cp's OEIS Frontend

A215543 Number of standard Young tableaux of shape [3n,3].

%I A215543 #18 Aug 29 2025 03:55:43
%S A215543 0,5,48,154,350,663,1120,1748,2574,3625,4928,6510,8398,10619,13200,
%T A215543 16168,19550,23373,27664,32450,37758,43615,50048,57084,64750,73073,
%U A215543 82080,91798,102254,113475,125488,138320,151998,166549,182000,198378,215710,234023,253344
%N A215543 Number of standard Young tableaux of shape [3n,3].
%C A215543 Also the number of binary words with 3n 1's and 3 0's such that for every prefix the number of 1's is >= the number of 0's.  The a(1) = 5 words are: 101010, 101100, 110010, 110100, 111000.
%H A215543 Alois P. Heinz, <a href="/A215543/b215543.txt">Table of n, a(n) for n = 0..1000</a>
%H A215543 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>.
%H A215543 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A215543 G.f.: (2*x^3-8*x^2+28*x+5)*x/(x-1)^4.
%F A215543 a(n) = (3*n-2)*(3*n+2)*(n+1)/2 for n>0, a(0) = 0.
%F A215543 From _Amiram Eldar_, Aug 29 2025: (Start)
%F A215543 Sum_{n>=1} 1/a(n) = 7/20 + sqrt(3)*Pi/10 - 3*log(3)/5.
%F A215543 Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(3)*Pi/5 - 4*log(2)/5 - 7/20. (End)
%p A215543 a:= n-> max(0, (3*n-2)*(3*n+2)*(n+1)/2):
%p A215543 seq(a(n), n=0..40);
%t A215543 LinearRecurrence[{4,-6,4,-1},{0,5,48,154,350},50] (* _Harvey P. Dale_, Dec 16 2017 *)
%o A215543 (PARI) a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,4,-6,4]^n*[0;5;48;154])[1,1] \\ _Charles R Greathouse IV_, Oct 18 2022
%Y A215543 Row n=3 of A214776.
%K A215543 nonn,easy,changed
%O A215543 0,2
%A A215543 _Alois P. Heinz_, Aug 15 2012