A243863 -id:A243863 - 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

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.

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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 = 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.

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

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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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Formula

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.