A030266 -id:A030266 - cp's OEIS frontend

A030277 Shifts left 2 places under COMPOSE transform.

1, 1, 1, 2, 4, 10, 28, 86, 286, 1019, 3857, 15418, 64782, 285041, 1309355, 6263085, 31127475, 160432093, 856039863, 4721638083, 26883787141, 157812750656, 953995201996, 5932516347277, 37913110456751, 248768719522450, 1674488577654500, 11553041105489900

Offset: 1

Christian G. Bower

N. J. A. Sloane, Transforms

Cf. A030266.

Nest[x + x^2 + x^2 (# /. x -> #) &, O[x], 20][[3]] (* Vladimir Reshetnikov, Aug 08 2019 *)

G.f. A(x) satisfies: A(x) = x + x^2 * (1 + A(A(x))). - Ilya Gutkovskiy, May 10 2019

A120976 G.f. A(x) satisfies A(x/A(x)^5) = 1 + x ; thus A(x) = 1 + series_reversion(x/A(x)^5).

Original entry on oeis.org

1, 1, 5, 60, 1060, 23430, 602001, 17281760, 541258390, 18210836060, 651246905140, 24566101401035, 971933892729980, 40156993344526860, 1726753293393763625, 77065076699967844390, 3561820706538663354320, 170162336673835615653925, 8389644485709060522744640

Offset: 0

Paul D. Hanna, Jul 20 2006

nonn

Cf. A120977; variants: A120970, A120972, A120974, A030266, A067145, A107096.

{a(n)=local(A=[1,1]);for(i=2,n,A=concat(A,0); A[ #A]=-Vec(subst(Ser(A),x,x/Ser(A)^5))[ #A]);A[n+1]}

b(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(5*n+k, j)/(5*n+k)*b(n-j, 5*j)));
a(n) = if(n==0, 1, b(n-1, 5)); \\ Seiichi Manyama, Jun 04 2025

G.f. satisfies: A(x) = 1 + x*B(x)^5 = 1 + (1 + x*C(x)^5 )^5 where B(x) and C(x) satisfy: C(x) = B(x)*B(A(x)-1), B(x) = A(A(x)-1), B(A(x)-1) = A(B(x)-1), B(x/A(x)^5) = A(x), B(x) = A(x*B(x)^5) and B(x) is g.f. of A120977.
From Seiichi Manyama, Jun 04 2025: (Start)
Let b(n,k) = [x^n] B(x)^k, where B(x) is the g.f. of A120977.
b(n,0) = 0^n; b(n,k) = k * Sum_{j=0..n} binomial(5*n+k,j)/(5*n+k) * b(n-j,5*j).
a(n) = b(n-1,5) for n > 0. (End)

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.

A182953 G.f. satisfies: A(x) = 1 + xA(x) A( x*A(x) )^3.

Original entry on oeis.org

1, 1, 4, 25, 197, 1797, 18178, 198937, 2318858, 28487593, 366129764, 4896068759, 67843403960, 971032668429, 14319735032679, 217136949146091, 3379973833321141, 53936100582832901, 881318215466710693, 14731508761600217914

Offset: 0

Paul D. Hanna, Dec 15 2010

nonn

			G.f.: A(x) = 1 + x + 4*x^2 + 25*x^3 + 197*x^4 + 1797*x^5 +...
Related expansions:
A(x)^3 = 1 + 3*x + 15*x^2 + 100*x^3 + 801*x^4 + 7296*x^5 + 73174*x^6 +...
A(x*A(x)) = 1 + x + 5*x^2 + 37*x^3 + 333*x^4 + 3389*x^5 + 37634*x^6 +...
A(x*A(x))^3 = 1 + 3*x + 18*x^2 + 142*x^3 + 1311*x^4 + 13461*x^5 +...
The g.f. satisfies:
log(A(x)) = A(x)^3*x + {d/dx x*A(x)^6}*x^2/2! + {d^2/dx^2 x^2*A(x)^9}*x^3/3! + {d^3/dx^3 x^3*A(x)^12}*x^4/4! +...

Cf. A147664, A030266, A121687, A182954, A182955.

{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^(3*m),k)*x^k)*x^m/m)+x*O(x^n)));polcoeff(A,n)}

{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, 3*k))))}

G.f. A(x) satisfies:
* A(x) = exp( Sum_{m>=0} {d^m/dx^m x^m*A(x)^(3m+3)} * x^(m+1)/(m+1)! );
* A(x) = exp( Sum_{m>=1} [Sum_{k>=0} C(m+k-1,k)*{[y^k] A(y)^(3m)}*x^k]*x^m/m);
which are equivalent.
Recurrence:
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,3k).
Given g.f. A(x), then G(x) = 1 + x*A(x)^3 satisfies G(x/G(x)) = 1 + x*G(x)^2 and G(x) is the g.f. of A147664.

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

A196523 G.f. satisfies A(x) = x + x*A(A(A(A(x)))).

