A258314 -id:A258314 - 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.

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

A258315 G.f. C(x) satisfies: C(x) = 1 + 2xA(x)B(x) where A(x) = B(x)C(x) and B(x) = 1 + xA(x)C(x).

Original entry on oeis.org

1, 2, 8, 46, 304, 2178, 16456, 129086, 1041248, 8582274, 71964232, 611954286, 5264786448, 45741886786, 400776143752, 3537136653566, 31417018218688, 280616550025218, 2518975669228936, 22712641808517166, 205612543320237808, 1868112977079278594, 17028815533533595080

Offset: 0

Paul D. Hanna, May 25 2015

nonn

			G.f.: C(x) = 1 + 2*x + 8*x^2 + 46*x^3 + 304*x^4 + 2178*x^5 + 16456*x^6 +...
where C(x) = 1 + 2*x*A(x)*B(x):
A(x) = 1 + 3*x + 15*x^2 + 93*x^3 + 641*x^4 + 4719*x^5 + 36335*x^6 +...
B(x) = 1 + x + 5*x^2 + 29*x^3 + 193*x^4 + 1389*x^5 + 10525*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. A258313 (A(x)), A258314 (B(x)).

{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(C,n)}
for(n=0,30,print1(a(n),", "))

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

G.f. C(x) satisfies:
(1) C(x) = 1 + 2*x*C(x)*(1 - C(x) + C(x)^2) + x^2*C(x)^4*(1 - C(x)).
(2) C(x) = (1/x)*Series_Reversion( x^2/(x + 2*Series_Reversion( x*(1-2*x^2)/(1+x) )^2) ).
(3) x = (sqrt(1 - 2*C(x) + 2*C(x)^2) - (1 - C(x) + C(x)^2)) / (C(x)^3*(1 - C(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).

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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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).

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A258315 G.f. C(x) satisfies: C(x) = 1 + 2xA(x)B(x) where A(x) = B(x)C(x) and B(x) = 1 + xA(x)C(x).

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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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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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A258315 G.f. C(x) satisfies: C(x) = 1 + 2*x*A(x)*B(x) where A(x) = B(x)*C(x) and B(x) = 1 + x*A(x)*C(x).

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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.

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).

A258315 G.f. C(x) satisfies: C(x) = 1 + 2xA(x)B(x) where A(x) = B(x)C(x) and B(x) = 1 + xA(x)C(x).