A191489 Number of compositions of even natural numbers into 6 parts <= n.

1, 32, 365, 2048, 7813, 23328, 58825, 131072, 265721, 500000, 885781, 1492992, 2413405, 3764768, 5695313, 8388608, 12068785, 17006112, 23522941, 32000000, 42883061, 56689952, 74017945, 95551488, 122070313, 154457888

Offset

0,2

Comments

Number of ways of placing of an even number of indistinguishable objects in 6 distinguishable boxes with 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 (6,-14,14,0,-14,14,-6,1).

Formula

a(n) = ((n + 1)^6 + (1+(-1)^n)/2 )/2.

G.f.: (x^2 + 10*x + 1)*(x^4 + 16*x^3 + 26*x^2 + 16*x + 1) / ( (1+x)*(1-x)^7 ). - R. J. Mathar, Jun 06 2011

a(2n) = A175113(n). - R. J. Mathar, Jun 07 2011

Example

a(1)=32 compositions of even natural numbers in 6 parts <= 1 are
:(0,0,0,0,0,0)--> 6!/(6!0!) =  1
:(0,0,0,0,1,1)--> 6!/(4!2!) = 15
:(0,0,1,1,1,1)--> 6!/(2!4!) = 15
:(1,1,1,1,1,1)--> 6!/(0!6!) =  1
a(2)=365 compositions of even natural numbers in 6 parts <= 2 are
:(0,0,0,0,0,0)--> 6!/(6!0!0!) =  1
:(0,0,0,0,1,1)--> 6!/(4!2!0!) = 15
:(0,0,0,0,0,2)--> 6!/(5!0!1!) =  6
:(0,0,1,1,1,1)--> 6!/(2!4!0!) = 15
:(0,0,0,1,1,2)--> 6!/(3!2!1!) = 60
:(0,0,0,0,2,2)--> 6!/(4!0!2!) = 15
:(0,1,1,1,1,2)--> 6!/(1!4!1!) = 30
:(0,0,0,2,2,2)--> 6!/(3!0!3!) = 20
:(0,0,1,1,2,2)--> 6!/(2!2!2!) = 90
:(1,1,1,1,1,1)--> 6!/(0!6!0!) =  1
:(0,1,1,2,2,2)--> 6!/(1!2!3!) = 60
:(0,0,2,2,2,2)--> 6!/(2!0!4!) = 15
:(1,1,1,1,2,2)--> 6!/(0!4!2!) = 15
:(0,2,2,2,2,2)--> 6!/(1!0!5!) =  6
:(1,1,2,2,2,2)--> 6!/(0!2!4!) = 15
:(2,2,2,2,2,2)--> 6!/(0!0!6!) =  1

Mathematica

Table[1/2*((n + 1)^6 + (1 + (-1)^n)*1/2), {n, 0, 25}]

Prog

(Magma) [((n + 1)^6 + (1+(-1)^n)/2 )/2: n in [0..40]]; // Vincenzo Librandi, Jun 16 2011

Crossrefs

Cf. A036486 (3 parts), A171714 (4 parts), A191484 (5 parts), A191494 (7 parts), A191495 (8 parts).

Sequence in context: A303511 A240787 A191901 * A055752 A362234 A250170

Adjacent sequences: A191486 A191487 A191488 * A191490 A191491 A191492

Keyword

nonn,easy

Author

Adi Dani, Jun 03 2011

Status

approved