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A191680 Number of compositions of odd natural numbers into 9 parts <= n.

%I A191680 #23 Sep 08 2022 08:45:57
%S A191680 0,256,9841,131072,976562,5038848,20176803,67108864,193710244,
%T A191680 500000000,1178973845,2579890176,5302249686,10330523392,19221679687,
%U A191680 34359738368,59293938248,99179645184,161343848889,256000000000,397140023290,603634608896,900576330731,1320903770112,1907348632812,2714751839488
%N A191680 Number of compositions of odd natural numbers into 9 parts <= n.
%H A191680 Vincenzo Librandi, <a href="/A191680/b191680.txt">Table of n, a(n) for n = 0..1000</a>
%H A191680 Adi Dani, <a href="https://oeis.org/wiki/User:Adi_Dani_/Restricted_compositions_of_natural_numbers">Restricted compositions of natural numbers</a>
%F A191680 a(n) = ((n + 1)^9 - (1 + (-1)^n)/2)/2.
%F A191680 G.f.: x*(256 + 7537*x + 51463*x^2 + 122149*x^3 + 122275*x^4 + 51379*x^5 + 7573*x^6 + 247*x^7 + x^8) / ( (1+x)*(x-1)^10 ). - _R. J. Mathar_, Jun 29 2011
%F A191680 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). - _R. J. Mathar_, Jun 29 2011
%F A191680 a(2n) = A191496(2n) - 1. a(2n+1) = A191496(2n+1). - _R. J. Mathar_, Jun 29 2011
%e A191680 a(1)=256 compositions of odd numbers into 9 parts <= 1:
%e A191680   1: (0,0,0,0,0,0,0,0,1) --> 9!/(8!1!) =   9
%e A191680   3: (0,0,0,0,0,0,1,1,1) --> 9!/(6!3!) =  84
%e A191680   5: (0,0,0,0,1,1,1,1,1) --> 9!/(4!5!) = 126
%e A191680   7: (0,0,1,1,1,1,1,1,1) --> 9!/(2!7!) =  36
%e A191680   9: (1,1,1,1,1,1,1,1,1) --> 9!/(0!9!) =   1
%e A191680   ------------------------------------------
%e A191680                                          256
%t A191680 Table[Floor[1/2*((n + 1)^9 - (1 + (-1)^n)/2)], {n, 0, 25}]
%o A191680 (Magma) [1/2*((n + 1)^9 - (1 + (-1)^n)/2): n in [0..30]]; // _Vincenzo Librandi_, Jun 16 2011
%K A191680 nonn
%O A191680 0,2
%A A191680 _Adi Dani_, Jun 11 2011
%E A191680 Offset changed from 1 to 0 by _Vincenzo Librandi_, Jun 16 2011