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 A268647 #22 Jan 26 2019 11:37:37
%S A268647 0,1,2,5,4,1,48,124,120,55,12,1,2160,6012,6636,3829,1260,238,24,1,
%T A268647 161280,478656,582080,387260,157080,40593,6720,690,40,1,18144000,
%U A268647 56772000,74396520,54801076,25494150,7927205,1690920,248523,24750,1595,60,1,2874009600,9397658880,13075800192,10415648880,5357255904,1893627736,476011536,86550035,11423412,1084083,72072,3185,84,1,610248038400,2071437822720,3028563232128,2569081620624,1429040500160,556365173000,157528627256,33179499353,5260335080,629597540,56560504,3753022,178360,5740,112,1
%N A268647 G.f.: C(x,y)^2 - S(x,y)^2 = Sum_{n>=0} x^(2*n)*y/[Sum_{k=0..2*n+1} T(n,k)*y^k], where C(x,y) = Sum_{n>=0} x^(2*n) / Product_{k=1..2*n} (k + y) and S(x,y) = Sum_{n>=0} x^(2*n+1) / Product_{k=1..2*n+1} (k + y).
%C A268647 This triangle illustrates the following identity.
%C A268647 Given
%C A268647 C(x,y) = Sum_{n>=0} x^(2*n) / Product_{k=1..2*n} (k + y)
%C A268647 S(x,y) = Sum_{n>=0} x^(2*n+1) / Product_{k=1..2*n+1} (k + y)
%C A268647 then
%C A268647 C(x,y)^2 - S(x,y)^2 = Sum_{n>=0} x^(2*n) * y / ((n + y) * Product_{k=1..2*n} (k + y)).
%H A268647 Paul D. Hanna, <a href="/A268647/b268647.txt">Table of n, a(n) for n = 0..991 terms in rows 0..30 of this triangle in flattened form.</a>
%F A268647 G.f. of row n: (n + y) * Product_{k=1..2*n} (k + y) = Sum_{k=0..2*n+1} T(n,k)*y^k, for n>=0.
%F A268647 Row sums equal A002674 (with offset): A002674(n+1) = (n+1)*(2*n+1)!.
%e A268647 Define C(x,y) by the series:
%e A268647 C(x,y) = 1 + x^2/((1+y)*(2+y)) + x^4/((1+y)*(2+y)*(3+y)*(4+y)) + x^6/((1+y)*(2+y)*(3+y)*(4+y)*(5+y)*(6+y)) + x^8/((1+y)*(2+y)*(3+y)*(4+y)*(5+y)*(6+y)*(7+y)*(8+y)) +...
%e A268647 and define S(x,y) by the series:
%e A268647 S(x,y) = x/(1+y) + x^3/((1+y)*(2+y)*(3+y)) + x^5/((1+y)*(2+y)*(3+y)*(4+y)*(5+y)) + x^7/((1+y)*(2+y)*(3+y)*(4+y)*(5+y)*(6+y)*(7+y)) + x^9/((1+y)*(2+y)*(3+y)*(4+y)*(5+y)*(6+y)*(7+y)*(8+y)*(9+y)) +...
%e A268647 then the g.f. of this triangle begins:
%e A268647 C(x,y)^2 - S(x,y)^2 = 1 + x^2*y/((1+y) * (1+y)*(2+y)) + x^4*y/((2+y) * (1+y)*(2+y)*(3+y)*(4+y)) + x^6*y/((3+y) * (1+y)*(2+y)*(3+y)*(4+y)*(5+y)*(6+y)) + x^8*y/((4+y) * (1+y)*(2+y)*(3+y)*(4+y)*(5+y)*(6+y)*(7+y)*(8+y)) +...
%e A268647 where the rows of this triangle are formed from the coefficients in the denominators of coefficients of x^(2*n) in C(x,y)^2 - S(x,y)^2, as more clearly seen in the expansion:
%e A268647 C(x,y)^2 - S(x,y)^2 = y/(0 + y) + x^2 * y/(2 + 5*y + 4*y^2 + y^3) +
%e A268647 x^4 * y/(48 + 124*y + 120*y^2 + 55*y^3 + 12*y^4 + y^5) +
%e A268647 x^6 * y/(2160 + 6012*y + 6636*y^2 + 3829*y^3 + 1260*y^4 + 238*y^5 + 24*y^6 + y^7) +
%e A268647 x^8 * y/(161280 + 478656*y + 582080*y^2 + 387260*y^3 + 157080*y^4 + 40593*y^5 + 6720*y^6 + 690*y^7 + 40*y^8 + y^9) +...
%e A268647 This triangle begins:
%e A268647 0, 1;
%e A268647 2, 5, 4, 1;
%e A268647 48, 124, 120, 55, 12, 1;
%e A268647 2160, 6012, 6636, 3829, 1260, 238, 24, 1;
%e A268647 161280, 478656, 582080, 387260, 157080, 40593, 6720, 690, 40, 1;
%e A268647 18144000, 56772000, 74396520, 54801076, 25494150, 7927205, 1690920, 248523, 24750, 1595, 60, 1;
%e A268647 2874009600, 9397658880, 13075800192, 10415648880, 5357255904, 1893627736, 476011536, 86550035, 11423412, 1084083, 72072, 3185, 84, 1;
%e A268647 610248038400, 2071437822720, 3028563232128, 2569081620624, 1429040500160, 556365173000, 157528627256, 33179499353, 5260335080, 629597540, 56560504, 3753022, 178360, 5740, 112, 1; ...
%o A268647 (PARI) /* C(x,y)^2 - S(x,y)^2 = Sum_{n>=0} x^(2*n)*y/[Sum_{k=0..2*n+1} T(n,k)*y^k] */
%o A268647 {T(n,k) = my(C=1,S=x); C = sum(m=0,n+1, x^(2*m)/prod(k=1,2*m, k + y) +x*O(x^(2*n)));
%o A268647 S = sum(m=1,n+1, x^(2*m-1)/prod(k=1,2*m-1, k + y) +x*O(x^(2*n)));
%o A268647 polcoeff( y/polcoeff( C^2 - S^2, 2*n, x), k, y)}
%o A268647 for(n=0,10, for(k=0,2*n+1, print1(T(n,k),", "));print(""))
%o A268647 (PARI) /* (n + y)*Product_{k=1..2*n} (k + y) = Sum_{k=0..2*n+1} T(n,k)*y^k */
%o A268647 {T(n,k) = polcoeff((n + y)*prod(k=1,2*n, k + y), k, y)}
%o A268647 for(n=0,10, for(k=0,2*n+1, print1(T(n,k),", "));print(""))
%Y A268647 Cf. A322627 (diagonal).
%K A268647 nonn,tabf
%O A268647 0,3
%A A268647 _Paul D. Hanna_, Mar 01 2016