 -52257*n^5
%p +98271*n^4 -131592*n^3 +141765*n^2 -101538*n +28512) *a(n-3) +(-315*n^8
%p +4821*n^7 -29287*n^6 +88832*n^5 -133626*n^4 +62593*n^3 +79108*n^2
%p -110766*n +38016) *a(n-2) +(63*n^8 -498*n^7 -706*n^6 +18915*n^5
%p -77705*n^4 +148961*n^3 -149588*n^2 +76326*n -15768) *a(n-1))/
%p (n^2 *(63*n^5 -813*n^4 +3989*n^3 -9250*n^2 +10040*n -4029)))
%p end:
%p seq(a(n), n=0..40); # _Alois P. Heinz_, Aug 07 2012
%t a[n_] := n!*Sum[ Boole[i+j+k <= n] / (i!*j!*k!), {i, 0, n}, {j, i, n}, {k, j, n}]; Table[a[n], {n, 0, 22}] (* _Jean-François Alcover_, Jun 18 2013 *)
%o (PARI) a(n)=n!*sum(i=0,n,sum(j=0,i,sum(k=0,j,(if(i+j+k-n,0,1)+if(sign(i+j+k-n)+1,0,1))/i!/j!/k!)))
%Y Cf. A076176.
%K nonn
%O 0,2
%A _Benoit Cloitre_, Nov 01 2002
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