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A191495 Number of compositions of even natural numbers into 8 parts <=n 3
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 (list; graph; refs; listen; history; text; internal format)
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

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).

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(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, 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). - Harvey P. Dale, Mar 13 2013

EXAMPLE

a(1)=128 compositions of even natural numbers i 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

Sequence in context: A022150 A269001 A191899 * A250172 A188303 A283813

Adjacent sequences:  A191492 A191493 A191494 * A191496 A191497 A191498

KEYWORD

nonn,easy

AUTHOR

Adi Dani, Jun 03 2011

STATUS

approved

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Last modified June 23 09:36 EDT 2018. Contains 305693 sequences. (Running on oeis4.)