login
The number of multinomial coefficients over partitions with value equal to 4.
6

%I #23 Feb 10 2024 12:18:34

%S 0,0,0,0,1,1,2,1,2,3,3,2,4,4,4,4,5,5,6,5,6,7,7,6,8,8,8,8,9,9,10,9,10,

%T 11,11,10,12,12,12,12,13,13,14,13,14,15,15,14,16,16,16,16,17,17,18,17,

%U 18,19,19,18,20,20,20,20,21,21,22,21,22,23,23,22,24

%N The number of multinomial coefficients over partitions with value equal to 4.

%C The number of multinomial coefficients such that multinomial(t_1+t_2+..._+t_n,t_1,t_2,...,t_n)=4 and t_1+2*t_2+...+n*t_n=n, where t_1, t_2, ... , t_n are nonnegative integers.

%H Vincenzo Librandi, <a href="/A230128/b230128.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,1,1,0,0,-1).

%F a(n) = floor((1/3)*(n-1)) + floor((1/4)*(n-1)) - floor((1/4)*n).

%F a(n) = a(n-3)+a(n-4)-a(n-7). G.f.: x^5*(2*x^2+x+1) / ((x-1)^2*(x+1)*(x^2+1)*(x^2+x+1)). - _Colin Barker_, Mar 06 2014

%e The number 10 has three partitions such that a(10)=4: 1+1+1+7, 1+3+3+3 and 2+2+2+4.

%p seq(floor((n-1)*(1/3))+floor((n-1)*(1/4))-floor((1/4)*n),n=1..50)

%t Table[Floor[(1/3) (n-1)] + Floor[(1/4) (n-1)] - Floor[(1/4) n], {n, 1, 100}] (* _Vincenzo Librandi_, Oct 11 2013 *)

%o (Magma) [Floor((1/3)*(n-1))+Floor((1/4)*(n-1))-Floor((1/4)*n): n in [1..100]]; // _Vincenzo Librandi_, Oct 11 2013

%Y Cf. A230149, A230167, A230197, A230198, A230257, A230258.

%K nonn,easy

%O 1,7

%A _Mircea Merca_, Oct 10 2013

%E More terms from _Vincenzo Librandi_, Oct 11 2013