A278880 -id:A278880 - cp's OEIS frontend

A278881 Triangle where g.f. C = C(x,m) and related series S = S(x,m) and D = D(x,m) satisfy S = xCD, C = 1 + xSD, and D = 1 + mxS*C, as read by rows of coefficients T(n,k) of x^(2n)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

Paul D. Hanna, Nov 29 2016

			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 +...

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.

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.

{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(""))

/* 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

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.

A278882 Triangle where g.f. D = D(x,m) and related series S = S(x,m) and C = C(x,m) satisfy S = xCD, C = 1 + xSD, and D = 1 + mxS*C, as read by rows of coefficients T(n,k) of x^(2n)m^k in C(x,m) for n>=1, k=0..n.

Original entry on oeis.org

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, 0, 7, 336, 2310, 3600, 1386, 112, 1, 0, 8, 588, 6552, 18150, 14520, 3276, 168, 1, 0, 9, 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

Offset: 0

Paul D. Hanna, Nov 29 2016

			This triangle of coefficients of x^(2*n)*m^k in D(x,m) for n>=0, k=0..n, begins:
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;
0, 7, 336, 2310, 3600, 1386, 112, 1;
0, 8, 588, 6552, 18150, 14520, 3276, 168, 1;
0, 9, 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; ...
Generating function:
D(x,m) = 1 + m*x^2 + (2*m + m^2)*x^4 + (3*m + 8*m^2 + m^3)*x^6 +
(4*m + 30*m^2 + 20*m^3 + m^4)*x^8 +
(5*m + 80*m^2 + 147*m^3 + 40*m^4 + m^5)*x^10 +
(6*m + 175*m^2 + 672*m^3 + 504*m^4 + 70*m^5 + m^6)*x^12 +
(7*m + 336*m^2 + 2310*m^3 + 3600*m^4 + 1386*m^5 + 112*m^6 + m^7)*x^14 +
(8*m + 588*m^2 + 6552*m^3 + 18150*m^4 + 14520*m^5 + 3276*m^6 + 168*m^7 + m^8)*x^16 +...
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,
such that
C = C^2 - S^2,
D = D^2 - m*S^2.
The square of the g.f. begins:
D(x,m)^2 = 1 + 2*m*x^2 + (3*m^2 + 4*m)*x^4 +
(4*m^3 + 20*m^2 + 6*m)*x^6 +
(5*m^4 + 60*m^3 + 70*m^2 + 8*m)*x^8 +
(6*m^5 + 140*m^4 + 392*m^3 + 180*m^2 + 10*m)*x^10 +
(7*m^6 + 280*m^5 + 1512*m^4 + 1680*m^3 + 385*m^2 + 12*m)*x^12 +
(8*m^7 + 504*m^6 + 4620*m^5 + 9900*m^4 + 5544*m^3 + 728*m^2 + 14*m)*x^14 +
(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 +
(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 +...

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

Cf. A278880 (S(x,m)), A278881 (C(x,m)), A278884 (central terms).

Cf. A001764 (row sums), A000108, A258315, A243863.

{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)}
for(n=0,15, for(k=0,n, print1(T(n,k),", "));print(""))

/* Explicit formula for T(n, k) */
{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)!) ))}
for(n=0, 15, for(k=0, n, print1(T(n, k), ", ")); print("")) \\ Paul D. Hanna, Dec 11 2016

G.f. D = D(x,m), and related functions S = S(x,m) and C = C(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) = (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
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
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) = A258315(n-1) for n>=0.
Sum_{k=0..n} (-1)^k * T(n,k) = (-1)^n * A243863(n) for n>=0.

A278745 G.f. satisfies: A(x) = x(1 - x^2A(x)^2)/(1 + x^2*A(x)^2)^2.

