A001028 -id:A001028 - cp's OEIS frontend

A030266 Shifts left under COMPOSE transform with itself.

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

A144002 E.g.f. A(x) satisfies: A(x) = 1 + Series_Reversion( Integral 1/A(x)^2 dx ).

Original entry on oeis.org

1, 1, 2, 10, 88, 1152, 20448, 464608, 12998176, 435443328, 17106187520, 775347933312, 40025403691136, 2328514989726720, 151324140857050624, 10904257049278844416, 865717992565002800640, 75309304802558209263616, 7143418423952431605493760, 735668180680897524348745728

Offset: 0

Paul D. Hanna, Sep 07 2008

nonn

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 10*x^3/3! + 88*x^4/4! + 1152*x^5/5! + ...

Paul D. Hanna, Table of n, a(n) for n = 0..100

Cf. A001028, A144003, A144004.

{a(n) = my(A=1+x+x*O(x^n)); for(i=0,n, A = 1 + serreverse(intformal(1/A^2))); n!*polcoef(A,n)}
for(n=0,20,print1(a(n),", "))

E.g.f. A(x) satisfies: A'(x) = A(A(x) - 1)^2. - Paul D. Hanna, Nov 25 2014 [corrected by Paul D. Hanna, Sep 07 2024]

A035049 E.g.f. satisfies A(x) = x*(1+A(A(x))), A(0)=0.

Original entry on oeis.org

1, 2, 12, 144, 2760, 74880, 2676240, 120234240, 6571393920, 426547296000, 32283270835200, 2808028566604800, 277433852555059200, 30836115140589158400, 3824551325912308992000, 525674251444773150720000, 79591811594194480508928000, 13205626859810397006618624000

Offset: 1

Christian G. Bower, Oct 15 1998

Alois P. Heinz, Table of n, a(n) for n = 1..283

Cf. A001028, A030266.

A:= proc(n) option remember; `if`(n=0, 0, (T-> unapply(
      convert(series(x*(1+T(T(x))), x, n+1), polynom), x))(A(n-1)))
    end:
a:= n-> coeff(A(n)(x), x, n)*n!:
seq(a(n), n=1..20);  # Alois P. Heinz, Aug 23 2008
# second Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1, add(k*
      a(j)*b(n-j, k-1)*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> `if`(n=0, 1, b(n-1, n)):
seq(a(n), n=1..20);  # Alois P. Heinz, Aug 21 2019

T[n_, m_] := T[n, m] = If[n == m, 1, m/n*Sum[Sum[T[n-m, i]*Binomial[i-1, k-1]*(-1)^i, {i, k, n-m}]*(-1)^k*Binomial[n+k-1, n-1], {k, 1, n-m}]]; Table[n!*T[n, 1], {n, 1, 16}] (* Jean-François Alcover, Feb 12 2014, after Vladimir Kruchinin *)

T(n,m):=if n=m then 1 else m/n*sum(sum(T(n-m,i)*binomial(i-1,k-1)*(-1)^i,i,k,n-m)*(-1)^k*binomial(n+k-1,n-1),k,1,n-m); makelist(n!*T(n,1),n,1,10); /* Vladimir Kruchinin, May 06 2012 */

a(n) = n!*T(n,1), T(n,m) = m/n*sum(k=1..n-m, sum(i=k..n-m, T(n-m,i) * C(i-1,k-1)*(-1)^i)*(-1)^k*C(n+k-1,n-1)), n>m, T(n,n)=1. - Vladimir Kruchinin, May 06 2012

More terms from Alois P. Heinz, Aug 23 2008

A144003 E.g.f. A(x) satisfies: A(x) = 1 + Series_Reversion( Integral 1/A(x)^3 dx ).

Original entry on oeis.org

1, 1, 3, 24, 339, 7101, 200961, 7256277, 321662502, 17029233774, 1054682936433, 75199620036177, 6094256204678922, 555527437385512095, 56468189426338157580, 6353824422205136494044, 786458781488123265873519

