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%I #35 Dec 11 2016 22:30:07
%S 1,0,1,0,2,1,0,3,8,1,0,4,30,20,1,0,5,80,147,40,1,0,6,175,672,504,70,1,
%T 0,7,336,2310,3600,1386,112,1,0,8,588,6552,18150,14520,3276,168,1,0,9,
%U 960,16170,72072,102245,48048,6930,240,1,0,10,1485,35904,240240,546546,455455,137280,13464,330,1,0,11,2200,73359,700128,2382380,3179904,1701700,350064,24453,440,1,0,12,3146,140140,1833975,8868288,17672928,15148224,5542680,815100,42042,572,1
%N Triangle where g.f. D = D(x,m) and related series S = S(x,m) and C = C(x,m) satisfy S = x*C*D, C = 1 + x*S*D, and D = 1 + m*x*S*C, as read by rows of coefficients T(n,k) of x^(2*n)*m^k in C(x,m) for n>=1, k=0..n.
%H Paul D. Hanna, <a href="/A278882/b278882.txt">Table of n, a(n) for n = 0..1080 for rows 0..45 of the flattened form of this triangle.</a>
%F G.f. D = D(x,m), and related functions S = S(x,m) and C = C(x,m) satisfy:
%F (1.a) S = x*C*D.
%F (1.b) C = 1 + x*S*D.
%F (1.c) D = 1 + m*x*S*C.
%F ...
%F (2.a) C = C^2 - S^2.
%F (2.b) D = D^2 - m*S^2.
%F (2.c) C = (1 + sqrt(1 + 4*S^2))/2.
%F (2.d) D = (1 + sqrt(1 + 4*m*S^2))/2.
%F ...
%F (3.a) S = x*(1 + x*S)*(1 + m*x*S) / (1 - m*x^2*S^2)^2.
%F (3.b) C = (1 + x*S) / (1 - m*x^2*S^2).
%F (3.c) D = (1 + m*x*S) / (1 - m*x^2*S^2).
%F (3.d) S = x/((1 - x^2*D^2)*(1 - m*x^2*C^2)).
%F (3.e) C = 1/(1 - x^2*D^2).
%F (3.f) D = 1/(1 - m*x^2*C^2).
%F ...
%F (4.a) x = m^2*x^4*S^5 - 2*m*x^2*S^3 - m*x^3*S^2 + (1 - (m+1)*x^2)*S.
%F (4.b) 0 = 1 - (1-x^2)*C - 2*m*x^2*C^2 + 2*m*x^2*C^3 + m^2*x^4*C^4 - m^2*x^4*C^5.
%F (4.c) 0 = 1 - (1-m*x^2)*D - 2*x^2*D^2 + 2*x^2*D^3 + x^4*D^4 - x^4*D^5.
%F ...
%F (5.a) S(x,m) = Series_Reversion( x*G(-x^2)*G(-m*x^2) ), where G(x) = 1 + x*G(x)^2 is the g.f. of the Catalan numbers (A000108).
%F Logarithmic derivatives.
%F (6.a) C'/C = 2*S*S' / (C^2 + S^2).
%F (6.b) D'/D = 2*m*S*S' / (D^2 + m*S^2).
%F ...
%F T(n,k) = (n-k+1) * A082680(n+1,n-k+1) for n>=0 with T(0,0) = 1 and T(n,0) = 0 for n>0. - _Paul D. Hanna_, Dec 11 2016
%F T(n,k) = (2*n-k)!*(n+k-1)!/(k!*(n-k)!*(2*k-1)!*(2*n-2*k+1)!) for n>k>0 with T(n,0) = 1 and T(n,n) = 0 for n>0. - _Paul D. Hanna_, Dec 11 2016
%F Row sums yield A001764(n) = binomial(3*n,n)/(2*n+1).
%F Central terms: T(2*n,n) = binomial(3*n-1,n) * binomial(3*n,n)/(2*n+1).
%F Sum_{k=0..n} 2^k * T(n,k) = A258315(n-1) for n>=0.
%F Sum_{k=0..n} (-1)^k * T(n,k) = (-1)^n * A243863(n) for n>=0.
