A120010 -id:A120010 - cp's OEIS frontend

A120020 Coefficients of x^n in the n-th iteration of the g.f. of A120010: a(n) = [x^n] { (1-sqrt(1-4x))/2 o x/(1-nx) o (x-x^2) } for n>=1.

1, 2, 9, 68, 710, 9348, 148085, 2740672, 58033953, 1383923040, 36705564368, 1071911496576, 34179790156473, 1181725089179936, 44035415728886145, 1759481180119564288, 75042973200676887772, 3402984761691650083008

Offset: 1

Paul D. Hanna, Jun 14 2006

nonn

Main diagonal of A120019, the table of self-compositions of A120010.

For n>=1, n divides a(n): a(n)/n = A120022(n).

			Successive iterations of F(x), the g.f. of A120010, begin:
F(x) = (1)x + x^2 + x^3 + 2x^4 + 6x^5 + 18x^6 + 53x^7 + 158x^8 +...
F(F(x)) = x + (2)x^2 + 4x^3 + 10x^4 + 32x^5 + 116x^6 + 440x^7 +...
F(F(F(x))) = x + 3x^2 + (9)x^3 + 30x^4 + 114x^5 + 480x^6 + 2157x^7 +...
F(F(F(F(x)))) = x + 4x^2 + 16x^3 + (68)x^4 + 312x^5 + 1536x^6 +...
F(F(F(F(F(x))))) = x + 5x^2 + 25x^3 + 130x^4 + (710)x^5 + 4070x^6 +...
F(F(F(F(F(F(x)))))) = x + 6x^2 + 36x^3 + 222x^4 + 1416x^5 + (9348)x^6+..

Cf. A120010, A120019, A120021, A120022.

{a(n)=sum(j=1, n, binomial(2*n-2*j, n-j)/(n-j+1)* sum(i=1, j,(-1)^(j-i)*binomial(n-j+i, j-i)*binomial(n-j+i-1, i-1)*n^(i-1)))}
for(n=1,25,print1(a(n),", "))

a(n) = Sum_{j=1..n} Catalan(n-j) * [ Sum_{i=1..j} (-1)^(j-i) * n^(i-1) * C(n-j+i, j-i) * C(n-j+i-1, i-1) ];

a(n) = Sum_{j=0..n-1} n^j * [ Sum_{i=j..n-1} (-1)^(i-j) * Catalan(n-i-1) * C(n-i+j, i-j) * C(n-i+j-1, j) ], where Catalan(n) = A000108(n) = C(2n, n)/(n+1).

A120019 Square table, read by antidiagonals, of self-compositions of A120010.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 2, 1, 4, 9, 10, 6, 1, 5, 16, 30, 32, 18, 1, 6, 25, 68, 114, 116, 53, 1, 7, 36, 130, 312, 480, 440, 158, 1, 8, 49, 222, 710, 1536, 2157, 1708, 481, 1, 9, 64, 350, 1416, 4070, 8000, 10092, 6760, 1491, 1, 10, 81, 520, 2562, 9348, 24365, 43472, 48525

Offset: 1

Paul D. Hanna, Jun 14 2006

The g.f. of row n is the composition: (1-sqrt(1-4*x))/2 o x/(1-nx) o (x-x^2).

