cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 16 results. Next

A179422 E.g.f.: A(x) = G(G(x)) = x*G'(x) where G(x) is the g.f. of A179420.

Original entry on oeis.org

1, 4, 36, 528, 11000, 301680, 10379376, 433371008, 21434318496, 1232928216000, 81297809313600, 6074187611551488, 509351655073262976, 47554889211476564736, 4909859201019880800000, 557309205260654645145600
Offset: 1

Views

Author

Paul D. Hanna, Jul 28 2010

Keywords

Examples

			E.g.f.: A(x) = x + 4*x^2/2! + 36*x^3/3! + 528*x^4/4! + 11000*x^5/5! +...
Let G(x) be the g.f. of A179420, then
. G(x) = x + 2*x^2/2! + 12*x^3/3! + 132*x^4/4! + 2200*x^5/5! +...
. G(G(x)) = x + 4*x^2/2! + 36*x^3/3! + 528*x^4/4! + 11000*x^5/5! + ...

Links

Crossrefs

Cf. A179420, A179421.

Programs

  • PARI
    {a(n)=local(A=x+x^2+sum(m=3,n-1,a(m)*x^m/(m*m!))+x*O(x^n));if(n<3,n!*polcoeff(A,n),n*n!*polcoeff(subst(A,x,A),n)/(n-2))}

Formula

a(n) = n*A179420(n) = n^2*A179421(n-1).
E.g.f. satisfies: x*A'(x)/A(x) = G(A(x))/G(x) where G(x) is the g.f. of A179420.

A221019 Reduced numerators of A179420(n)/n!, where e.g.f. A(x) = Sum_{n>=0} A179420(n)/n! satisfies: A(A(x)) = x*A'(x) with A(0)=0, A'(0)=1.

Original entry on oeis.org

1, 1, 2, 11, 55, 419, 1471, 483673, 2756471, 1902667, 139975567, 79889883359, 2616245762827, 97206324428221, 2108611283335, 2036091547932503, 5773060629575464637, 737098816821260577403, 3053528216809788427627, 496374854736310558422419
Offset: 1

Views

Author

Paul D. Hanna, Dec 28 2012

Keywords

Comments

See A179420 for a description of the fascinating properties of the e.g.f. A(x) that satisfies: A(A(x)) = x*A'(x).

Examples

			E.g.f. A(x) of A179420 begins:
A(x) = x + 2*x^2/2! + 12*x^3/3! + 132*x^4/4! + 2200*x^5/5! +...+ A179420(n)/n!*x^n +...
or, equivalently,
A(x) = x + 1/1*x^2 + 2/1*x^3 + 11/2*x^4 + 55/3*x^5 + 419/6*x^6 + 1471/5*x^7 + 483673/360*x^8 + 2756471/420*x^9 + 1902667/56*x^10 +...+ a(n)/A221020(n)*x^n +...
which satisfies: A(A(x)) = x*A'(x) where:
A'(x) = 1 + 2*x + 12*x^2/2! + 132*x^3/3! + 2200*x^4/4! +...
A(A(x)) = x + 4*x^2/2! + 36*x^3/3! + 528*x^4/4! + 11000*x^5/5! +...

Links

Crossrefs

Cf. A221020, A179420.

Programs

  • PARI
    {A179420(n)=local(A=x+x^2+sum(m=3, n-1, A179420(m)*x^m/m!)+x*O(x^n)); if(n<3, n!*polcoeff(A, n),
n!*polcoeff(subst(A, x, A), n)/(n-2))}
{a(n)=numerator(A179420(n)/n!)}
for(n=1,25,print1(a(n),","))

Formula

a(n)/A221020(n) = A179420(n)/n!.

A221020 Reduced denominators of A179420(n)/n!, where e.g.f. A(x) = Sum_{n>=0} A179420(n)/n! satisfies: A(A(x)) = x*A'(x) with A(0)=0, A'(0)=1.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 5, 360, 420, 56, 756, 75600, 415800, 2494800, 8424, 1223040, 504504000, 9081072000, 5145940800, 111152321280, 754247894400, 37712394720000, 430747632000, 14454741869568, 319672175961600, 4080179409546240, 14011605115200000, 1653814216454400000
Offset: 1

Views

Author

Paul D. Hanna, Dec 28 2012

Keywords

Comments

See A179420 for a description of the fascinating properties of the e.g.f. A(x) that satisfies: A(A(x)) = x*A'(x).