Offset: 0

Paul D. Hanna, Sep 07 2008

nonn

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 24*x^3/3! + 339*x^4/4! + 7101*x^5/5! + 200961*x^6/6! + 7256277*x^7/7! + 321662502*x^8/8! + ...
where A(x) = 1 + Series_Reversion( Integral 1/A(x)^3 dx ).
RELATED SERIES.
Integral 1/A(x)^3 dx = x - 3*x^2/2! + 3*x^3/3! - 24*x^4/4! - 261*x^5/5! - 6543*x^6/6! - 202671*x^7/7! - 7911351*x^8/8! + ...
where Integral 1/A(x)^3 dx = Series_Reversion(A(x) - 1).
A(A(x) - 1) = 1 + x + 6*x^2/2! + 75*x^3/3! + 1479*x^4/4! + 40617*x^5/5! + 1447785*x^6/6! + 64027656*x^7/7! + 3404869020*x^8/8! + ...
A(A(x) - 1)^3 = 1 + 3*x + 24*x^2/2! + 339*x^3/3! + 7101*x^4/4! + ...
where A(A(x) - 1)^3 = d/dx A(x).

Paul D. Hanna, Table of n, a(n) for n = 0..201

Cf. A001028, A144002, A144004.

{a(n) = my(A=1+x+x*O(x^n)); for(i=0,n, A = 1 + serreverse(intformal(1/A^3))); n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

E.g.f. A(x) satisfies: A'(x) = A(A(x) - 1)^3. - Paul D. Hanna, Aug 26 2024

A144004 E.g.f. A(x) satisfies: A(x) = 1 + Series_Reversion( Integral 1/A(x)^4 dx ).

Original entry on oeis.org

1, 1, 4, 44, 856, 24664, 958592, 47463936, 2881313024, 208638075392, 17654019768320, 1717961286944768, 189836122499649536, 23574107397852049408, 3261667682403085852672, 499151625979680748978176

Offset: 0

Paul D. Hanna, Sep 07 2008

nonn

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 44*x^3/3! + 856*x^4/4! + 24664*x^5/5! + 958592*x^6/6! + 47463936*x^7/7! + 2881313024*x^8/8! + ...

Cf. A001028, A144002, A144003.

{a(n) = my(A=1+x+x*O(x^n)); for(i=0,n, A = 1 + serreverse(intformal(1/A^4))); n!*polcoef(A,n)}
for(n=0,20,print1(a(n),", "))

E.g.f. A(x) satisfies: A'(x) = A(A(x) - 1)^4. - Paul D. Hanna, Sep 07 2024

A193100 E.g.f. A(x) satisfies: A’(x) = x + A(A(x)), where A(x) = Sum_{n>=0} a(n)x^(3n+2)/(3*n+2)!.

Original entry on oeis.org

1, 3, 63, 6804, 1990170, 1145276496, 1172421884088, 1981846981092069, 5166650461467914874, 19710026486212156729362, 105613632141369240315500892, 768455476842781911036557334267, 7380326961188107570497477933701847

Offset: 0

Paul D. Hanna, Jul 15 2011

nonn

			E.g.f.: A(x) = x^2/2! + 3*x^5/5! + 63*x^8/8! + 6804*x^11/11! + 1990170*x^14/14! + 1145276496*x^17/17! + 1172421884088*x^20/20! +...
where A'(x) = x + 3*x^4/4! + 63*x^7/7! + 6804*x^10/10! +...
and A(A(x)) = 3*x^4/4! + 63*x^7/7! + 6804*x^10/10! + 1990170*x^13/13! +...

Cf. A001028, A179420.

{a(n)=local(A=x^2/2);for(i=1,n,A=intformal(x+subst(A,x,A+O(x^(3*n+3)))));(3*n+2)!*polcoeff(A,3*n+2)}

A193098 E.g.f. A(x) satisfies: A'(x) = 1 + A(A(A(x))).

Original entry on oeis.org

1, 1, 3, 18, 171, 2283, 39942, 874944, 23243829, 731486637, 26782956144, 1124838704976, 53567894139165, 2865318598843281, 170774893724336223, 11264050942430761881, 817374450539598433587, 64917115563124199691834

Offset: 1

Paul D. Hanna, Jul 15 2011

nonn

			E.g.f.: A(x) = x + x^2/2! + 3*x^3/3! + 18*x^4/4! + 171*x^5/5! + 2283*x^6/6! +...
where the derivative of the e.g.f. begins:
A'(x) = 1 + x + 3*x^2/2! + 18*x^3/3! + 171*x^4/4! + 2283*x^5/5! +...
Related expansions.
A(A(x)) = x + 2*x^2/2! + 9*x^3/3! + 69*x^4/4! + 777*x^5/5! + 11802*x^6/6! + 229047*x^7/7! + 5472600*x^8/8! +...
A(A(A(x))) = x + 3*x^2/2! + 18*x^3/3! + 171*x^4/4! + 2283*x^5/5! +...