			Square table begins:
1, 1, 1, 2, 6, 18, 53, 158, 481, 1491, ...
1, 2, 4, 10, 32, 116, 440, 1708, 6760, 27232, ...
1, 3, 9, 30, 114, 480, 2157, 10092, 48525, 238143, ...
1, 4, 16, 68, 312, 1536, 8000, 43472, 243808, 1400448, ...
1, 5, 25, 130, 710, 4070, 24365, 151330, 968785, 6355795, ...
1, 6, 36, 222, 1416, 9348, 63768, 448188, 3234216, 23875296, ...
1, 7, 49, 350, 2562, 19236, 148085, 1167488, 9409645, 77367087, ...
1, 8, 64, 520, 4304, 36320, 312512, 2740672, 24476800, 222358528, ...
1, 9, 81, 738, 6822, 64026, 610245, 5906502, 58033953, 578488563, ...
1, 10, 100, 1010, 10320, 106740, 1117880, 11855660, 127313320, ...
Successive self-compositions of F(x), the g.f. of A120010, begin:
F(x) = x + x^2 + x^3 + 2x^4 + 6x^5 + 18x^6 + 53x^7 + 158x^8 +...
F(F(x)) = x + 2x^2 + 4x^3 + 10x^4 + 32x^5 + 116x^6 + 440x^7 +...
F(F(F(x))) = x + 3x^2 + 9x^3 + 30x^4 + 114x^5 + 480x^6 + 2157x^7 +...
F(F(F(F(x)))) = x + 4x^2 + 16x^3 + 68x^4 + 312x^5 + 1536x^6 +...
F(F(F(F(F(x))))) = x + 5x^2 + 25x^3 + 130x^4 + 710x^5 + 4070x^6 +...
F(F(F(F(F(F(x)))))) = x + 6x^2 + 36x^3 + 222x^4 + 1416x^5 + 9348x^6 +...

Rows: A120010, A120017, A120018; Diagonals: A120020, A120021. Variant: A120013.

{T(n,k)=sum(j=1, k, binomial(2*k-2*j, k-j)/(k-j+1)* sum(i=1, j,(-1)^(j-i)*binomial(k-j+i, j-i)*binomial(k-j+i-1, i-1)*n^(i-1)))}

T(n, k) = Sum_{j=1..k}Catalan(k-j)*[Sum_{i=1..j}(-1)^(j-i)*n^(i-1)*C(k-j+i, j-i)*C(k-j+i-1, i-1)]; Also, T(n, k) = Sum_{j=0..k-1}n^j*[Sum_{i=j..k-1}(-1)^(i-j)*Catalan(k-i-1)*C(k-i+j, i-j)*C(k-i+j-1, j)]; where Catalan(n) = A000108(n) = C(2n, n)/(n+1).

A120018 The third self-composition of A120010; g.f.: A(x) = G(G(G(x))), where G(x) = g.f. of A120010.

Original entry on oeis.org

1, 3, 9, 30, 114, 480, 2157, 10092, 48525, 238143, 1187952, 6006171, 30710553, 158535975, 825143145, 4325320191, 22814398392, 120999555588, 644878190175, 3451975941243, 18550877091063, 100047282676491, 541314936448764

Offset: 1

Paul D. Hanna, Jun 14 2006

nonn

Row 3 of A120019, the square table of self-compositions of A120010.

			A(x) = x + 3*x^2 + 9*x^3 + 30*x^4 + 114*x^5 + 480*x^6 + 2157*x^7 +...
G(x) = x + x^2 + x^3 + 2*x^4 + 6*x^5 + 18*x^6 + 53*x^7 + 158*x^8 +...
where G(x) is the g.f. of A120010 and G(G(G(x))) = A(x).

Vincenzo Librandi, Table of n, a(n) for n = 1..300

Cf. A120010, A120017 (2nd self-composition), A120019.

CoefficientList[Series[(1 - Sqrt[1 - 4 x (1-x) / (1 -3 x + 3 x^2)]) / x / 2,  {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 24 2012 *)

{a(n)=polcoeff((1 - sqrt(1 - 4*x*(1-x)/(1-3*x+3*x^2+x*O(x^n)) ))/2, n)}

G.f.: A(x) = (1 - sqrt(1 - 4*x*(1-x)/(1-3*x+3*x^2) ))/2.
Recurrence: n*a(n) = 2*(5*n-6)*a(n-1) - (31*n-66)*a(n-2) + 42*(n-3)*a(n-3) - 21*(n-4)*a(n-4). - Vaclav Kotesovec, Oct 24 2012
a(n) ~ sqrt(14*sqrt(21)-42)*((7+sqrt(21))/2)^n/(16*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 24 2012

Typo in Mma program fixed by Vincenzo Librandi, May 22 2013

A120021 Coefficients of x^n in the (n+1)-th self-composition of the g.f. of A120010: a(n) = [x^n] { (1-sqrt(1-4x))/2 o x/(1-(n+1)x) o (x-x^2) } for n>=1.

