A191496 Number of compositions of even numbers into 9 parts <= n.

1, 256, 9842, 131072, 976563, 5038848, 20176804, 67108864, 193710245, 500000000, 1178973846, 2579890176, 5302249687, 10330523392, 19221679688, 34359738368, 59293938249, 99179645184, 161343848890, 256000000000, 397140023291, 603634608896, 900576330732, 1320903770112, 1907348632813, 2714751839488

Offset: 0

Author

Adi Dani, Jun 03 2011

Keywords

• nonn

Comments

Number of ways of placing an even number of indistinguishable objects in 9 distinguishable boxes with the condition that each box can hold most n objects.

Examples

			a(1)=256: there are 256 compositions of even numbers into 9 parts <= 1:
   0: (0,0,0,0,0,0,0,0,0) --> 9!/9!0! =   1
   2: (0,0,0,0,0,0,0,1,1) --> 9!/7!2! =  36
   4: (0,0,0,0,0,1,1,1,1) --> 9!/5!4! = 126
   8: (0,0,0,1,1,1,1,1,1) --> 9!/3!6! =  84
  10: (0,1,1,1,1,1,1,1,1) --> 9!/1!8! =   9
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Adi Dani, Restricted compositions of natural numbers

Programs

Magma

[( (n+1)^9 + (1+(-1)^n)/2 )/2: n in [0..30]]; // Vincenzo Librandi, Jun 16 2011

Mathematica

Table[1/2*((n + 1)^9 + (1 + (-1)^n)*1/2), {n, 0, 25}]

Formula

a(n)= ( (n+1)^9 + (1 + (-1)^n)/2 )/2.

a(n) = 9*a(n-1) - 35*a(n-2) + 75*a(n-3) - 90*a(n-4) + 42*a(n-5) + 42*a(n-6) - 90*a(n-7) + 75*a(n-8) - 35*a(n-9) + 9*a(n-10) - a(n-11).

G.f.: (1 + 247*x + 7573*x^2 + 51379*x^3 + 122275*x^4 + 122149*x^5 + 51463*x^6 + 7537*x^7 + 256*x^8) / ( (1+x)*(1-x)^10 ). - R. J. Mathar, Jun 06 2011