Cf. A001028, A193099, A193100, A179420.

{a(n)=local(A=x); for(i=1, n, A=intformal(1+subst(A, x, subst(A, x, A+O(x^(n+1)))))); n!*polcoeff(A, n)}

A193099 E.g.f. A(x) satisfies: A'(x) = 1 + A(A(A(A(x)))).

Original entry on oeis.org

1, 1, 4, 34, 466, 9044, 230827, 7388781, 287044354, 13212057907, 707417718215, 43431362340153, 3022050938855344, 236053437141340206, 20532456001485751429, 1975258248906891145913, 208928124926501980596761, 24172548454436633069025270

Offset: 1

Paul D. Hanna, Jul 15 2011

nonn

			E.g.f.: A(x) = x + x^2/2! + 4*x^3/3! + 34*x^4/4! + 466*x^5/5! + 9044*x^6/6! +...
where the derivative of the e.g.f. begins:
A'(x) = 1 + x + 4*x^2/2! + 34*x^3/3! + 466*x^4/4! + 9044*x^5/5! +...
Related expansions.
A(A(x)) = x + 2*x^2/2! + 11*x^3/3! + 111*x^4/4! + 1702*x^5/5! + 35854*x^6/6! +...
A(A(A(x))) = x + 3*x^2/2! + 21*x^3/3! + 249*x^4/4! + 4303*x^5/5! + 99650*x^6/6! +...
A(A(A(A(x)))) = x + 4*x^2/2! + 34*x^3/3! + 466*x^4/4! + 9044*x^5/5! +...

Cf. A001028, A193098, A193100, A179420.

{a(n)=local(A=x); for(i=1, n, A=intformal(1+subst(A, x, subst(A, x, subst(A, x, A+O(x^(n+1))))))); n!*polcoeff(A, n)}

A231868 E.g.f. satisfies: A'(x) = A(x*A(x)) with A(0)=1.

Original entry on oeis.org

1, 1, 1, 3, 12, 84, 774, 9468, 146052, 2764980, 62759736, 1678881096, 52185496464, 1862666455104, 75581146734912, 3456542059903296, 176834245093202736, 10053690187338014256, 631507398302281340736, 43596564604477924096512, 3292312674449093132923488

Offset: 0

Paul D. Hanna, Nov 14 2013

nonn

			E.g.f.: A(x) = 1 + x + x^2/2! + 3*x^3/3! + 12*x^4/4! + 84*x^5/5! + 774*x^6/6! + ...
such that
A(x*A(x)) = A'(x) = 1 + x + 3*x^2/2! + 12*x^3/3! + 84*x^4/4! + 774*x^5/5! + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..175

Cf. A001028, A231619.

{a(n)=local(A=1+x);for(i=1,n,A=1+intformal(subst(A,x,x*A +x*O(x^n))));n!*polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

E.g.f. satisfies: A(x) = 1 + Integral( A(x*A(x)) dx).

cp's OEIS Frontend

A030266 Shifts left under COMPOSE transform with itself.

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A144002 E.g.f. A(x) satisfies: A(x) = 1 + Series_Reversion( Integral 1/A(x)^2 dx ).

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A035049 E.g.f. satisfies A(x) = x*(1+A(A(x))), A(0)=0.

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A144003 E.g.f. A(x) satisfies: A(x) = 1 + Series_Reversion( Integral 1/A(x)^3 dx ).

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A144004 E.g.f. A(x) satisfies: A(x) = 1 + Series_Reversion( Integral 1/A(x)^4 dx ).

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A193100 E.g.f. A(x) satisfies: A’(x) = x + A(A(x)), where A(x) = Sum_{n>=0} a(n)*x^(3*n+2)/(3*n+2)!.

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PARI

A193098 E.g.f. A(x) satisfies: A'(x) = 1 + A(A(A(x))).

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PARI

A193099 E.g.f. A(x) satisfies: A'(x) = 1 + A(A(A(A(x)))).

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A193100 E.g.f. A(x) satisfies: A’(x) = x + A(A(x)), where A(x) = Sum_{n>=0} a(n)x^(3n+2)/(3*n+2)!.