Original entry on oeis.org

1, 3, 16, 130, 1416, 19236, 312512, 5906502, 127313320, 3082645951, 82848394752, 2447576485341, 78846484722208, 2750891289611235, 103344880800464896, 4159577854374314795, 178587276548655542112, 8147334149686335230068

Offset: 1

Paul D. Hanna, Jun 14 2006

nonn

Secondary diagonal of A120019, the table of self-compositions of A120010.

			Successive self-compositions of F(x), the g.f. of A120010, begin:
F(x) = x + x^2 + x^3 + 2x^4 + 6x^5 + 18x^6 + 53x^7 + 158x^8 +...
F(F(x)) = (1)x + 2x^2 + 4x^3 + 10x^4 + 32x^5 + 116x^6 + 440x^7 +...
F(F(F(x))) = x + (3)x^2 + 9x^3 + 30x^4 + 114x^5 + 480x^6 + 2157x^7 +...
F(F(F(F(x)))) = x + 4x^2 + (16)x^3 + 68x^4 + 312x^5 + 1536x^6 +...
F(F(F(F(F(x))))) = x + 5x^2 + 25x^3 + (130)x^4 + 710x^5 + 4070x^6 +...
F(F(F(F(F(F(x)))))) = x + 6x^2 + 36x^3 + 222x^4 + (1416)x^5 + 9348x^6+..

Cf. A120010, A120019, A120020.

a(n)=polcoeff((1-sqrt(1-4*x*(1-x)/(1-(n+1)*x*(1-x)+x*O(x^n))))/2, n, x)

/* Alternative Formula: */ a(n)=sum(j=1, n, binomial(2*n-2*j, n-j)/(n-j+1)*sum(i=1, j,(-1)^(j-i)*(n+1)^(i-1)*binomial(n-j+i, j-i)*binomial(n-j+i-1, i-1)))

a(n) = Sum_{j=1..n} Catalan(n-j)*[Sum_{i=1..j} (-1)^(j-i)*(n+1)^(i-1)*C(n-j+i, j-i)*C(n-j+i-1, i-1)], where Catalan(n) = A000108(n) = C(2n, n)/(n+1).

A120017 The 2nd self-composition of A120010; g.f.: A(x) = G(G(x)), where G(x) = g.f. of A120010.

Original entry on oeis.org

1, 2, 4, 10, 32, 116, 440, 1708, 6760, 27232, 111392, 461536, 1933024, 8170400, 34807232, 149304080, 644298592, 2795216576, 12184415360, 53338632256, 234393350912, 1033614750080, 4572427361536, 20285780245120, 90238113332992

Offset: 1

Paul D. Hanna, Jun 14 2006

nonn

			A(x) = x + 2*x^2 + 4*x^3 + 10*x^4 + 32*x^5 + 116*x^6 + 440*x^7 +...
G(x) = x + x^2 + x^3 + 2*x^4 + 6*x^5 + 18*x^6 + 53*x^7 + 158*x^8 +...
where G(x) is the g.f. of A120010 and G(G(x)) = A(x).

Cf. A120010, A120018 (3rd self-composition).

{a(n)=polcoeff((1 - sqrt(1 - 4*x*(1-x)/(1-2*x+2*x^2+x*O(x^n)) ))/2, n)}

G.f.: A(x) = (1 - sqrt(1 - 4*x*(1-x)/(1-2*x+2*x^2) ))/2.

