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
Examples
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
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).
Crossrefs
Programs
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Magma
[((n + 1)^6 + (1+(-1)^n)/2 )/2: n in [0..40]]; // Vincenzo Librandi, Jun 16 2011
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Mathematica
Table[1/2*((n + 1)^6 + (1 + (-1)^n)*1/2), {n, 0, 25}]
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
Comments