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URL: https://oeis.org/A191495

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A191495
Number of compositions of even natural numbers into 8 parts <= n.
5
1, 128, 3281, 32768, 195313, 839808, 2882401, 8388608, 21523361, 50000000, 107179441, 214990848, 407865361, 737894528, 1281445313, 2147483648, 3487878721, 5509980288, 8491781521, 12800000000, 18911429681, 27437936768, 39155492641, 55037657088, 76293945313, 104413532288
OFFSET
0,2
COMMENTS
Number of ways of placing an even number of indistinguishable objects in 8 distinguishable boxes with the condition that in each box can be at most n objects.
FORMULA
a(n) = ( (n+1)^8 + (1 + (-1)^n)/2 )/2.
G.f.: ( -1 - 120*x - 2284*x^2 - 9928*x^3 - 15654*x^4 - 9928*x^5 - 2284*x^6 - 120*x^7 - x^8 ) / ( (1+x)*(x-1)^9 ). - R. J. Mathar, Jun 06 2011
a(n) = 8*a(n-1) - 27*a(n-2) + 48*a(n-3) - 42*a(n-4) + 42*a(n-6) - 48*a(n-7) + 27*a(n-8) - 8*a(n-9) + a(n-10); a(0)=1, a(1)=128, a(2)=3281, a(3)=32768, a(4)=195313, a(5)=839808, a(6)=2882401, a(7)=8388608, a(8)=21523361, a(9)=50000000. - Harvey P. Dale, Mar 13 2013
EXAMPLE
a(1)=128 compositions of even natural numbers into 8 parts no greater than 1 are
(0,0,0,0,0,0,0,0) --> 8!/(8!0!) = 1
(0,0,0,0,0,0,1,1) --> 8!/(6!2!) = 28
(0,0,0,0,1,1,1,1) --> 8!/(4!4!) = 70
(0,0,1,1,1,1,1,1) --> 8!/(2!6!) = 28
(1,1,1,1,1,1,1,1) --> 8!/(0!8!) = 1
MATHEMATICA
Table[1/2*((n + 1)^8 + (1 + (-1)^n)*1/2), {n, 0, 25}]
LinearRecurrence[{8, -27, 48, -42, 0, 42, -48, 27, -8, 1}, {1, 128, 3281, 32768, 195313, 839808, 2882401, 8388608, 21523361, 50000000}, 30] (* Harvey P. Dale, Mar 13 2013 *)
PROG
(Magma) [ 1/2*((n + 1)^8 + (1 + (-1)^n)*1/2): n in [0..35]]; // Vincenzo Librandi, Jun 06 2011
(PARI) a(n)=((n+1)^8+1)>>1 \\ Charles R Greathouse IV, Jun 06 2011
CROSSREFS
Cf. A036486 (3 parts), A171714 (4 parts), A191484 (5 parts), A191489 (6 parts), A191494 (7 parts).
Sequence in context: A022150 A269001 A191899 * A250172 A188303 A283813
KEYWORD
nonn,easy
AUTHOR
Adi Dani, Jun 03 2011
STATUS
approved