A191495 Number of compositions of even natural numbers into 8 parts <= n.
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
Examples
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
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Adi Dani, Restricted compositions of natural numbers
- Index entries for linear recurrences with constant coefficients, signature (8,-27,48,-42,0,42,-48,27,-8,1).
Crossrefs
Programs
-
Magma
[ 1/2*((n + 1)^8 + (1 + (-1)^n)*1/2): n in [0..35]]; // Vincenzo Librandi, Jun 06 2011
-
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 *) -
PARI
a(n)=((n+1)^8+1)>>1 \\ Charles R Greathouse IV, Jun 06 2011
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
Comments