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 A132339 #30 Sep 23 2024 04:20:32
%S A132339 1,-1,-1,0,2,0,0,-2,-2,0,0,2,10,2,0,0,-2,-28,-28,-2,0,0,2,60,168,60,2,
%T A132339 0,0,-2,-110,-660,-660,-110,-2,0,0,2,182,2002,4290,2002,182,2,0,0,-2,
%U A132339 -280,-5096,-20020,-20020,-5096,-280,-2,0,0,2,408,11424,74256,136136,74256,11424,408,2,0
%N A132339 Array T(n, k) = (-1)^(n+k)*(n+k-2)!*(2*n+2*k-2)!/(n!*k!*(2*n-1)!*(2*k-1)!), with T(0, 0) = 1, T(0, 1) = T(1, 0) = -1, read by antidiagonals.
%H A132339 G. C. Greubel, <a href="/A132339/b132339.txt">Antidiagonals n = 0..50, flattened</a>
%H A132339 G. Kreweras, <a href="http://www.numdam.org/numdam-bin/item?id=BURO_1965__6__9_0">Sur une classe de problèmes de dénombrement liés au treillis des partitions des entiers</a>, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, 6 (1965), circa p. 82.
%F A132339 T(n, k) = (-1)^(n+k)*(n+k-2)!*(2*n+2*k-2)!/(n!*k!*(2*n-1)!*(2*k-1)!), with T(0, 0) = 1, T(0, 1) = T(1, 0) = -1.
%F A132339 A(n, k) = T(n-k, k) (antidiagonals).
%F A132339 A(n, n-k) = A(n, k).
%F A132339 A(2*n, n) = A132341(n).
%e A132339 Array (T(n,k)) begins:
%e A132339 1, -1, 0, 0, 0, 0, 0 ... A154955(k)
%e A132339 -1, 2, -2, 2, -2, 2, -2 ... (-1)^(k+1)*A040000(k)
%e A132339 0, -2, 10, -28, 60, -110, 182 ... (-1)^k*A006331(k)
%e A132339 0, 2, -28, 168, -660, 2002, -5096 ... (-1)^k*A006332(k)
%e A132339 0, -2, 60, -660, 4290, -20020, 74256 ... (-1)^k*A006333(k)
%e A132339 0, 2, -110, 2002, -20020, 136136, -705432 ... (-1)^k*A006334(k)
%e A132339 0, -2, 182, -5096, 74256, -705432, 4938024 ...
%e A132339 0, 2, -280, 11424, -232560, 2984520, -27457584 ...
%e A132339 Antidiagonal (A(n,k)) triangle begins as:
%e A132339 1;
%e A132339 -1, -1;
%e A132339 0, 2, 0;
%e A132339 0, -2, -2, 0;
%e A132339 0, 2, 10, 2, 0;
%e A132339 0, -2, -28, -28, -2, 0;
%e A132339 0, 2, 60, 168, 60, 2, 0;
%e A132339 0, -2, -110, -660, -660, -110, -2, 0;
%e A132339 0, 2, 182, 2002, 4290, 2002, 182, 2, 0;
%e A132339 0, -2, -280, -5096, -20020, -20020, -5096, -280, -2, 0;
%e A132339 0, 2, 408, 11424, 74256, 136136, 74256, 11424, 408, 2, 0;
%t A132339 Flatten[{{1}, {-1, -1}}~Join~Table[(2(-1)^(#+k)*(#+k-1)!*(2#+2k-3)!)/(#!*k!*(2# - 1)!*(2k-1)!) &@(n-k), {n,2,12}, {k,0,n}]] (* _Michael De Vlieger_, Mar 26 2016 *)
%o A132339 (Sage)
%o A132339 f=factorial
%o A132339 def T(n,k):
%o A132339 if (k==0): return bool(n==0) - bool(n==1)
%o A132339 elif (n==0): return bool(k==0) - bool(k==1)
%o A132339 else: return (-1)^(n+k)*f(n+k-2)*f(2*n+2*k-2)/(f(n)*f(k)*f(2*n-1)*f(2*k-1))
%o A132339 flatten([[T(n-k, k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Dec 14 2021
%Y A132339 Cf. A006331, A006332, A006333, A006334, A040000, A132341, A154955.
%K A132339 sign,tabl,easy
%O A132339 0,5
%A A132339 _N. J. A. Sloane_, Nov 08 2007
%E A132339 More terms from _Max Alekseyev_, Sep 12 2009