A110447 -id:A110447 - cp's OEIS frontend

A379598 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A110447.

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 6, 0, 1, 4, 9, 16, 23, 0, 1, 5, 14, 31, 62, 104, 0, 1, 6, 20, 52, 123, 278, 531, 0, 1, 7, 27, 80, 213, 552, 1398, 2982, 0, 1, 8, 35, 116, 340, 964, 2750, 7718, 18109, 0, 1, 9, 44, 161, 513, 1561, 4784, 14976, 46083, 117545, 0

Offset: 0

Seiichi Manyama, Feb 27 2025

			Square array begins:
  1,   1,    1,    1,    1,    1,     1, ...
  0,   1,    2,    3,    4,    5,     6, ...
  0,   2,    5,    9,   14,   20,    27, ...
  0,   6,   16,   31,   52,   80,   116, ...
  0,  23,   62,  123,  213,  340,   513, ...
  0, 104,  278,  552,  964, 1561,  2400, ...
  0, 531, 1398, 2750, 4784, 7755, 11987, ...

Columns k=0..1 give A000007, A110447 (A030266(n+1)).

Cf. A379599, A380178.

a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(n+k, j)/(n+k)*a(n-j, j)));

See A030266.

A111448 a(n) = A110447(n)^(1/3).

Original entry on oeis.org

0, 1, 8, 7, 4, 5, 6, 3, 2, 9, 71, 8, 17, 6, 73, 39, 41, 47, 24, 5, 3, 12, 9, 11, 18, 77, 46, 33, 79, 81, 7, 14, 63, 22, 49, 51, 28, 37, 36, 93, 19, 21, 67, 4, 23, 32, 89, 91, 38, 97, 15, 26, 53, 59, 61, 27, 44, 83, 42, 29, 31, 48, 57, 16, 13, 99, 701, 887, 134, 843, 269, 471, 58

Offset: 0

Amarnath Murthy, Aug 03 2005

Cf. A077744.

More terms from Franklin T. Adams-Watters, Jul 27 2006

A030266 Shifts left under COMPOSE transform with itself.

Original entry on oeis.org

0, 1, 1, 2, 6, 23, 104, 531, 2982, 18109, 117545, 808764, 5862253, 44553224, 353713232, 2924697019, 25124481690, 223768976093, 2062614190733, 19646231085928, 193102738376890, 1956191484175505, 20401540100814142, 218825717967033373, 2411606083999341827

Offset: 0

Christian G. Bower

			G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 23*x^5 + 104*x^6 + ...
A(A(x)) = x + 2*x^2 + 6*x^3 + 23*x^4 + 104*x^5 + 531*x^6 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..220
David Callan, A Combinatorial Interpretation of the Eigensequence for Composition, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.4.
David Callan, Lagrange Inversion Counts 3bar-5241-Avoiding Permutations, J. Int. Seq. 14 (2011) # 11.9.4.
N. J. A. Sloane, Transforms

Cf. A001028, A035049.
Cf. A110447.
Cf. A128325, A121687.

A:= proc(n) option remember;
      unapply(`if`(n=0, x,
      A(n-1)(x)+coeff(A(n-1)(A(n-1)(x)), x, n) *x^(n+1)), x)
    end:
a:= n-> coeff(A(n)(x),x,n):
seq(a(n), n=0..20);  # Alois P. Heinz, Feb 24 2012

A[0] = Identity; A[n_] := A[n] = Function[x, Evaluate[A[n-1][x]+Coefficient[A[n-1][A[n-1][x]], x, n]*x^(n+1)]]; a[n_] := Coefficient[A[n][x], x, n]; Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)
CoefficientList[Nest[x + x (# /. x -> #) &, O[x], 30], x] (* Vladimir Reshetnikov, Aug 08 2019 *)

{a(n)=local(A=1+x);for(i=0,n,A=1+x*A*subst(A,x,x*A+x*O(x^n)));polcoeff(A,n)} \\ Paul D. Hanna, Mar 10 2007

{a(n)=local(A=sum(i=1,n-1,a(i)*x^i)+x*O(x^n));if(n==0,0,polcoeff((1+A)^n/n,n-1))} \\ Paul D. Hanna, Nov 18 2008

{a(n,m=1)=if(n==0,1,if(m==0,0^n,sum(k=0,n,m*binomial(n+m,k)/(n+m)*a(n-k,k))))} \\ Paul D. Hanna, Jul 09 2009