A120022 a(n) = A120020(n)/n = coefficient of x^n in the n-th self-composition of the g.f. of A120010, divided by n, for n>=1.

Original entry on oeis.org

1, 1, 3, 17, 142, 1558, 21155, 342584, 6448217, 138392304, 3336869488, 89325958048, 2629214627421, 84408934941424, 2935694381925743, 109967573757472768, 4414292541216287516, 189054708982869449056

Offset: 1

Paul D. Hanna, Jun 14 2006

nonn

Cf. A120010, A120019, A120020, A120021, A120016.

a[n_] := Sum[((-1)^(j-i) n^(i-2) Binomial[2n-2j, n-j] Binomial[n+i-j, j-i] Binomial[n+i-j-1, i-1])/(n-j+1), {j, 1, n}, {i, 1, j}]; Array[a, 18] (* Jean-François Alcover, Nov 14 2016 *)

a(n)=polcoeff((1-sqrt(1-4*x*(1-x)/(1-(n+1)*x*(1-x)+x*O(x^n))))/2, n)/n

/* Alternate Formula: */ a(n)=sum(j=1, n, binomial(2*n-2*j, n-j)/(n-j+1)* sum(i=1, j,(-1)^(j-i)*binomial(n-j+i, j-i)*binomial(n-j+i-1, i-1)*n^(i-2)))

A120009 G.f.: A(x) = (x-x^2) o x/(1-x) o (1-sqrt(1-4*x))/2, a composition of functions involving the Catalan function and its inverse.

Original entry on oeis.org

1, 1, 1, 0, -6, -33, -143, -572, -2210, -8398, -31654, -118864, -445740, -1671525, -6273135, -23571780, -88704330, -334347090, -1262330850, -4773905760, -18083762580, -68611922730, -260725306374, -992233959480, -3781513867796, -14431491699548, -55147299002348

Offset: 1

Paul D. Hanna, Jun 03 2006

The n-th self-composition of A(x) is: (x-x^2) o x/(1-n*x) o (1-sqrt(1-4*x))/2. See A120010 for the transpose of the composition of the same functions.

			A(x) = x + x^2 + x^3 - 6*x^5 - 33*x^6 - 143*x^7 - 572*x^8 - 2210*x^9 + ...
A(x) = x*C(x)^2 - x^2*C(x)^4 where C(x) is Catalan function so that:
x*C(x)^2 = x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 + ...
x^2*C(x)^4 = x^2 + 4*x^3 + 14*x^4 + 48*x^5 + 165*x^6 + 572*x^7 + ...

Cf. A120010 (composition transpose), A000108 (Catalan), A000245.

Cf. A003517 (|a(n+1)|-|a(n)|). - Olivier Gérard, Oct 11 2012

[3*Catalan(n) - Catalan(n+1): n in [1..30]]; // Vincenzo Librandi, Jan 02 2025

f[n_]:=3*CatalanNumber[n] -CatalanNumber[n+1];Array[f,30,1] (* Vincenzo Librandi, Jan 02 2025 *)

a(n)=binomial(2*n,n)/(n+1)-4*binomial(2*n-1,n-2)/(n+2)

G.f.: A(x) = ((1-3*x)*sqrt(1-4*x) - (1-x)*(1-4*x))/(2*x^2) = x*C(x)^2 - x^2*C(x)^4 where C(x) is the Catalan function (A000108).
a(n) = C(2*n,n)/(n+1) - 4*C(2*n-1,n-2)/(n+2).
a(n) = 3*Catalan(n) - Catalan(n+1). - David Callan, Nov 21 2006
D-finite with recurrence: (n+2)*a(n) + (-7*n-2)*a(n-1) + 6*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Jan 20 2020, corrected Feb 16 2020
From Peter Bala, Feb 02 2024: (Start)
a(n) = 3*(-1)^n*Sum_{k = 0..n} (-4)^(n-k)*binomial(n,k)*(2*k + 2)!/((k + 3)!*k!).
G.f.: x/(1 - 4*x)*c(-x/(1 - 4*x))^3, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)
E.g.f.: exp(2*x)*(2*BesselI(0, 2*x) - 3*BesselI(1, 2*x) + BesselI(2, 2*x)) - 2. - Stefano Spezia, Dec 31 2024

