This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A215541 #21 Aug 29 2025 03:55:29 %S A215541 1,4,35,350,3705,40480,451269,5101360,58261125,670609940,7766844470, %T A215541 90404916420,1056658719675,12393263030400,145787921878840, %U A215541 1719353829326880,20322351313767965,240674861588534100,2855214354095519625,33924757188414045330,403641797464597415570 %N A215541 a(n) = binomial(5*n,n)*(3*n+1)/(4*n+1). %C A215541 Number of standard Young tableaux of shape [4n,n]. Also the number of binary words with 4n 1's and n 0's such that for every prefix the number of 1's is >= the number of 0's. The a(1) = 4 words are: 10111, 11011, 11101, 11110. %H A215541 Alois P. Heinz, <a href="/A215541/b215541.txt">Table of n, a(n) for n = 0..285</a> %H A215541 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>. %F A215541 a(n) = C(5*n,n)*(3*n+1)/(4*n+1). %F A215541 a(n) = [x^n] ((1 - sqrt(1 - 4*x))/(2*x))^(3*n+1). - _Ilya Gutkovskiy_, Nov 01 2017 %F A215541 Recurrence: 8*n*(2*n - 1)*(3*n - 2)*(4*n - 1)*(4*n + 1)*a(n) = 5*(3*n + 1)*(5*n - 4)*(5*n - 3)*(5*n - 2)*(5*n - 1)*a(n-1). - _Vaclav Kotesovec_, Feb 03 2018 %F A215541 a(n) ~ 3 * 5^(5*n+1/2) / (2^(8*n+7/2) * sqrt(Pi*n)). - _Amiram Eldar_, Aug 29 2025 %p A215541 a:= n-> binomial(5*n,n)*(3*n+1)/(4*n+1): %p A215541 seq(a(n), n=0..25); %t A215541 a[n_] := Binomial[5*n,n]*(3*n+1)/(4*n+1); Array[a, 21, 0] (* _Amiram Eldar_, Aug 29 2025 *) %Y A215541 Column k=4 of A214776. %K A215541 nonn,changed %O A215541 0,2 %A A215541 _Alois P. Heinz_, Aug 15 2012