{a(n)=local(A=1+sum(i=1,n-1,a(i)*x^i+x*O(x^n)));
for(i=1,n,A=exp(sum(m=1,n,sum(k=0,n-m,binomial(m+k-1,k)*polcoeff(A^(2*m),k)*x^k)*x^m/m)+x*O(x^n)));polcoeff(A,n)} \\ Paul D. Hanna, Dec 15 2010

G.f. A(x) satisfies the functional equation: A(x)-x = x*A(A(x)). - Paul D. Hanna, Aug 04 2002
G.f.: A(x/(1+A(x))) = x. - Paul D. Hanna, Dec 04 2003
Suppose the functions A=A(x), B=B(x), C=C(x), etc., satisfy: A = 1 + xAB, B = 1 + xABC, C = 1 + xABCD, D = 1 + xABCDE, etc., then B(x)=A(x*A(x)), C(x)=B(x*A(x)), D(x)=C(x*A(x)), etc., where A(x) = 1 + x*A(x)*A(x*A(x)) and x*A(x) is the g.f. of this sequence (see table A128325). - Paul D. Hanna, Mar 10 2007
G.f. A(x) = x*F(x,1) where F(x,n) satisfies: F(x,n) = F(x,n-1)*(1 + x*F(x,n+1)) for n>0 with F(x,0)=1. - Paul D. Hanna, Apr 16 2007
a(n) = [x^(n-1)] [1 + A(x)]^n/n for n>=1 with a(0)=0; i.e., a(n) equals the coefficient of x^(n-1) in [1+A(x)]^n/n for n >= 1. - Paul D. Hanna, Nov 18 2008
From Paul D. Hanna, Jul 09 2009: (Start)
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n with a(0,m)=1, then
a(n,m) = Sum_{k=0..n} m*C(n+m,k)/(n+m) * a(n-k,k).
(End)
G.f. satisfies:
* A(x) = x*exp( Sum_{m>=0} {d^m/dx^m A(x)^(m+1)/x} * x^(m+1)/(m+1)! );
* A(x) = x*exp( Sum_{m>=1} [Sum_{k>=0} C(m+k-1,k)*{[y^k] A(y)^m/y^m}*x^k]*x^m/m );
which are equivalent. - Paul D. Hanna, Dec 15 2010
The g.f. satisfies:
log(A(x)/x) = A(x) + {d/dx A(x)^2/x}*x^2/2! + {d^2/dx^2 A(x)^3/x}*x^3/3! + {d^3/dx^3 A(x)^4/x}*x^4/4! + ... - Paul D. Hanna, Dec 15 2010

A120971 G.f. A(x) satisfies A(x) = 1 + xA(x)^2 A( x*A(x)^2 )^2.

Original entry on oeis.org

1, 1, 4, 26, 218, 2151, 23854, 289555, 3783568, 52624689, 772928988, 11918181144, 192074926618, 3224153299106, 56213565222834, 1015694652332437, 18982833869517376, 366384235565593176, 7292660345274942402

Offset: 0

Paul D. Hanna, Jul 20 2006

nonn

			G.f.: A(x) = 1 + x + 4*x^2 + 26*x^3 + 218*x^4 + 2151*x^5 + 23854*x^6 +...
From _Paul D. Hanna_, Apr 16 2007: G.f. A(x) is the unique solution to variable A in the infinite system of simultaneous equations:
A = 1 + x*B^2;
B = A*(1 + x*C^2);
C = B*(1 + x*D^2);
D = C*(1 + x*E^2);
E = D*(1 + x*F^2); ...
The above series begin:
B(x) = 1 + 2*x + 11*x^2 + 87*x^3 + 841*x^4 + 9288*x^5 + 113166*x^6 +...
C(x) = 1 + 3*x + 21*x^2 + 198*x^3 + 2204*x^4 + 27431*x^5 + 371102*x^6 +...
D(x) = 1 + 4*x + 34*x^2 + 374*x^3 + 4747*x^4 + 66350*x^5 + 996943*x^6 +...
E(x) = 1 + 5*x + 50*x^2 + 630*x^3 + 9015*x^4 + 140510*x^5 + 2334895*x^6 +...
F(x) = 1 + 6*x + 69*x^2 + 981*x^3 + 15658*x^4 + 270016*x^5 + 4933294*x^6 +...

