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 A084534 #52 Aug 24 2024 05:53:26
%S A084534 1,1,2,1,4,2,1,6,9,2,1,8,20,16,2,1,10,35,50,25,2,1,12,54,112,105,36,2,
%T A084534 1,14,77,210,294,196,49,2,1,16,104,352,660,672,336,64,2,1,18,135,546,
%U A084534 1287,1782,1386,540,81,2,1,20,170,800,2275,4004,4290,2640,825,100,2
%N A084534 Triangle read by rows: row #n has n+1 terms. T(n,0)=1, T(n,n)=2, T(n,m) = T(n-1,m-1) + Sum_{k=0..m} T(n-1-k,m-k).
%C A084534 Sum of row #n = A000204(2n). (But sum of row #0 = 1.)
%C A084534 Row #n has the unsigned coefficients of the monic polynomial whose roots are 2 cos(Pi*(2k-1)/(4n)) for k=1..2n. [Comment corrected by Barry Brent, Jan 03 2006]
%C A084534 The positive roots are some diagonal lengths of a regular (4n)-gon, inscribed in the unit circle.
%C A084534 Polynomial of row #n = Sum_{m=0..n} (-1)^m * T(n,m) x^(2*n-2*m).
%C A084534 This is the unsigned version of the coefficient table for scaled Chebyshev T(2*n,x) polynomials. - _Wolfdieter Lang_, Mar 07 2007
%C A084534 Reversed A127677 (cf. A156308, A217476, A263916). - _Tom Copeland_, Nov 07 2015
%D A084534 I. Kaplansky and J. Riordan, The problème des ménages, Scripta Math. 12, (1946), 113-124. See p. 118.
%D A084534 Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990. p. 37, eq.(1.96) and p. 4. eq.(1.10).
%H A084534 G. C. Greubel, <a href="/A084534/b084534.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A084534 I. Kaplansky and J. Riordan, <a href="/A000166/a000166_1.pdf">The problème des ménages</a>, Scripta Math. 12, (1946), 113-124. [Scan of annotated copy]
%F A084534 T(n,m) = binomial(2*n-m, m)*2*n/(2*n-m) for n > 0. - _Andrew Howroyd_, Dec 18 2017
%F A084534 Signed version from _Wolfdieter Lang_, Mar 07 2007: (Start)
%F A084534 a(n,m)=0 if n<m, a(0,0)=1 else a(n,m) = (-1)^m*binomial(2*n-m, m)*2*n/(2*n-m).
%F A084534 a(n,m)=0 if n<m, a(0,0)=1 else a(n,m) = (-1)^m*Sum_{l=0..n-m} binomial(m+l,l)*binomial(2*n,2*(l+m))/2^(2*(n-m)-1).
%F A084534 a(n,m)=0 if n<m, a(0,0)=1 else a(n,m) = A127674(n,n-m)/2^(2*(n-m)-1) (scaled coefficients of Chebyshev's T(2*n,x), decreasing even powers). [corrected by _Johannes W. Meijer_, May 31 2018] (End)
%e A084534 First few Chebyshev T(2*n,x) polynomials:
%e A084534 T(2*0,x) = 1;
%e A084534 T(2*1,x) = x^2 - 2;
%e A084534 T(2*2,x) = x^4 - 4*x^2 + 2;
%e A084534 T(2*3,x) = x^6 - 6*x^4 + 9*x^2 - 2;
%e A084534 T(2*4,x) = x^8 - 8*x^6 + 20*x^4 - 16*x^2 + 2;
%e A084534 T(2*5,x) = x^10 - 10*x^8 + 35*x^6 - 50*x^4 + 25*x^2 - 2;
%e A084534 Triangle begins as:
%e A084534 1;
%e A084534 1, 2;
%e A084534 1, 4, 2;
%e A084534 1, 6, 9, 2;
%e A084534 1, 8, 20, 16, 2;
%e A084534 1, 10, 35, 50, 25, 2;
%e A084534 1, 12, 54, 112, 105, 36, 2;
%p A084534 T := proc(n, m): if n=0 then 1 else binomial(2*n-m, m)*2*n/(2*n-m) fi: end: seq(seq(T(n,m),m=0..n),n=0..10); # _Johannes W. Meijer_, May 31 2018
%t A084534 a[n_, m_] := Binomial[2n-m, m]*2n/(2n-m); a[0, 0] = 1; Table[a[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* _Jean-François Alcover_, Apr 12 2016, after _Wolfdieter Lang_ *)
%o A084534 (PARI) T(n,m) = if(n==0, m==0, binomial(2*n-m, m)*2*n/(2*n-m)) \\ _Andrew Howroyd_, Dec 18 2017
%o A084534 (Magma)
%o A084534 A084534:= func< n,k | k eq 0 select 1 else 2*(n/k)*Binomial(2*n-k-1, k-1) >;
%o A084534 [A084534(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 02 2022
%o A084534 (Sage)
%o A084534 def A084534(n,k): return 1 if (k==0) else 2*(n/k)*binomial(2*n-k-1, k-1)
%o A084534 flatten([[A084534(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 02 2022
%Y A084534 Row sums are A005248 for n > 0.
%Y A084534 Companion triangle A082985.
%Y A084534 Cf. A082985 (unsigned scaled coefficient table for Chebyshev's T(2*n+1, x) polynomials).
%Y A084534 Cf. A127677, A156308, A217476, A263916.
%K A084534 nonn,tabl,easy
%O A084534 0,3
%A A084534 _Gary W. Adamson_, May 29 2003
%E A084534 Edited by _Don Reble_, Nov 12 2005