Original entry on oeis.org

1, -3, 23, -232, 2671, -33247, 435732, -5923596, 82761455, -1181085841, 17143012047, -252288796800, 3755832135428, -56459641712052, 855828940166728, -13066760979482436, 200764834403473647, -3101861571115286485, 48161808069368073765, -751107354803633628504, 11760546724914570170423, -184805245095048170080367, 2913533082844307942651984, -46070266558711138024672784, 730480047034266200626268676

Offset: 1

Paul D. Hanna, Dec 01 2016

sign

			G.f.: A(x) = x - 3*x^5 + 23*x^9 - 232*x^13 + 2671*x^17 - 33247*x^21 + 435732*x^25 - 5923596*x^29 + 82761455*x^33 - 1181085841*x^37 + 17143012047*x^41 +...
such that A(x) = x*(1 - x^2*A(x)^2)/(1 + x^2*A(x)^2)^2.
RELATED SERIES.
A(x)^2 = x^2 - 6*x^6 + 55*x^10 - 602*x^14 + 7263*x^18 - 93192*x^22 + 1247636*x^26 - 17230290*x^30 + 243669007*x^34 - 3511010950*x^38 + 51361157967*x^42 +...
G.f. A(x) = x*C(x)*D(x) where
C(x) = (1 + x*A(x))/(1 + x^2*A(x)^2) = 1 + x^2 - x^4 - 4*x^6 + 7*x^8 + 33*x^10 - 68*x^12 - 344*x^14 + 767*x^16 + 4035*x^18 +...+ A243863(n)*x^(2*n) +...
D(x) = (1 - x*A(x))/(1 + x^2*A(x)^2) = 1 - x^2 - x^4 + 4*x^6 + 7*x^8 - 33*x^10 - 68*x^12 + 344*x^14 + 767*x^16 - 4035*x^18 +...+ (-1)^n*A243863(n)*x^(2*n) +...
such that C(x)^2 - A(x)^2 = C(x) and D(x)^2 + A(x)^2 = D(x).

Paul D. Hanna, Table of n, a(n) for n = 1..200
Jun Yan, Lattice paths enumerations weighted by ascent lengths, arXiv:2501.01152 [math.CO], 2025. See p. 8.

Cf. A278880, A243863.

{a(n) = my(A=x); for(i=0,4*n, A = x*(1 - x^2*A^2)/(1 + x^2*A^2 +x*O(x^(4*n)))^2 ); polcoeff(A,4*n-3)}
for(n=1,30,print1(a(n),", "))

/* Explicit formula from triangle A278880 */
{a(n) = sum(k=0,2*n-2, (-1)^k * (4*n-3)/((4*n-2*k-3)*(2*k+1)) * binomial(4*n-k-4, k) * binomial(2*n+k-2, 2*n-k-2) )}
for(n=1,30,print1(a(n),", "))

G.f. A(x) satisfies:
(1) A(x) = x/( G(A(x)^2) * G(-A(x)^2) ) where G(x) = 1 + x*G(x)^2.
(2) A(x) = x - x^3*A(x)^2 - 2*x^2*A(x)^3 - x^4*A(x)^5.
(3) A(x) = Series_Reversion( x*(1 + x^2)^2 / (1 - x^2) ).
(4) A(x) = x*C(x)*D(x) where
(4.a) C(x) = C(x)^2 - A(x)^2.
(4.b) D(x) = D(x)^2 + A(x)^2.
(4.c) C(x) = (1 + x*A(x))/(1 + x^2*A(x)^2).
(4.d) D(x) = (1 - x*A(x))/(1 + x^2*A(x)^2).
(4.e) C(x) = (1 + sqrt(1 + 4*A(x)^2))/2.
(4.f) D(x) = (1 + sqrt(1 - 4*A(x)^2))/2.
(4.g) C(x) = 1/G(-A(x)^2) where G(x) = 1 + x*G(x)^2.
(4.h) D(x) = 1/G(A(x)^2) where G(x) = 1 + x*G(x)^2 is the g.f. of Catalan numbers (A000108).
a(n) = Sum_{k=0..2*n-2} (-1)^k * A278880(2*n-1,k) for n>=1.
a(n) = Sum_{k=0..2*n-2} (-1)^k * (4*n-3)/((4*n-2*k-3)*(2*k+1)) * binomial(4*n-k-4, k) * binomial(2*n+k-2, 2*n-k-2). - Paul D. Hanna, Dec 08 2016
D-finite with recurrence -256*(n-1)*(4*n-5)*(2*n-1)*(142049551*n -178081473) *(4*n-3)*a(n) +16*(-1155885932064*n^5 +6748253449456*n^4 -14295401330216*n^3 +11571204221621*n^2 +77734459403*n -3289778607450)*a(n-1) +6*(3234453621264*n^5 -46690598461608*n^4 +268825512890063*n^3 -771308050258028*n^2 +1102485156931319*n -627947169605910)*a(n-2) -3*(n-3) *(2231943393*n -5530565638)*(3*n-10) *(2*n-7)*(3*n-11)*a(n-3)=0. - R. J. Mathar, Nov 22 2024

A258313 G.f. A(x) satisfies: A(x) = B(x)C(x) where B(x) = 1 + xA(x)C(x) and C(x) = 1 + 2xA(x)B(x).

