A278881 Triangle where g.f. C = C(x,m) and related series S = S(x,m) and D = D(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>=0, k=0..n.

1, 1, 0, 1, 2, 0, 1, 8, 3, 0, 1, 20, 30, 4, 0, 1, 40, 147, 80, 5, 0, 1, 70, 504, 672, 175, 6, 0, 1, 112, 1386, 3600, 2310, 336, 7, 0, 1, 168, 3276, 14520, 18150, 6552, 588, 8, 0, 1, 240, 6930, 48048, 102245, 72072, 16170, 960, 9, 0, 1, 330, 13464, 137280, 455455, 546546, 240240, 35904, 1485, 10, 0, 1, 440, 24453, 350064, 1701700, 3179904, 2382380, 700128, 73359, 2200, 11, 0, 1, 572, 42042, 815100, 5542680, 15148224, 17672928, 8868288, 1833975, 140140, 3146, 12, 0

Offset

0,5

Links

Paul D. Hanna, Table of n, a(n) for n = 0..1080 for rows 0..45 of the flattened form of this triangle.

Thomas Einolf, Robert Muth, and Jeffrey Wilkinson, Injectively k-colored rooted forests, arXiv:2107.13417 [math.CO], 2021.

Tad White, Quota Trees, arXiv:2401.01462 [math.CO], 2024. See p. 20.

Formula

G.f. C = C(x,m), and related functions S = S(x,m) and D = D(x,m) satisfy:

(1.a) S = x*C*D.

(1.b) C = 1 + x*S*D.

(1.c) D = 1 + m*x*S*C.

...

(2.a) C = C^2 - S^2.

(2.b) D = D^2 - m*S^2.

(2.c) C = (1 + sqrt(1 + 4*S^2))/2.

(2.d) D = (1 + sqrt(1 + 4*m*S^2))/2.

...

(3.a) S = x*(1 + x*S)*(1 + m*x*S) / (1 - m*x^2*S^2)^2.

(3.b) C = (1 + x*S) / (1 - m*x^2*S^2).

(3.c) D = (1 + m*x*S) / (1 - m*x^2*S^2).

(3.d) S = x/((1 - x^2*D^2)*(1 - m*x^2*C^2)).

(3.e) C = 1/(1 - x^2*D^2).

(3.f) D = 1/(1 - m*x^2*C^2).

...

(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.

(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.

(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.

...

(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).

Logarithmic derivatives.

(6.a) C'/C = 2*S*S' / (C^2 + S^2).

(6.b) D'/D = 2*m*S*S' / (D^2 + m*S^2).

...

T(n,k) = (k+1) * A082680(n+1,k+1) for n>=0 with T(0,0) = 1 and T(n,n) = 1 for n>0. - Paul D. Hanna, Dec 11 2016

T(n,k) = (n+k)!*(2*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

Row sums yield A001764(n) = binomial(3*n,n)/(2*n+1).

Central terms: T(2*n,n) = binomial(3*n-1,n) * binomial(3*n,n)/(2*n+1).

Sum_{k=0..n} 2^k * T(n,k) = A258314(n-1) for n>=0.

Sum_{k=0..n} (-1)^k * T(n,k) = A243863(n) for n>=0.

Example

This triangle of coefficients of x^(2*n)*m^k in C(x,m) for n>=0, k=0..n, begins:
1;
1, 0;
1, 2, 0;
1, 8, 3, 0;
1, 20, 30, 4, 0;
1, 40, 147, 80, 5, 0;
1, 70, 504, 672, 175, 6, 0;
1, 112, 1386, 3600, 2310, 336, 7, 0;
1, 168, 3276, 14520, 18150, 6552, 588, 8, 0;
1, 240, 6930, 48048, 102245, 72072, 16170, 960, 9, 0;
1, 330, 13464, 137280, 455455, 546546, 240240, 35904, 1485, 10, 0;
1, 440, 24453, 350064, 1701700, 3179904, 2382380, 700128, 73359, 2200, 11, 0;
1, 572, 42042, 815100, 5542680, 15148224, 17672928, 8868288, 1833975, 140140, 3146, 12, 0; ...
Generating function:
C(x,m) = 1 + x^2 + (1 + 2*m)*x^4 + (1 + 8*m + 3*m^2)*x^6 +
(1 + 20*m + 30*m^2 + 4*m^3)*x^8 +
(1 + 40*m + 147*m^2 + 80*m^3 + 5*m^4)*x^10 +
(1 + 70*m + 504*m^2 + 672*m^3 + 175*m^4 + 6*m^5)*x^12 +
(1 + 112*m + 1386*m^2 + 3600*m^3 + 2310*m^4 + 336*m^5 + 7*m^6)*x^14 +
(1 + 168*m + 3276*m^2 + 14520*m^3 + 18150*m^4 + 6552*m^5 + 588*m^6 + 8*m^7)*x^16 +...
where g.f. C = C(x,m) and related series S = S(x,m) and D = D(x,m) satisfy
S = x*C*D, C = 1 + x*S*D, and D = 1 + m*x*S*C,
such that
C = C^2 - S^2,
D = D^2 - m*S^2.
The square of the g.f. begins:
C(x,m)^2 = 1 + 2*x^2 + (4*m + 3)*x^4 + (6*m^2 + 20*m + 4)*x^6 +
(8*m^3 + 70*m^2 + 60*m + 5)*x^8 +
(10*m^4 + 180*m^3 + 392*m^2 + 140*m + 6)*x^10 +
(12*m^5 + 385*m^4 + 1680*m^3 + 1512*m^2 + 280*m + 7)*x^12 +
(14*m^6 + 728*m^5 + 5544*m^4 + 9900*m^3 + 4620*m^2 + 504*m + 8)*x^14 +
(16*m^7 + 1260*m^6 + 15288*m^5 + 47190*m^4 + 43560*m^3 + 12012*m^2 + 840*m + 9)*x^16 +
(18*m^8 + 2040*m^7 + 36960*m^6 + 180180*m^5 + 286286*m^4 + 156156*m^3 + 27720*m^2 + 1320*m + 10)*x^18 +...

Prog

(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(C, 2*n, x), k, m)}
for(n=0, 15, for(k=0, n, print1(T(n, k), ", ")); print(""))
(PARI) /* Explicit formula for T(n, k) */
{T(n, k) = if(k==0, 1, if(n==k, 0, (n+k)!*(2*n-k-1)!/(k!*(n-k)!*(2*k+1)!*(2*n-2*k-1)!) ))}
for(n=0, 15, for(k=0, n, print1(T(n, k), ", ")); print("")) \\ Paul D. Hanna, Dec 11 2016

Crossrefs

Cf. A278880 (S(x,m)), A278882 (D(x,m)), A278884 (central terms).

Cf. A001764 (row sums), A000108, A258314 (C(x,m) at m=2), A243863.

Sequence in context: A356265 A309993 A248673 * A335095 A344069 A337444

Adjacent sequences: A278878 A278879 A278880 * A278882 A278883 A278884

Keyword

nonn,tabl

Author

Paul D. Hanna, Nov 29 2016

Status

approved