Original entry on oeis.org

1, 1, 4, 28, 262, 2944, 37666, 532276, 8151322, 133562194, 2320621222, 42475263136, 814932467836, 16326188799508, 340479903535258, 7373196169450312, 165453350568966163, 3840489521467649158, 92072430090995120044, 2276807696679096394552

Offset: 1

Paul D. Hanna, Oct 03 2011

nonn

Conjecture: a(n) == 1 (mod 3) for n >= 1. - Paul D. Hanna, Dec 01 2024

			G.f.: A(x) = x + x^2 + 4*x^3 + 28*x^4 + 262*x^5 + 2944*x^6 + 37666*x^7 +...
Given g.f. A(x), A(x)/x is the unique solution to variable A in the infinite system of simultaneous equations:
A = 1 + x*D;
B = A*(1 + x*E);
C = B*(1 + x*F);
D = C*(1 + x*G);
E = D*(1 + x*H); ...
The solution to the variables in the system of equations are:
A=A(x)/x, B=A(A(x))/x, C=A(A(A(x)))/x, D=A(A(A(A(x))))/x, etc.,
where iterations begin:
A(x) = x + x^2 + 4*x^3 + 28*x^4 + 262*x^5 + 2944*x^6 + 37666*x^7 +...
A(A(x)) = x + 2*x^2 + 10*x^3 + 77*x^4 + 760*x^5 + 8846*x^6 + 116140*x^7 +...
A(A(A(x))) = x + 3*x^2 + 18*x^3 + 153*x^4 + 1608*x^5 + 19566*x^6 +...
A(A(A(A(x)))) = x + 4*x^2 + 28*x^3 + 262*x^4 + 2944*x^5 + 37666*x^6 +...
A(A(A(A(A(x))))) = x + 5*x^2 + 40*x^3 + 410*x^4 + 4930*x^5 + 66530*x^6 +...
A(A(A(A(A(A(x)))))) = x + 6*x^2 + 54*x^3 + 603*x^4 + 7752*x^5 + 110484*x^6 +...
ALTERNATE GENERATING METHOD.
The g.f. A(x) equals the sum of products of {3*k}-iterations of A(x):
A(x) = x + x*A_3(x) + x*A_3(x)*A_6(x) + x*A_3(x)*A_6(x)*A_9(x) + x*A_3(x)*A_6(x)*A_9(x)*A_12(x) +...+ Product_{k=0..n} A_{3*k}(x) +...
where A_n(x) = A_{n-1}(A(x)) is the n-th iteration of A(x) with A_0(x)=x.
Related expansions.
x*A_3(x) = x^2 + 3*x^3 + 18*x^4 + 153*x^5 + 1608*x^6 + 19566*x^7 +...
x*A_3(x)*A_6(x) = x^3 + 9*x^4 + 90*x^5 + 1026*x^6 + 13059*x^7 +...
x*A_3(x)*A_6(x)*A_9(x) = x^4 + 18*x^5 + 279*x^6 + 4320*x^7 +...
x*A_3(x)*A_6(x)*A_9(x)*A_12(x) = x^5 + 30*x^6 + 675*x^7 +...

Paul D. Hanna, Table of n, a(n) for n = 1..400

Cf. A030266, A091713.

Nest[x + x (# /. x -> # /. x -> # /. x -> #) &, O[x], 30][[3]] (* Vladimir Reshetnikov, Aug 08 2019 *)

/* Define the n-th iteration of function F: */
{ITERATE(n, F, p)=local(G=x); for(i=1, n, G=subst(F, x, G+x*O(x^p))); G}
/* A(x) results from nested iterations of shifted series: */

{a(n)=local(A=x); for(i=1,n, A=x+x*ITERATE(4,A, n)); polcoeff(A, n)}

{a(n)=local(A=x); for(i=1,n, A=x+x*sum(m=1,n,prod(k=1,m,ITERATE(3*k,A,n)))); polcoeff(A, n)}

G.f.: A(x) = A(A(x))/(1 + A(A(A(A(A(x)))))).
G.f.: A(x) = Series_Reversion[ x/(1 + A(A(A(x)))) ].
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+3)) for n>0 with F(x,0)=1; further, x*F(x,n) is the n-th iteration of A(x).
G.f. satisfies: A(x) = Sum_{n>=0} Product_{k=0..n} A_{3*k}(x), where A_n(x) denotes the n-th iteration of A(x) with A_0(x)=x.

A347080 G.f. A(x) satisfies: A(x) = x + x * A(A(-x)).

Original entry on oeis.org

0, 1, -1, -2, 2, 13, -16, -161, 170, 2647, -1711, -51248, 5711, 1103710, 599246, -25521869, -33907174, 620323849, 1410745127, -15678980390, -53746958146, 411344661913, 1998823108706, -11256049308869, -75003366373495, 323285264486686, 2904292324907387

Offset: 0

Ilya Gutkovskiy, Jan 27 2022

sign

Cf. A030266, A030277, A171212.

nmax = 26; A[] = 0; Do[A[x] = x + x A[A[-x]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

A171214 G.f. A(x) satisfies A(x) = x + xA(A(x/3)) = Sum_{n>=1} a(n)x^n/3^(n*(n+1)/2).