A155839 A ratio of two Catalan arrays.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 2, 3, 0, 1, 0, 4, 7, 6, 0, 1, 0, 8, 18, 16, 10, 0, 1, 0, 16, 45, 51, 30, 15, 0, 1, 0, 32, 110, 152, 115, 50, 21, 0, 1, 0, 64, 264, 436, 396, 225, 77, 28, 0, 1, 0, 128, 624, 1212, 1300, 876, 399, 112, 36, 0, 1

Offset: 0

Paul Barry, Jan 28 2009

			Triangle begins
  1;
  0,  1;
  0,  0,   1;
  0,  1,   0,   1;
  0,  2,   3,   0,   1;
  0,  4,   7,   6,   0,  1;
  0,  8,  18,  16,  10,  0,  1;
  0, 16,  45,  51,  30, 15,  0, 1;
  0, 32, 110, 152, 115, 50, 21, 0, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000108, A033184, A120010 (row sums), A124644.

A155839:= func< n,k | (&+[(-1)^(n-j)*Binomial(j+1, n-j)*Binomial(j, k)*Catalan(j-k) : j in [k..n]]) >;
[A155839(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 04 2021

T[n_, k_] = Sum[(-1)^j*Binomial[n-j, k]*Binomial[n-j+1, j]*CatalanNumber[n-k-j], {j, 0, n-k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 04 2021 *)

def A155839(n,k): return sum( (-1)^j*binomial(n-j,k)*binomial(n-j+1,j)*catalan_number(n-k-j) for j in (0..n-k))
flatten([[A155839(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 04 2021

T(n, k) = Sum_{j=k..n} (-1)^(n-j)*binomial(j+1, n-j)*binomial(j, k)*A000108(j-k).
Sum_{k=0..n} T(n, k) = A120010(n+1).
Equals A033184^{-1}*A124644.

cp's OEIS Frontend

A120020 Coefficients of x^n in the n-th iteration of the g.f. of A120010: a(n) = [x^n] { (1-sqrt(1-4*x))/2 o x/(1-n*x) o (x-x^2) } for n>=1.

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A120019 Square table, read by antidiagonals, of self-compositions of A120010.

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A120018 The third self-composition of A120010; g.f.: A(x) = G(G(G(x))), where G(x) = g.f. of A120010.

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A120021 Coefficients of x^n in the (n+1)-th self-composition of the g.f. of A120010: a(n) = [x^n] { (1-sqrt(1-4*x))/2 o x/(1-(n+1)*x) o (x-x^2) } for n>=1.

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A120017 The 2nd self-composition of A120010; g.f.: A(x) = G(G(x)), where G(x) = g.f. of A120010.

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A120022 a(n) = A120020(n)/n = coefficient of x^n in the n-th self-composition of the g.f. of A120010, divided by n, for n>=1.

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A120009 G.f.: A(x) = (x-x^2) o x/(1-x) o (1-sqrt(1-4*x))/2, a composition of functions involving the Catalan function and its inverse.

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A155839 A ratio of two Catalan arrays.

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A120020 Coefficients of x^n in the n-th iteration of the g.f. of A120010: a(n) = [x^n] { (1-sqrt(1-4x))/2 o x/(1-nx) o (x-x^2) } for n>=1.

A120021 Coefficients of x^n in the (n+1)-th self-composition of the g.f. of A120010: a(n) = [x^n] { (1-sqrt(1-4x))/2 o x/(1-(n+1)x) o (x-x^2) } for n>=1.