%e This triangle of coefficients of x^(2*n)*m^k in D(x,m) for n>=0, k=0..n, begins:
%e 1;
%e 0, 1;
%e 0, 2, 1;
%e 0, 3, 8, 1;
%e 0, 4, 30, 20, 1;
%e 0, 5, 80, 147, 40, 1;
%e 0, 6, 175, 672, 504, 70, 1;
%e 0, 7, 336, 2310, 3600, 1386, 112, 1;
%e 0, 8, 588, 6552, 18150, 14520, 3276, 168, 1;
%e 0, 9, 960, 16170, 72072, 102245, 48048, 6930, 240, 1;
%e 0, 10, 1485, 35904, 240240, 546546, 455455, 137280, 13464, 330, 1;
%e 0, 11, 2200, 73359, 700128, 2382380, 3179904, 1701700, 350064, 24453, 440, 1;
%e 0, 12, 3146, 140140, 1833975, 8868288, 17672928, 15148224, 5542680, 815100, 42042, 572, 1; ...
%e Generating function:
%e D(x,m) = 1 + m*x^2 + (2*m + m^2)*x^4 + (3*m + 8*m^2 + m^3)*x^6 +
%e (4*m + 30*m^2 + 20*m^3 + m^4)*x^8 +
%e (5*m + 80*m^2 + 147*m^3 + 40*m^4 + m^5)*x^10 +
%e (6*m + 175*m^2 + 672*m^3 + 504*m^4 + 70*m^5 + m^6)*x^12 +
%e (7*m + 336*m^2 + 2310*m^3 + 3600*m^4 + 1386*m^5 + 112*m^6 + m^7)*x^14 +
%e (8*m + 588*m^2 + 6552*m^3 + 18150*m^4 + 14520*m^5 + 3276*m^6 + 168*m^7 + m^8)*x^16 +...
%e where g.f. D = D(x,m) and related series S = S(x,m) and C = C(x,m) satisfy
%e S = x*C*D, C = 1 + x*S*D, and D = 1 + m*x*S*C,
%e such that
%e C = C^2 - S^2,
%e D = D^2 - m*S^2.
%e The square of the g.f. begins:
%e D(x,m)^2 = 1 + 2*m*x^2 + (3*m^2 + 4*m)*x^4 +
%e (4*m^3 + 20*m^2 + 6*m)*x^6 +
%e (5*m^4 + 60*m^3 + 70*m^2 + 8*m)*x^8 +
%e (6*m^5 + 140*m^4 + 392*m^3 + 180*m^2 + 10*m)*x^10 +
%e (7*m^6 + 280*m^5 + 1512*m^4 + 1680*m^3 + 385*m^2 + 12*m)*x^12 +
%e (8*m^7 + 504*m^6 + 4620*m^5 + 9900*m^4 + 5544*m^3 + 728*m^2 + 14*m)*x^14 +
%e (9*m^8 + 840*m^7 + 12012*m^6 + 43560*m^5 + 47190*m^4 + 15288*m^3 + 1260*m^2 + 16*m)*x^16 +
%e (10*m^9 + 1320*m^8 + 27720*m^7 + 156156*m^6 + 286286*m^5 + 180180*m^4 + 36960*m^3 + 2040*m^2 + 18*m)*x^18 +...
%o (PARI) {T(n,k) = my(S=x,C=1,D=1); for(i=0,2*n, S = x*C*D + O(x^(2*n+2)); C = 1 + x*S*D; D = 1 + m*x*S*C;); polcoeff(polcoeff(D,2*n,x),k,m)}
%o for(n=0,15, for(k=0,n, print1(T(n,k),", "));print(""))
%o (PARI) /* Explicit formula for T(n, k) */
%o {T(n, k) = if(n==k, 1, if(k==0, 0, (2*n-k)!*(n+k-1)!/(k!*(n-k)!*(2*k-1)!*(2*n-2*k+1)!) ))}
%o for(n=0, 15, for(k=0, n, print1(T(n, k), ", ")); print("")) \\ _Paul D. Hanna_, Dec 11 2016
%Y Cf. A278880 (S(x,m)), A278881 (C(x,m)), A278884 (central terms).
%Y Cf. A001764 (row sums), A000108, A258315, A243863.
%K nonn,tabl
%O 0,5
%A _Paul D. Hanna_, Nov 29 2016
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