Original entry on oeis.org

1, 1, 2, 10, 137, 5296, 588365, 190088818, 179954321171, 501722122937995, 4134242130461174144, 100943613343624534183723, 7317423203727305175501741434, 1577227642328692213664066391691150

Offset: 1

Paul D. Hanna, Dec 08 2009

nonn

More generally, if F(x) = x + x*F(F(qx)), then
F(x) = x + x*F(qx) + x*F(qx)*F(qF(qx)) + x*F(qx)*F(qF(qx))*F(qF(qF(qx))) +...
with a simple solution at q=1/2:
F(x) = x/(1-x/2) satisfies F(x) = x + x*F(F(x/2)).
At q=1, F(x,q=1) is the g.f. of A030266.
QUESTIONS regarding convergence of F(x,q) as a power series in x.
(1) What is Q, the maximum q below which a radius of convergence exists? Is Q=1?
(2) What is the radius of convergence for a given q < Q?

			G.f.: A(x) = x + x^2/3 + 2*x^3/3^3 + 10*x^4/3^6 + 137*x^5/3^10 + 5296*x^6/3^15 +...+ a(n)*x^n/3^(n(n-1)/2) +...
A(x) = x + x*A(x/3) + x*A(x/3)*A(A(x/3)/3) + x*A(x/3)*A(A(x/3)/3)*A(A(A(x/3)/3)/3) +...
A(A(x)) = x + 2*x^2/3 + 10*x^3/3^3 + 137*x^4/3^6 + 5296*x^5/3^10 +...
SUMS OF SERIES at certain arguments.
A(1) = 1.423879975541542344910599787693637973194...
A(1/3) = 0.373293286580877833612329400906044642790...
A(A(1/3)) = A(1) - 1 = 0.42387997554...
A(A(1)) = 2.387414460111728675082753594461076041830...
A(3) = 3 + 3*A(A(1)) = 10.16224338033518602524826...

Cf. A171212 (q=2), A171213 (q=3), A030266 (q=1).

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

A182955 G.f. satisfies: A(x) = 1 + xA(x) A( x*A(x) )^5.

Original entry on oeis.org

1, 1, 6, 56, 651, 8671, 126997, 1997798, 33260799, 580270730, 10534337521, 197986746949, 3837397114948, 76473239154148, 1563252546786254, 32716989219013821, 699959257347957763, 15288884723649589585

Offset: 0

Paul D. Hanna, Dec 15 2010

nonn

			G.f.: A(x) = 1 + x + 6*x^2 + 56*x^3 + 651*x^4 + 8671*x^5 +...
Related expansions:
A(x*A(x)) = 1 + x + 7*x^2 + 74*x^3 + 953*x^4 + 13846*x^5 +...
A(x*A(x))^5 = 1 + 5*x + 45*x^2 + 520*x^3 + 6950*x^4 + 102481*x^5 +...
The g.f. satisfies:
log(A(x)) = A(x)^5*x + {d/dx x*A(x)^10}*x^2/2! + {d^2/dx^2 x^2*A(x)^15}*x^3/3! + {d^3/dx^3 x^3*A(x)^20}*x^4/4! +...

Cf. A030266, A121687, A182953, A182954.

{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^(5*m),k)*x^k)*x^m/m)+x*O(x^n)));polcoeff(A,n)}

{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, 5*k))))}

G.f. A(x) satisfies:
* A(x) = exp( Sum_{m>=0} {d^m/dx^m x^m*A(x)^(5m+5)} * x^(m+1)/(m+1)! );
* A(x) = exp( Sum_{m>=1} [Sum_{k>=0} C(m+k-1,k)*{[y^k] A(y)^(5m)}*x^k]*x^m/m);
which are equivalent.
Recurrence:
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,5k).

cp's OEIS Frontend

A030277 Shifts left 2 places under COMPOSE transform.

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A120976 G.f. A(x) satisfies A(x/A(x)^5) = 1 + x ; thus A(x) = 1 + series_reversion(x/A(x)^5).

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A128327 Row 2 of table A128325.

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A128328 Row 3 of table A128325.

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A182953 G.f. satisfies: A(x) = 1 + x*A(x) * A( x*A(x) )^3.

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

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A196523 G.f. satisfies A(x) = x + x*A(A(A(A(x)))).

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A347080 G.f. A(x) satisfies: A(x) = x + x * A(A(-x)).

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A171214 G.f. A(x) satisfies A(x) = x + x*A(A(x/3)) = Sum_{n>=1} a(n)*x^n/3^(n*(n+1)/2).

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A182953 G.f. satisfies: A(x) = 1 + xA(x) A( x*A(x) )^3.

A171214 G.f. A(x) satisfies A(x) = x + xA(A(x/3)) = Sum_{n>=1} a(n)x^n/3^(n*(n+1)/2).

A182955 G.f. satisfies: A(x) = 1 + xA(x) A( x*A(x) )^5.