A076177 a(n) = n! * Sum_{ 0<=i<=j<=k<=n, i+j+k<=n } 1/(i!*j!*k!).

1, 2, 7, 31, 147, 801, 5028, 35757, 287703, 2594737, 25961688, 285620919, 3427588046, 44559016789, 623827340321, 9357413642355, 149718628050447, 2545216705948425, 45813900799435848, 870464115447489783, 17409282309719616078, 365594928506541029523, 8043088427150753393871

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..449

Formula

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

a(n) ~ c * n!, where c = 7.1557679887402719497137033299521416531568... - Vaclav Kotesovec, Feb 25 2014

Maple

a:=proc(n) option remember; `if`(n<7, [1, 2, 7, 31, 147, 801, 5028][n+1],
    ((-15309*n^8 +396576*n^7 -4332204*n^6 +25987635*n^5 -93262671*n^4
     +203936049*n^3 -263303136*n^2 +181302300*n -49863600)*a(n-7)
     +(-5103*n^8 +152604*n^7 -1863729*n^6 +12224196*n^5 -47180232*n^4
     +109510056*n^3 -148441896*n^2 +106270704*n -30093120) *a(n-6)
     +(8505*n^8 -181197*n^7 +1629585*n^6 -8044083*n^5 +23717421*n^4
     -42527862*n^3 +44992341*n^2 -25476606*n +5861160) *a(n-5) +(-1134*n^8
     +12366*n^7 -9135*n^6 -449289*n^5 +2794014*n^4 -7745031*n^3 +11267883*n^2
     -8231706*n +2317248) *a(n-4) +(189*n^8 -2817*n^7 +16755*n^6 -52257*n^5
     +98271*n^4 -131592*n^3 +141765*n^2 -101538*n  +28512) *a(n-3) +(-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) +(63*n^8 -498*n^7 -706*n^6 +18915*n^5
     -77705*n^4 +148961*n^3 -149588*n^2 +76326*n -15768) *a(n-1))/
     (n^2 *(63*n^5 -813*n^4 +3989*n^3 -9250*n^2 +10040*n -4029)))
   end:
seq(a(n), n=0..40);  # Alois P. Heinz, Aug 07 2012

Mathematica

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 *)

Prog

(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!)))

Crossrefs

Cf. A076176.

Sequence in context: A349769 A358963 A055836 * A335868 A126033 A369622

Adjacent sequences: A076174 A076175 A076176 * A076178 A076179 A076180

Keyword

nonn

Author

Benoit Cloitre, Nov 01 2002

Status

approved