Examples

			E.g.f. A(x) of A179420 begins:
A(x) = x + 2*x^2/2! + 12*x^3/3! + 132*x^4/4! + 2200*x^5/5! +...+ A179420(n)/n!*x^n +...
or, equivalently,
A(x) = x + 1/1*x^2 + 2/1*x^3 + 11/2*x^4 + 55/3*x^5 + 419/6*x^6 + 1471/5*x^7 + 483673/360*x^8 + 2756471/420*x^9 + 1902667/56*x^10 +...+ A221019(n)/A221020(n)*x^n +...
which satisfies: A(A(x)) = x*A'(x) where:
A'(x) = 1 + 2*x + 12*x^2/2! + 132*x^3/3! + 2200*x^4/4! +...
A(A(x)) = x + 4*x^2/2! + 36*x^3/3! + 528*x^4/4! + 11000*x^5/5! +...

Links

Crossrefs

Cf. A221019, A179420.

Programs

  • PARI
    {A179420(n)=local(A=x+x^2+sum(m=3, n-1, A179420(m)*x^m/m!)+x*O(x^n)); if(n<3, n!*polcoeff(A, n),
n!*polcoeff(subst(A, x, A), n)/(n-2))}
{a(n)=denominator(A179420(n)/n!)}
for(n=1,25,print1(a(n),","))

Formula

A221019(n)/A221020(n) = A179420(n)/n!.

A179421 E.g.f. A(x) satisfies: x*A(x) equals column 0 in the matrix log of the Riordan array (A(x), x*A(x)).

Original entry on oeis.org

1, 1, 4, 33, 440, 8380, 211824, 6771422, 264621216, 12329282160, 671882721600, 42181858413552, 3013915118776704, 242626985772839616, 21821596448977248000, 2176989083049432207600, 239420370429753669425664
Offset: 0

Views

Author

Paul D. Hanna, Jul 15 2010

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 33*x^3/3! + 440*x^4/4! +...
x*A(x) = x + 2*x^2/2! + 12*x^3/3! + 132*x^4/4! + 2200*x^5/5! +...
...
The RIORDAN ARRAY (A(x), x*A(x)) begins:
. 1;
. 1, 1;
. 4/2!, 2, 1;
. 33/3!, 10/2!, 3, 1;
. 440/4!, 90/3!, 18/2!, 4, 1;
. 8380/5!, 1240/4!, 177/3!, 28/2!, 5, 1;
. 211824/6!, 23800/5!, 2544/4!, 300/3!, 40/2!, 6, 1;
. 6771422/7!, 598788/6!, 49680/5!, 4520/4!, 465/3!, 54/2!, 7, 1; ...
where the e.g.f. of column k = A(x)^(k+1) for k>=0.
...
The MATRIX LOG of the above Riordan array (A(x), x*A(x)) begins:
. 0;
. 1, 0;
. 2/2!, 2, 0;
. 12/3!, 4/2!, 3, 0;
. 132/4!, 24/3!, 6/2!, 4, 0;
. 2200/5!, 264/4!, 36/3!, 8/2!, 5, 0;
. 50280/6!, 4400/5!, 396/4!, 48/3!, 10/2!, 6, 0;
. 1482768/7!, 100560/6!, 6600/5!, 528/4!, 60/3!, 12/2!, 7, 0; ...
where the e.g.f. of column k = (k+1)*x*A(x) for k>=0.

Links

Crossrefs

Cf. A179420, A179422, A179423, A179424.

Programs

  • PARI
    {a(n)=local(A=1+x+sum(m=2,n-1,a(m)*x^m/m!)+x*O(x^n));if(n<2,n!*polcoeff(A,n),n!*polcoeff(subst(x*A,x,x*A)/x,n)/(n-1))}

Formula

a(n) = A179420(n+1)/(n+1).
E.g.f. A(x) satisfies: A(x) = x*A'(x)/(A(x*A(x)) - 1).
Let L_{n}(x) denote the n-th iteration of L=L(x)=x*A(x) then:
. L(L(x)) = x*L'(x),
. L_{n+1}(x) = L(x)*x*[d/dx L_n(x)]/L_n(x),
. L(x)/x = 1 + L + L*Dx(L)/2! + L*Dx(L*Dx(L))/3! + L*Dx(L*Dx(L*Dx(L)))/4! +...
. L_{n}(x)/x = 1 + n*L + n^2*L*Dx(L)/2! + n^3*L*Dx(L*Dx(L))/3! + n^4*L*Dx(L*Dx(L*Dx(L)))/4! +...
where Dx(F) = d/dx(x*F).