Cf. A120970; variants: A110447, A120973, A120975, A120977.

Cf. A002449, A087949, A088714, A088717, A091713.

m = 19; A[] = 0; Do[A[x] = 1 + x A[x]^2 A[x A[x]^2]^2 + O[x]^m, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Nov 07 2019 *)

{a(n)=local(A,G=[1,1]);for(i=1,n,G=concat(G,0); G[ #G]=-Vec(subst(Ser(G),x,x/Ser(G)^2))[ #G]); A=Vec(((Ser(G)-1)/x)^(1/2));A[n+1]}

a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(2*n+k, j)/(2*n+k)*a(n-j, 2*j))); \\ Seiichi Manyama, Mar 01 2025

G.f. A(x) satisfies:
(1) A(x) = G(G(x)-1),
(2) A(G(x)-1) = G(A(x)-1),
(3) A(x) = G(x*A(x)^2),
(4) A(x/G(x)^2) = G(x),
where G(x) is the g.f. of A120970 and satisfies G(x/G(x)^2) = 1 + x.
G.f. A(x) = F(x,1) where F(x,n) satisfies: F(x,n) = F(x,n-1)*(1 + x*F(x,n+1)^2) for n>0 with F(x,0)=1. - Paul D. Hanna, Apr 16 2007
Let B(x) = Sum_{n>=0} a(n)*x^(2*n+1), then B( x/(1+B(x)^2) ) = x. - Paul D. Hanna, Oct 30 2013
From Seiichi Manyama, Mar 01 2025: (Start)
Let a(n,k) = [x^n] A(x)^k.
a(n,0) = 0^n; a(n,k) = k * Sum_{j=0..n} binomial(2*n+k,j)/(2*n+k) * a(n-j,2*j). (End)

A120973 G.f. A(x) satisfies A(x) = 1 + xA(x)^3 A(x*A(x)^3)^3.

Original entry on oeis.org

1, 1, 6, 60, 776, 11802, 201465, 3759100, 75404151, 1608036861, 36172106112, 853346084343, 21021015647574, 538868533164995, 14336235065928966, 394957784033440194, 11246848201518516044, 330520280036501809758

Offset: 0

Paul D. Hanna, Jul 20 2006

nonn

Cf. A120972; variants: A110447, A120971, A120975, A120977.

{a(n)=local(A,G=[1,1]);for(i=1,n,G=concat(G,0); G[ #G]=-Vec(subst(Ser(G),x,x/Ser(G)^3))[ #G]); A=Vec(((Ser(G)-1)/x)^(1/3));A[n+1]}

a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(3*n+k, j)/(3*n+k)*a(n-j, 3*j))); \\ Seiichi Manyama, Mar 01 2025

G.f. A(x) satisfies: A(x) = G(G(x)-1), A(G(x)-1) = G(A(x)-1), A(x) = G(x*A(x)^3) and A(x/G(x)^3) = G(x), where G(x) is the g.f. of A120972 and satisfies G(x/G(x)^3) = 1 + x.
From Seiichi Manyama, Mar 01 2025: (Start)
Let a(n,k) = [x^n] A(x)^k.
a(n,0) = 0^n; a(n,k) = k * Sum_{j=0..n} binomial(3*n+k,j)/(3*n+k) * a(n-j,3*j). (End)

A120975 G.f. A(x) satisfies A(x) = 1 + xA(x)^4 A(x*A(x)^4)^4.

Original entry on oeis.org

1, 1, 8, 108, 1888, 38798, 894308, 22517256, 609112756, 17507219813, 530495478900, 16850219461706, 558608940038072, 19263089278722726, 689119527976265884, 25519081467271687938, 976447764170903902364

Offset: 0

Paul D. Hanna, Jul 20 2006

nonn

Cf. A120974; variants: A110447, A120971, A120973, A120977.

