A191901 - OEIS

%I #22 Sep 08 2022 08:45:57

%S 0,32,364,2048,7812,23328,58824,131072,265720,500000,885780,1492992,

%T 2413404,3764768,5695312,8388608,12068784,17006112,23522940,32000000,

%U 42883060,56689952,74017944,95551488,122070312,154457888,193710244,240945152,297411660,364500000,443751840

%N Number of compositions of odd natural numbers into 6 parts <= n.

%H Vincenzo Librandi, <a href="/A191901/b191901.txt">Table of n, a(n) for n = 0..10000</a>

%H Adi Dani, <a href="https://oeis.org/wiki/User:Adi_Dani_/Restricted_compositions_of_natural_numbers">Restricted compositions of natural numbers</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (6,-14,14,0,-14,14,-6,1)

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

%F G.f. -4*x*(8+43*x+78*x^2+43*x^3+8*x^4) / ( (1+x)*(x-1)^7 ). - _R. J. Mathar_, Jun 26 2011

%F a(0)=0, a(1)=32, a(2)=364, a(3)=2048, a(4)=7812, a(5)=23328, a(6)=58824, a(7)=131072, a(n)=6*a(n-1)-14*a(n-2)+14*a(n-3)-14*a(n-5)+ 14*a(n-6)- 6*a(n-7)+a(n-8). - _Harvey P. Dale_, Apr 12 2015

%e a(1)=32 compositions of odd numbers into 6 parts <=1.

%e 1:(0,0,0,0,0,1)-->6!/(5!1!)= 6

%e 3:(0,0,0,1,1,1)-->6!/(3!3!)=20

%e 5:(0,1,1,1,1,1)-->6!/(1!5!)= 6

%e -------------------------------------

%e                             32

%p A191901:=n->((n + 1)^6 - (1 + (-1)^n)/2)/2: seq(A191901(n), n=0..50); # _Wesley Ivan Hurt_, Apr 10 2017

%t Table[Floor[1/2*((n + 1)^6 - (1 + (-1)^n)/2)], {n, 0, 30}]

%t LinearRecurrence[{6,-14,14,0,-14,14,-6,1},{0,32,364,2048,7812,23328,58824,131072},40] (* _Harvey P. Dale_, Apr 12 2015 *)

%o (Magma) [((n + 1)^6 - (1 + (-1)^n)/2)/2 : n in [0..25]]; // _Vincenzo Librandi_, Jul 03 2011

%K nonn,easy

%O 0,2

%A _Adi Dani_, Jun 19 2011