A179497 E.g.f. satisfies: A(A(x))^2 = A(x)^2 * A'(x).

Original entry on oeis.org

1, 2, 18, 312, 8240, 297000, 13705776, 776778688, 52511234688, 4143702216960, 375403993060800, 38537107042934016, 4435139176244554752, 567238312617468850176, 80029364113424328422400
Offset: 1

Views

Author

Paul D. Hanna, Jul 31 2010

Keywords

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 18*x^3/3! + 312*x^4/4! + 8240*x^5/5! +..
Related expansions:
. A(x)/x = 1 + x + 6*x^2/2! + 78*x^3/3! + 1648*x^4/4! + 49500*x^5/5! +..
. A(x)^2/x = x + 4*x^2/2! + 42*x^3/3! + 768*x^4/4! + 20680*x^5/5! +..
. A'(x) = 1 + 2*x + 18*x^2/2! + 312*x^3/3! + 8240*x^4/4! +...
. A(A(x)) = x + 4*x^2/2! + 48*x^3/3! + 1008*x^4/4! + 30880*x^5/5! +...
. A(A(x))^2 = 2*x^2/2! + 24*x^3/3! + 480*x^4/4! + 13920*x^5/5! +...
Illustrate a main property of the iterations A_n(x) of A(x) by:
. [A_3(x)]^2 = A(x)^2 * A_2'(x);
. [A_4(x)]^2 = A(x)^2 * A_3'(x);
. [A_5(x)]^2 = A(x)^2 * A_4'(x); ...
which can be shown to hold by the chain rule of differentiation.
...
The RIORDAN ARRAY (A(x)/x, A(x)) begins:
. 1;
. 1, 1;
. 6/2!, 2, 1;
. 78/3!, 14/2!, 3, 1;
. 1648/4!, 192/3!, 24/2!, 4, 1;
. 49500/5!, 4136/4!, 348/3!, 36/2!, 5, 1;
. 1957968/6!, 124840/5!, 7680/4!, 552/3!, 50/2!, 6, 1;
. 97097336/7!, 4928256/6!, 233940/5!, 12520/4!, 810/3!, 66/2!, 7, 1; ...
where the e.g.f. of column k = [A(x)/x]^(k+1) for k>=0.
...
The MATRIX LOG of the above Riordan array (A(x)/x, A(x)) begins:
. 0;
. 1, 0;
. 4/2!, 2, 0;
. 42/3!, 8/2!, 3, 0;
. 768/4!, 84/3!, 12/2!, 4, 0;
. 20680/5!, 1536/4!, 126/3!, 16/2!, 5, 0;
. 749040/6!, 41360/5!, 2304/4!, 168/3!, 20/2!, 6, 0;
. 34497792/7!, 1498080/6!, 62040/5!, 3072/4!, 210/3!, 24/2!, 7, 0; ...
where the e.g.f. of column k = (k+1)*A(x)^2/x for k>=0.

Links

Crossrefs

Cf. A179498, A179499, A179420.

Programs

  • PARI
    {a(n)=local(A=x+1*x^2+sum(m=3,n-1,a(m)*x^m/m!)+O(x^(n+3)));if(n<3,n!*polcoeff(A,n),n!*polcoeff(subst(A,x,A)^2/A^2,n-1)/(n-2))}

Formula

E.g.f. A(x) satisfies: A(x)^2/x equals the e.g.f. of column 0 in the matrix log of the Riordan array (A(x)/x, A(x)).
Let A_n(x) denote the n-th iteration of e.g.f. A(x) with A_0(x)=x,
then L=A(x)^2/x satisfies the series:
. A(x)/x = 1 + L + L*Dx(L)/2! + L*Dx(L*Dx(L))/3! + L*Dx(L*Dx(L*Dx(L)))/4! +...
. A_{-1}(x)/x = 1 - L + L*Dx(L)/2! - L*Dx(L*Dx(L))/3! + L*Dx(L*Dx(L*Dx(L)))/4! -+...
. A_n(x)/x = 1 + n*L + n^2*L*Dx(L)/2! + n^3*L*Dx(L*Dx(L))/3! + n^4*L*Dx(L*Dx(L*Dx(L)))/4! +...
where Dx(F) = d/dx(x*F).
Further, we have:
. [A_{n+1}(x)]^2 = A(x)^2*A_n'(x)
which holds for all n.