Original entry on oeis.org

1, 3, 15, 93, 641, 4719, 36335, 289017, 2356321, 19586283, 165364799, 1414193205, 12224831937, 106645825047, 937685498271, 8301129707121, 73929906605249, 661919872559763, 5954449287679919, 53791836313371405, 487807821246726273, 4438980860105747967, 40521481906592540175

Offset: 0

Paul D. Hanna, May 25 2015

nonn

			G.f.: A(x) = 1 + 3*x + 15*x^2 + 93*x^3 + 641*x^4 + 4719*x^5 + 36335*x^6 +...
where A(x) = B(x)*C(x):
B(x) = 1 + x + 5*x^2 + 29*x^3 + 193*x^4 + 1389*x^5 + 10525*x^6 +...
C(x) = 1 + 2*x + 8*x^2 + 46*x^3 + 304*x^4 + 2178*x^5 + 16456*x^6 +...
Related series:
A(x)*B(x) = 1 + 4*x + 23*x^2 + 152*x^3 + 1089*x^4 + 8228*x^5 +...
A(x)*C(x) = 1 + 5*x + 29*x^2 + 193*x^3 + 1389*x^4 + 10525*x^5 +...

Cf. A258314 (B(x)), A258315 (C(x)), A278880.

Table[Sum[2^k*(2*n + 1)/((2*n - 2*k + 1)*(2*k + 1))*Binomial[2*n - k, k]*Binomial[n + k, n - k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 09 2016, after Paul D. Hanna *)

{a(n)=local(A=1+x,B=1+x,C=1+2*x);for(i=1,n, A = B*C +x*O(x^n); B = 1 + x*A*C + x*O(x^n); C = 1 + 2*x*A*B + x*O(x^n)); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

{a(n)=local(A=1); A = (1/x) * serreverse( x*(1-2*x^2)^2 / ((1+x)*(1+2*x) +x*O(x^n)) ); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

/* Explicit formula from triangle A278880 */
{a(n) = sum(k=0,n, 2^k * (2*n+1)/((2*n-2*k+1)*(2*k+1)) * binomial(2*n-k, k) * binomial(n+k, n-k) )}
for(n=0,30,print1(a(n),", ")) \\ Paul D. Hanna, Dec 08 2016

G.f. A(x) satisfies:
(1) A(x) = 1 + 3*x*A(x) + 2*x^2*A(x)^2*(1 + 2*A(x)) - 4*x^4*A(x)^5.
(2) A(x) = (1 + x*A(x))*(1 + 2*x*A(x)) / (1 - 2*x^2*A(x)^2)^2.
(3) A(x) = (1/x) * Series_Reversion( x*(1-2*x^2)^2 / ((1+x)*(1+2*x)) ).
Other relations involving A=A(x), B=B(x), and C=C(x) are:
(a) B = (1 + x*A) / (1 - 2*x^2*A^2).
(b) C = (1 + 2*x*A) / (1 - 2*x^2*A^2).
(c) B = 1/(1 - x*C^2).
(d) C = 1/(1 - 2*x*B^2).
a(n) = Sum_{k=0..n} 2^k * (2*n+1)/((2*n-2*k+1)*(2*k+1)) * binomial(2*n-k, k) * binomial(n+k, n-k). - Paul D. Hanna, Dec 08 2016
Recurrence: 16*n*(n+1)*(2*n-1)*(2*n+1)*(78144*n^5 - 638176*n^4 + 2009556*n^3 - 3030476*n^2 + 2162967*n - 571095)*a(n) = 6*n*(2*n - 1)*(3750912*n^7 - 30632448*n^6 + 95859584*n^5 - 141041184*n^4 + 91266236*n^3 - 10305348*n^2 - 11143087*n + 2769495)*a(n-1) + 18*(1875456*n^9 - 19067136*n^8 + 81388448*n^7 - 188788320*n^6 + 255050924*n^5 - 194874764*n^4 + 66686587*n^3 + 7734535*n^2 - 12646725*n + 2646000)*a(n-2) + 18*(n-2)*(1875456*n^8 - 18129408*n^7 + 70578528*n^6 - 140304800*n^5 + 146662564*n^4 - 69042202*n^3 - 184198*n^2 + 11212005*n - 2646000)*a(n-3) + 3*(n-3)*(n-2)*(3*n - 10)*(3*n - 5)*(78144*n^5 - 247456*n^4 + 238292*n^3 - 49424*n^2 - 31301*n + 10920)*a(n-4). - Vaclav Kotesovec, Dec 09 2016

