A128325 -id:A128325 - cp's OEIS frontend

A128329 Main diagonal of table A128325.

1, 1, 4, 30, 321, 4389, 72512, 1399755, 30865353, 764755508, 21024535960, 634924059276, 20890221475598, 743727414390456, 28484480606420928, 1167761832049224515, 51022550712426870397, 2366859765773183488674

Offset: 0

Paul D. Hanna, Mar 11 2007

nonn

Cf. A030266; A128325 (table), A128326 (row 1), A128327 (row 2), A128328 (row 3).

{a(n)=local(A=1+x,B);for(i=0,n,A=1+x*A*subst(A,x,x*A+x*O(x^n))); B=A;for(i=1,n,B=subst(B,x,x*A+x*O(x^n)));polcoeff(B,n)}

a(n) = [x^n] {1 + H(x)}, where H(x) is the (n+2)-th self-composition of G(x) and G(x) = x + x*G(G(x)) is the g.f. of A030266.

A128327 Row 2 of table A128325.

Original entry on oeis.org

1, 1, 4, 20, 114, 712, 4772, 33896, 253102, 1975610, 16054568, 135413280, 1182664740, 10675334958, 99437919664, 954581258020, 9433732288486, 95883201181772, 1001411775057322, 10738668800872594, 118151145186400408

Offset: 0

Paul D. Hanna, Mar 11 2007

nonn

Cf. A030266; A128325 (table), A128326 (row 1), A128328 (row 3), A128329 (main diagonal).

{a(n)=local(A=1+x,B);for(i=0,n,A=1+x*A*subst(A,x,x*A+x*O(x^n))); B=A;for(i=1,2,B=subst(B,x,x*A+x*O(x^n)));polcoeff(B,n)}

G.f.: A(x) = 1 + G(G(G(G(x)))) = B(G(x)), where B(x) is the g.f. of A128326 and G(x) = x + x*G(G(x)) is the g.f. of A030266.

A128328 Row 3 of table A128325.

Original entry on oeis.org

1, 1, 5, 30, 200, 1435, 10900, 86799, 720074, 6196295, 55135043, 506125404, 4784680169, 46516469860, 464550190798, 4761343733469, 50044839978614, 539051253692777, 5946806890025709, 67156408547628636, 775935817487472046

Offset: 0

Paul D. Hanna, Mar 11 2007

nonn

Cf. A030266; A128325 (table), A128326 (row 1), A128327 (row 2), A128329 (main diagonal).

{a(n)=local(A=1+x,B);for(i=0,n,A=1+x*A*subst(A,x,x*A+x*O(x^n))); B=A;for(i=1,3,B=subst(B,x,x*A+x*O(x^n)));polcoeff(B,n)}

G.f.: A(x) = 1 + G(G(G(G(G(x))))) = B(G(x)), where B(x) is the g.f. of A128327 and G(x) = x + x*G(G(x)) is the g.f. of A030266.

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

A128326 G.f.: A(x) = 1 + G(G(G(x))), where G(x) = x + x*G(G(x)) is the g.f. of A030266.

Original entry on oeis.org

1, 1, 3, 12, 57, 305, 1787, 11269, 75629, 535960, 3987913, 31021693, 251445581, 2117993712, 18499513147, 167246537937, 1562556275281, 15066167302802, 149737897716757, 1532313152898208, 16129331500727047

Offset: 0

Paul D. Hanna, Mar 11 2007

nonn

Equals row 1 of table A128325.

Cf. A030266; A128325 (table), A128327 (row 2), A128328 (row 3), A128329 (main diagonal).

{a(n)=local(A=1+x,B);for(i=0,n,A = 1 + x*A * subst(A,x,x*A+x*O(x^n))); B=A;B=subst(B,x,x*A+x*O(x^n));polcoeff(B,n)}
for(n=0, 30, print1(a(n), ", "))

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

G.f. satisfies: A(x) = x/(1 - A( x/(1 - A(x)) )) when offset is taken to be 1. - Paul D. Hanna, Dec 20 2014

A141141 The main diagonal in the table of coefficients of iterations of G(x), where G(x) = x + x*G(G(x)) = g.f. of A030266.

Original entry on oeis.org

1, 2, 12, 114, 1435, 22219, 406441, 8559852, 203792337, 5409449156, 158350300141, 5066765087000, 175908765569628, 6585443884172129, 264428161094825151, 11335716352419699208, 516717363793695685925, 24955728581736822645816

Offset: 1

Paul D. Hanna, Jun 10 2008

nonn

			a(n) = the n-th coefficient of the n-th iteration of G(x):
[x] G(x) = 1, [x^2] G(G(x)) = 2, [x^3] G(G(G(x))) = 12, etc.
The initial iterations (n=1..7) of G(x) are:
n=1: x + x^2 + 2*x^3 + 6*x^4 + 23*x^5 + 104*x^6 + 531*x^7 +...
n=2: x + 2*x^2 + 6*x^3 + 23*x^4 + 104*x^5 + 531*x^6 + 2982*x^7 +...
n=3: x + 3*x^2 + 12*x^3 + 57*x^4 + 305*x^5 + 1787*x^6 + 11269*x^7 +...
n=4: x + 4*x^2 + 20*x^3 + 114*x^4 + 712*x^5 + 4772*x^6 + 33896*x^7 +...
n=5: x + 5*x^2 + 30*x^3 + 200*x^4 + 1435*x^5 + 10900*x^6 + 86799*x^7+...
n=6: x + 6*x^2 + 42*x^3 + 321*x^4 + 2608*x^5 + 22219*x^6 + 196910*x^7+...
n=7: x + 7*x^2 + 56*x^3 + 483*x^4 + 4389*x^5 + 41531*x^6 + 406441*x^7+...
Notice the main diagonal of the table formed from these coefficients.

Cf. A030266, A128325, A128329.

{a(n)=local(A=x,B); if(n<1, 0, for(i=1, n, A=serreverse(x/(1+A +x*O(x^n)))); B=x;for(i=1,n,B=subst(A,x,B+x*O(x^n)));polcoeff(B,n))}

cp's OEIS Frontend

A128329 Main diagonal of table A128325.

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A128327 Row 2 of table A128325.

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A128328 Row 3 of table A128325.

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A030266 Shifts left under COMPOSE transform with itself.

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A128326 G.f.: A(x) = 1 + G(G(G(x))), where G(x) = x + x*G(G(x)) is the g.f. of A030266.

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A141141 The main diagonal in the table of coefficients of iterations of G(x), where G(x) = x + x*G(G(x)) = g.f. of A030266.

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