A179499 E.g.f. A(x) = G(x)^2/x where G(x) satisfies: G(G(x))^2 = G(x)^2 * G'(x) and G(x) is the g.f. of A179499.

Original entry on oeis.org

1, 4, 42, 768, 20680, 749040, 34497792, 1944626432, 130528288512, 10219233265920, 918320724657600, 93509880099305472, 10677268138244018688, 1355273477576934150144, 189834913883100796531200
Offset: 1

Views

Author

Paul D. Hanna, Jul 31 2010

Keywords

Examples

			E.g.f.: A(x) = x + 4*x^2/2! + 42*x^3/3! + 768*x^4/4! + 20680*x^5/5! +...

Crossrefs

Cf. A179497, A179498, A179420.

A179330 E.g.f. satisfies: A(x) = (1+x)/(1+3*x) * A(x*(1+x)^2).

Original entry on oeis.org

0, 2, -6, 42, -468, 7080, -133128, 2938824, -73169568, 1997384832, -58814501760, 1868053207680, -65311214042880, 2585560450337280, -115344597684718080, 5424254194395456000, -244310147229735014400, 10256126830544041574400
Offset: 0

Views

Author

Paul D. Hanna, Jul 21 2010

Keywords

Examples

			E.g.f.: A(x) = 2*x - 6*x^2/2! + 42*x^3/3! - 468*x^4/4! + 7080*x^5/5! - 133128*x^6/6! + 2938824*x^7/7! - 73169568*x^8/8! + 1997384832*x^9/9! - 58814501760*x^10/10! + 1868053207680*x^11/11! - 65311214042880*x^12/12! +...
...
A(x*(1+x)^2) = 2*x + 2*x^2/2! - 18*x^3/3! + 108*x^4/4! - 480*x^5/5! - 2808*x^6/6! + 162792*x^7/7! - 3940128*x^8/8! + 57267648*x^9/9! + 534366720*x^10/10! - 78703384320*x^11/11! + 2883142045440*x^12/12! +...
...
where A(x*(1+x)^2) = (1+3*x)/(1+x) * A(x).
...
Related expansions begin:
. A = 2*x - 6*x^2/2! + 42*x^3/3! - 468*x^4/4! + 7080*x^5/5! +...
. A*Dx(A)/2! = 8*x^2/2! - 90*x^3/3! + 1332*x^4/4! - 25200*x^5/5! +...
. A*Dx(A*Dx(A))/3! = 48*x^3/3! - 1248*x^4/4! + 32760*x^5/5! -+...
. A*Dx(A*Dx(A*Dx(A)))/4! = 384*x^4/4! - 18480*x^5/5! + 770400*x^6/6! -+...
. A*Dx(A*Dx(A*Dx(A*Dx(A))))/5! = 3840*x^5/5! - 300672*x^6/6! +-...
...
Sums of which generate the square of the g.f. of A001764:
. G001764(-x)^2 = 1 - A + A*Dx(A)/2! - A*Dx(A*Dx(A))/3! + A*Dx(A*Dx(A*Dx(A)))/4! -+...
. G001764(-x)^2 = 1 - 2*x + 7*x^2 - 30*x^3 + 143*x^4 - 728*x^5 +...+ A006013(n)*(-x)^n +...
...
The Riordan array ((1+x)^2, x*(1+x)^2) (cf. A116088) begins:
1;
2, 1;
1, 4, 1;
0, 6, 6, 1;
0, 4, 15, 8, 1;
0, 1, 20, 28, 10, 1;
0, 0, 15, 56, 45, 12, 1; ...
The matrix log of Riordan array ((1+x)^2, x*(1+x)^2) begins:
0;
2, 0;
-6/2!, 4, 0;
42/3!, -12/2!, 6, 0;
-468/4!, 84/3!, -18/2!, 8, 0;
7080/5!, -936/4!, 126/3!, -24/2!, 10, 0;
-133128/6!, 14160/5!, -1404/4!, 168/3!, -30/2!, 12, 0; ...
where the g.f. of the leftmost column equals the e.g.f. of this sequence.

Crossrefs

Cf. A179331, variants: A179320, A179420.

