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%I #13 Jan 25 2024 17:34:11
%S 0,14,275,1260,3705,8602,17199,31000,51765,81510,122507,177284,248625,
%T 339570,453415,593712,764269,969150,1212675,1499420,1834217,2222154,
%U 2668575,3179080,3759525,4416022,5154939,5982900,6906785,7933730,9071127,10326624,11708125
%N Number of standard Young tableaux of shape [4n,4].
%C Also the number of binary words with 4n 1's and 4 0's such that for every prefix the number of 1's is >= the number of 0's. The a(1) = 14 words are: 10101010, 10101100, 10110010, 10110100, 10111000, 11001010, 11001100, 11010010, 11010100, 11011000, 11100010, 11100100, 11101000, 11110000.
%H Alois P. Heinz, <a href="/A215544/b215544.txt">Table of n, a(n) for n = 0..1000</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F G.f.: (3*x^4-15*x^3-25*x^2-205*x-14)*x/(x-1)^5.
%F a(n) = (4*n-3)*(4*n+3)*(2*n+1)*(n+1)/3 for n>0, a(0) = 0.
%p a:= n-> max(0, (4*n+3)*(2*n+1)*(4*n-3)*(n+1)/3):
%p seq(a(n), n=0..40);
%t LinearRecurrence[{5,-10,10,-5,1},{0,14,275,1260,3705,8602},40] (* _Harvey P. Dale_, Jan 25 2024 *)
%o (PARI) a(n)=max((4*n-3)*(4*n+3)*(2*n+1)*(n+1)/3,0) \\ _Charles R Greathouse IV_, Oct 21 2022
%Y Row n=4 of A214776.
%K nonn,easy
%O 0,2
%A _Alois P. Heinz_, Aug 15 2012
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