{a(n)=local(A,G=[1,1]);for(i=1,n,G=concat(G,0); G[ #G]=-Vec(subst(Ser(G),x,x/Ser(G)^4))[ #G]); A=Vec(((Ser(G)-1)/x)^(1/4));A[n+1]}

a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(4*n+k, j)/(4*n+k)*a(n-j, 4*j))); \\ Seiichi Manyama, Mar 01 2025

G.f. A(x) satisfies: A(x) = G(G(x)-1), A(G(x)-1) = G(A(x)-1), A(x) = G(x*A(x)^4) and A(x/G(x)^4) = G(x), where G(x) is the g.f. of A120974 and satisfies G(x/G(x)^4) = 1 + x.
From Seiichi Manyama, Mar 01 2025: (Start)
Let a(n,k) = [x^n] A(x)^k.
a(n,0) = 0^n; a(n,k) = k * Sum_{j=0..n} binomial(4*n+k,j)/(4*n+k) * a(n-j,4*j). (End)

A120977 G.f. A(x) satisfies A(x) = 1 + xA(x)^5 A(x*A(x)^5)^5.

Original entry on oeis.org

1, 1, 10, 170, 3745, 96960, 2814752, 89221360, 3037327145, 109825686370, 4185287088735, 167139924222426, 6964610755602495, 301800832258018835, 13564159649547824735, 630916661388096564620, 30316241123672291911875

Offset: 0

Paul D. Hanna, Jul 20 2006

nonn

Cf. A120976; variants: A110447, A120971, A120973, A120975.

{a(n)=local(A,G=[1,1]);for(i=1,n,G=concat(G,0); G[ #G]=-Vec(subst(Ser(G),x,x/Ser(G)^5))[ #G]); A=Vec(((Ser(G)-1)/x)^(1/5));A[n+1]}

a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(5*n+k, j)/(5*n+k)*a(n-j, 5*j))); \\ Seiichi Manyama, Mar 01 2025

G.f. A(x) satisfies: A(x) = G(G(x)-1), A(G(x)-1) = G(A(x)-1), A(x) = G(x*A(x)^5) and A(x/G(x)^5) = G(x), where G(x) is the g.f. of A120976 and satisfies G(x/G(x)^5) = 1 + x.
From Seiichi Manyama, Mar 01 2025: (Start)
Let a(n,k) = [x^n] A(x)^k.
a(n,0) = 0^n; a(n,k) = k * Sum_{j=0..n} binomial(5*n+k,j)/(5*n+k) * a(n-j,5*j). (End)

A384145 G.f. A(x) satisfies A(x) = 1/( 1 - xA(xA(x)^3) ).

Original entry on oeis.org

1, 1, 2, 8, 44, 298, 2359, 21112, 209175, 2262121, 26431042, 331096188, 4419824468, 62565545535, 935341395343, 14716294925179, 242945752432294, 4197094127399756, 75698807290515322, 1422350601250404765, 27788515730656558613, 563512508612712699574, 11841983002490204813514

Offset: 0

Seiichi Manyama, Jun 06 2025

nonn

Column k=1 of A384652.

Cf. A110447, A162661, A384649, A384650.

a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(3*n-2*j+k, j)/(3*n-2*j+k)*a(n-j, j)));

See A384652.

A113882 Number of well-nested drawings of a rooted tree.

Original entry on oeis.org

1, 1, 2, 9, 64, 605, 6996, 94556, 1452928, 24921765, 471091360, 9720039120, 217285778700, 5230874655578, 134929133296972, 3713182459524270, 108605754921052880, 3364866315332574493, 110099293819641466488, 3794219154973411079432, 137375263325254329836460

Offset: 0

Marco Kuhlmann (kuhlmann(AT)ps.uni-sb.de), Jan 27 2006

The value a(n) also gives the number of non-crossing partitions on [n] such that (a) each block of the partition is a non-crossing partition itself (recursively) and (b) every partition in this recursion contains at least one singleton block (see A000108). Omitting the factor n in the equation for a(n) gives A110447.

			a(5)=605 because there are 605 possibilities to form 5 nodes into a rooted tree and order the nodes of this tree such that no two subtrees interleave. Two subtrees t1, t2 interleave if their roots are (tree-)disjoint and there are four nodes l1, r1 from t1 and l2, r2 from t2 such that l1 < l2 < r1 < r2.
Comment from _Paul D. Hanna_, Aug 08 2007 (revised Apr 28 2012): (Start)
Illustrate a(n) = [x^(n-1)] A(x)^n by the following generating method.
Form a table of coefficients in powers of the g.f. A(x):
A(x)^1: [(1), 1, 2, 9, 64, 605, 6996, 94556, ...];
A(x)^2: [1, (2), 5, 22, 150, 1374, 15539, 206676, ...];
A(x)^3: [1, 3, (9), 40, 264, 2346, 25937, 339294, ...];
A(x)^4: [1, 4, 14, (64), 413, 3568, 38558, 495848, ...];
A(x)^5: [1, 5, 20, 95, (605), 5096, 53840, 680365, ...];
A(x)^6: [1, 6, 27, 134, 849, (6996), 72302, 897558, ...];
A(x)^7: [1, 7, 35, 182, 1155, 9345, (94556), 1152936, ...]; ...
then the coefficients along the main diagonal form the initial terms of this sequence. (End)