Programs

  • PARI
    /* E.g.f. satisfies: A(x) = (1+x)/(1+3*x)*A(x*(1+x)^2): */
{a(n)=local(A=2*x, B); for(m=2, n, B=(1+x)/(1+3*x+O(x^(n+3)))*subst(A,x,x*(1+x)^2+O(x^(n+3))); A=A-polcoeff(B, m+1)*x^m/(m-1)/2); n!*polcoeff(A, n)}
  • PARI
    /* (1+x)^2 = 1 + A + A*Dx(A)/2! + A*Dx(A*Dx(A))/3! +...: */
{a(n)=local(A=0+sum(m=1,n-1,a(m)*x^m/m!),D=1,R=0);R=-((1+x)^2+x*O(x^n))+1+sum(m=1,n,(D=A*deriv(x*D+x*O(x^n)))/m!);-n!*polcoeff(R,n)}
  • PARI
    /* First column of the matrix log of triangle A116088: */
{a(n)=local(M=matrix(n+1, n+1, r, c, if(r>=c, polcoeff(((1+x)^2+x*O(x^n))^c,r-c))), LOG, ID=M^0); LOG=sum(m=1, n+1, -(ID-M)^m/m); n!*LOG[n+1, 1]}

Formula

E.g.f. A=A(x) satisfies:
. (1+x)^2 = 1 + A + A*Dx(A)/2! + A*Dx(A*Dx(A))/3! + A*Dx(A*Dx(A*Dx(A)))/4! +...
. (1+x)^2*(1+x*(1+x)^2)^2 = 1 + 2*A + 2^2*A*Dx(A)/2! + 2^3*A*Dx(A*Dx(A))/3! + 2^4*A*Dx(A*Dx(A*Dx(A)))/4! +...
. G001764(-x)^2 = 1 - A + A*Dx(A)/2! - A*Dx(A*Dx(A))/3! + A*Dx(A*Dx(A*Dx(A)))/4! -+...; G001764(x) = g.f. of A001764;
where Dx(F) = d/dx(x*F).
INVERSION FORMULA:
More generally, if A(x) = A(G(x)) * G(x)/(x*G'(x)) with G(0)=0, G'(0)=1,
then G(x) can be obtained from A=A(x) by the series:
. G(x)/x = 1 + A + A*Dx(A)/2! + A*Dx(A*Dx(A))/3! + A*Dx(A*Dx(A*Dx(A)))/4! +... where Dx(F) = d/dx(x*F).
ITERATION FORMULA:
Let G_n(x) denote the n-th iteration of G(x) = x*(1+x)^2, and A=A(x), then:
. A(x) = A(G_n(x)) * G_n(x)/(x*G_n'(x)) for all n;
. G_n(x)/x = 1 + n*A + n^2*A*Dx(A)/2! + n^3*A*Dx(A*Dx(A))/3! + n^4*A*Dx(A*Dx(A*Dx(A)))/4! +... where Dx(F) = d/dx(x*F).
...
MATRIX LOG OF RIORDAN ARRAY (G(x)/x, G(x)) where G(x) = x*(1+x)^2:
. k*A(x) = e.g.f. of column k of the matrix log of triangle A116088 for k>=0.

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)!.

Original entry on oeis.org

1, 3, 63, 6804, 1990170, 1145276496, 1172421884088, 1981846981092069, 5166650461467914874, 19710026486212156729362, 105613632141369240315500892, 768455476842781911036557334267, 7380326961188107570497477933701847
Offset: 0

Views

Author

Paul D. Hanna, Jul 15 2011

Keywords

Examples

			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! +...

Crossrefs

Cf. A001028, A179420.

Programs

  • PARI
    {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)}

A193202 E.g.f. A(x) satisfies: A(A(x)) = x*A'(A(x)).

Original entry on oeis.org

1, 2, 0, 12, -160, 3240, -86688, 2922640, -119971584, 5847901920, -332122243200, 21653202377664, -1601381638172160, 133036354347921024, -12314128238585510400, 1261212911036957548800, -142082122642808666185728, 17514853400850824425213440, -2351847513553411263501035520, 342599734607249938595012582400
Offset: 1