A278883 a(n) = (4n+1) ( binomial(3n,n)/(2n+1) )^2.

Original entry on oeis.org

1, 5, 81, 1872, 51425, 1565109, 50979600, 1742711616, 61765676577, 2251396558125, 83924761860225, 3186277484832000, 122829049870699536, 4796448751641900752, 189381233826675892800, 7549371503605704934656, 303473219026059360959265, 12289574902507266828093525, 500960076377670398672062425, 20540854991655352005504930000, 846696245823312839372671355025, 35068049224094584278339245227125, 1458752047374053912228252043321600

Offset: 0

Paul D. Hanna, Nov 29 2016

nonn

Central terms of triangle A278880; a(n) = A278880(2*n+1, n) for n>=0.

Cf. A278880.

{a(n) = (4*n+1) * ( binomial(3*n,n)/(2*n+1) )^2}
for(n=0,20,print1(a(n),", "))

A364757 The pyramidal array T(r,g,b) = (r+g+b)/((g+b)(r+b))C(r+g,b-1)C(g+b,r)C(r+b,g), where 1 <= b <= ceiling((r+g+b)/2) and 0 <= r,g <= floor((r+g+b)/2). Read first over the layers corresponding to fixed sum r+g+b, then over the diagonals corresponding to fixed b.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 2, 2, 1, 8, 1, 5, 15, 15, 1, 5, 1, 3, 3, 8, 54, 8, 1, 27, 27, 1, 7, 70, 70, 42, 168, 42, 1, 14, 14, 1, 4, 4, 30, 192, 30, 20, 400, 400, 20, 1, 64, 200, 64, 1, 9, 210, 210, 405, 1500, 405, 90, 900, 900, 90, 1, 30, 81, 30, 1, 5, 5, 80, 500, 80, 147, 2625, 2625, 147, 40, 1750, 5000, 1750, 40, 1, 125, 875, 875, 125, 1

Offset: 1

Robert Muth, Aug 05 2023

T(r,g,b) is the number of injectively 3-colored trees with r red vertices, g green vertices, and b blue vertices, including a root vertex which is colored blue.
Summing T(r,g,b) over all r,g,b such that r+g+b=n yields the n-th Catalan number, A000108(n).
Column (or row) sums within each fixed r+g+b=n layer yield the number of ordered trees on n edges containing a fixed number of nodes adjacent to a leaf, A108759(n).
Main antidiagonal (corresponding to maximal value b = ceiling((r+g+b)/2)) within each fixed odd (r+g+b) layer is the number of "fighting fish" with fixed numbers of left lower free and right lower free edges with a marked tail A278880.