Manuel Bodirsky, Marco Kuhlmann and Mathias Mohl: Well-Nested Drawings as Models of Syntactic Structure, 10th Conference on Formal Grammar and 9th Meeting on Mathematics of Language, Edinburgh, Scotland, UK

Manuel Bodirsky, Marco Kuhlmann and Mathias Mohl: Well-Nested Drawings as Models of Syntactic Structure, 10th Conference on Formal Grammar and 9th Meeting on Mathematics of Language, Edinburgh, Scotland, UK

Cf. A000108, A110447, A132070, A361048.

{a(n)=my(G=1+x+2*x^2); for(k=0,n,G=1+x*deriv(serreverse(x/(G+x^2*O(x^n) )))); polcoef(G,n)} \\ Paul D. Hanna, Aug 08 2007

a(0) = a(1) = 1; a(n) = n * F(n-1), where F(0) = F(1) = 1, F(n) = sum_{i=1}^{n} a(i) * F(n-i, i), where F(0, k) = 1; F(n, 1) = F(n), F(n, k) = sum_{i=0}^{n} F(i) * F(n-i, k-1).
Contribution from Paul D. Hanna, Aug 08 2007 (revised Apr 28 2012): (Start)
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
a(n) = [x^(n-1)] A(x)^n for n>=1;
a(n) = (n+1)*A132070(n+1) for n>=0;
A(x) = x / Series_Reversion(x*G(x)) = G(x/A(x)) where G(x) = A(x*G(x)) is the g.f. of A132070. (End)

Initial term added and offset changed to 0 by Paul D. Hanna, Apr 28 2012.

A384649 G.f. A(x) satisfies A(x) = 1/( 1 - xA(xA(x)^4) ).

Original entry on oeis.org

1, 1, 2, 9, 56, 432, 3935, 40820, 471633, 5980210, 82329140, 1220547845, 19359684220, 326799737576, 5844913732057, 110341722975077, 2191461358459051, 45656013573862832, 995196646595460516, 22644288881875546322, 536706817952488705651, 13225669497771610891404

Offset: 0

Seiichi Manyama, Jun 06 2025

nonn

Column k=1 of A384653.

Cf. A110447, A162661, A384145, A384650.

a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(4*n-3*j+k, j)/(4*n-3*j+k)*a(n-j, j)));

See A384653.

cp's OEIS Frontend

A379598 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A110447.

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A111448 a(n) = A110447(n)^(1/3).

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A030266 Shifts left under COMPOSE transform with itself.

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A120971 G.f. A(x) satisfies A(x) = 1 + x*A(x)^2 * A( x*A(x)^2 )^2.

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Mathematica

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A120973 G.f. A(x) satisfies A(x) = 1 + x*A(x)^3 * A(x*A(x)^3)^3.

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A120975 G.f. A(x) satisfies A(x) = 1 + x*A(x)^4 * A(x*A(x)^4)^4.

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A120977 G.f. A(x) satisfies A(x) = 1 + x*A(x)^5 * A(x*A(x)^5)^5.

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A384145 G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x*A(x)^3) ).

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A113882 Number of well-nested drawings of a rooted tree.

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A120971 G.f. A(x) satisfies A(x) = 1 + xA(x)^2 A( x*A(x)^2 )^2.

A120973 G.f. A(x) satisfies A(x) = 1 + xA(x)^3 A(x*A(x)^3)^3.

A120975 G.f. A(x) satisfies A(x) = 1 + xA(x)^4 A(x*A(x)^4)^4.

A120977 G.f. A(x) satisfies A(x) = 1 + xA(x)^5 A(x*A(x)^5)^5.

A384145 G.f. A(x) satisfies A(x) = 1/( 1 - xA(xA(x)^3) ).

A384649 G.f. A(x) satisfies A(x) = 1/( 1 - xA(xA(x)^4) ).