Views

Author

Paul D. Hanna, Jul 22 2011

Keywords

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 12*x^4/4! - 160*x^5/5! + 3240*x^6/6! - 86688*x^7/7! + 2922640*x^8/8! - 119971584*x^9/9! + 5847901920*x^10/10! +...
RELATED EXPANSIONS.
_ A(A(x)) = x + 4*x^2/2! + 12*x^3/3! + 48*x^4/4! + 40*x^5/5! + 2640*x^6/6! - 57456*x^7/7! + 2059904*x^8/8! - 85967136*x^9/9! + 4262310720*x^10/10! +...
_ A'(A(x)) = 1 + 2*x + 4*x^2/2! + 12*x^3/3! + 8*x^4/4! + 440*x^5/5! - 8208*x^6/6! +...
_ A(x)/A'(x) = x - 2*x^2/2! + 12*x^3/3! - 132*x^4/4! + 2200*x^5/5! - 50280*x^6/6! + 1482768*x^7/7! - 54171376*x^8/8! + 2381590944*x^9/9! - 123292821600*x^10/10! +...
Higher order iterations begin:
_ A_3(x) = x + 6*x^2/2! + 36*x^3/3! + 252*x^4/4! + 1800*x^5/5! + 16920*x^6/6! +...
_ A_4(x) = x + 8*x^2/2! + 72*x^3/3! + 768*x^4/4! + 9200*x^5/5! + 126720*x^6/6! +...
_ A_5(x) = x + 10*x^2/2! + 120*x^3/3! + 1740*x^4/4! + 29200*x^5/5! + 561000*x^6/6! +...
Illustrate a main property of the iterations A_n(x) by:
_ A(x)/A'(x) = A(x) * A(A(x)) / (x*d/dx A(A(x)));
_ A(x)/A'(x) = A_2(x) * A_3(x) / (x*d/dx A_3(x));
_ A(x)/A'(x) = A_3(x) * A_4(x) / (x*d/dx A_4(x));
_ A(x)/A'(x) = A_4(x) * A_5(x) / (x*d/dx A_5(x)); ...
which can be shown consistent by the chain rule of differentiation.

Links

Crossrefs

Cf. A179420.

Programs

  • PARI
    {a(n)=local(A=x+x^2+sum(m=3, n-1, a(m)*x^m/m!)+x*O(x^n)); if(n<3, n!*polcoeff(A, n), n!*polcoeff(subst(A,x,A)-x*subst(A',x,A), n)/(n-2))}

Formula

E.g.f. satisfies: A( A(x)/A'(x) ) = x.
E.g.f. satisfies: A(x) = Series_Reversion(-G(-x)) where G(x) = -A(-x)/A'(-x) is the e.g.f. of A179420 and satisfies: G(G(x)) = x*G'(x).
The inverse function of A(x), A(x)/A'(x), equals the g.f. of column 0 of the matrix log of the Riordan array (A(x)/x, A(x)).
Let A_n(x) denote the n-th iteration of e.g.f. A(x) with A_0(x)=x, then:
A(x)/A'(x) = A{n-1}(x) * A_n(x) / (x * d/dx A_n(x)) for all n.

A193264 E.g.f. A(x) satisfies: A(A(A(x))) = 2*x*A'(x) - A(x), where A(x) = Sum_{n>=1} a(n)*x^n, with a(1)=1, a(2)=2.

Original entry on oeis.org

1, 2, 18, 324, 9140, 359460, 18408600, 1174201280, 90423766800, 8215991163000, 865420074120800, 104218992780067440, 14188585798246317120, 2163608674997595229040, 366682177870608886473600, 68611838511981521881152000, 14093827998013078645611495680
Offset: 1

Views

Author

Paul D. Hanna, Jul 20 2011

Keywords

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 18*x^3/3! + 324*x^4/4! + 9140*x^5/5! + 359460*x^6/6! +...+ a(n)*x^n/n! +...
where
A(A(A(x))) = x + 6*x^2/2! + 90*x^3/3! + 2268*x^4/4! + 82260*x^5/5! + 3954060*x^6/6! +...+ (2*n-1)*a(n)*x^n/n! +...
which equals:
2*x*A'(x) - A(x) = x + 3*2*x^2/2! + 5*18*x^3/3! + 7*324*x^4/4! + 9*9140*x^5/5! +...

Links

Crossrefs

Cf. A193265, A179420.

Programs

  • PARI
    {a(n)=local(A=x);if(n<1,0,if(n<=2,n,A=x+sum(m=2,n-1,a(m)*x^m/m!)+x*O(x^n);
n!*polcoeff(subst(A,x,subst(A,x,A))-2*x*A',n)/(2*n-4)))}

Formula

E.g.f. A(x) satisfies: A(A(A(x))) = Sum_{n>=1} (2*n-1)*a(n)*x^n.
a(n) = n*A193265(n-1).
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