			The first few layers of the pyramidal array are:
-----------------------------------------------------------------------
      1           (r+g+b=1), (b=1)           T(0,0,1)
                                                         LAYER SUM:   1
-----------------------------------------------------------------------
     1 1          (r+g+b=2), (b=1)        T(0,1,1) T(1,0,1)
                                                         LAYER SUM:   2
-----------------------------------------------------------------------
      3           (r+g+b=3), (b=1)           T(1,1,1)
     1 1          (r+g+b=3), (b=2)        T(0,1,2) T(1,0,2)
                                                         LAYER SUM:   5
-----------------------------------------------------------------------
     2 2          (r+g+b=4), (b=1)        T(1,2,1) T(2,1,1)
    1 8 1         (r+g+b=4), (b=2)     T(0,2,2) T(1,1,2) T(2,0,2)
                                                         LAYER SUM:  14
-----------------------------------------------------------------------
      5           (r+g+b=5), (b=1)           T(2,2,1)
    15 15         (r+g+b=5), (b=2)        T(1,2,2) T(2,1,2)
   1  5  1        (r+g+b=5), (b=3)     T(0,2,3) T(1,1,3) T(2,0,3)
                                                         LAYER SUM:  42
-----------------------------------------------------------------------
     3  3         (r+g+b=6), (b=1)        T(2,3,1) T(3,2,1)
   8  54  8       (r+g+b=6), (b=2)     T(1,3,2) T(2,2,2) T(3,1,2)
 1  27  27  1     (r+g+b=6), (b=3)  T(0,3,3) T(1,2,3) T(2,1,3) T(3,0,3)
                                                         LAYER SUM: 132
-----------------------------------------------------------------------
      7           (r+g+b=7), (b=1)            T(3,3,1)
   70   70        (r+g+b=7), (b=2)        T(2,3,2) T(3,2,2)
 42  168  42      (r+g+b=7), (b=3)     T(1,3,3) T(2,2,3) T(3,1,3)
1   14  14   1    (r+g+b=7), (b=4)  T(0,3,4) T(1,2,4) T(2,1,4) T(3,0,4)
                                                         LAYER SUM: 429
-----------------------------------------------------------------------

T. Einolf, R. Muth and J. Wilkinson, Injectively k-colored rooted forests, arXiv:2107.13417 [math.CO], 2021, Remark 4.7.

Cf. A000108, A108759, A278880.

T(r,g,b) = ((r+g+b)/((g+b)*(r+b)))*C(r+g,b-1)*C(g+b,r)*C(r+b,g).

T(r,g,b) = ((r+g+b)/((g+b)*(r+b))) * ((r+g)!/((r+g-b+1)!*(b-1)!)) * ((g+b)!/((g+b-r)!*r!)) * ((r+b)!/((r+b-g)!*g!)).

cp's OEIS Frontend

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.

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A278882 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.

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A278745 G.f. satisfies: A(x) = x*(1 - x^2*A(x)^2)/(1 + x^2*A(x)^2)^2.

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A258313 G.f. A(x) satisfies: A(x) = B(x)*C(x) where B(x) = 1 + x*A(x)*C(x) and C(x) = 1 + 2*x*A(x)*B(x).

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A278883 a(n) = (4*n+1) * ( binomial(3*n,n)/(2*n+1) )^2.

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A364757 The pyramidal array T(r,g,b) = (r+g+b)/((g+b)*(r+b))*C(r+g,b-1)*C(g+b,r)*C(r+b,g), where 1 <= b <= ceiling((r+g+b)/2) and 0 <= r,g <= floor((r+g+b)/2). Read first over the layers corresponding to fixed sum r+g+b, then over the diagonals corresponding to fixed b.

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A278881 Triangle where g.f. C = C(x,m) and related series S = S(x,m) and D = D(x,m) satisfy S = xCD, C = 1 + xSD, and D = 1 + mxS*C, as read by rows of coefficients T(n,k) of x^(2n)m^k in C(x,m) for n>=0, k=0..n.

A278882 Triangle where g.f. D = D(x,m) and related series S = S(x,m) and C = C(x,m) satisfy S = xCD, C = 1 + xSD, and D = 1 + mxS*C, as read by rows of coefficients T(n,k) of x^(2n)m^k in C(x,m) for n>=1, k=0..n.

A278745 G.f. satisfies: A(x) = x(1 - x^2A(x)^2)/(1 + x^2*A(x)^2)^2.

A258313 G.f. A(x) satisfies: A(x) = B(x)C(x) where B(x) = 1 + xA(x)C(x) and C(x) = 1 + 2xA(x)B(x).

A278883 a(n) = (4n+1) ( binomial(3n,n)/(2n+1) )^2.

A364757 The pyramidal array T(r,g,b) = (r+g+b)/((g+b)(r+b))C(r+g,b-1)C(g+b,r)C(r+b,g), where 1 <= b <= ceiling((r+g+b)/2) and 0 <= r,g <= floor((r+g+b)/2). Read first over the layers corresponding to fixed sum r+g+b, then over the diagonals corresponding to fixed b.