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 1-7 of 7 results.

A179423 E.g.f. A(x) = F(x)^2 where F(x) is the e.g.f. of A179421.

Original entry on oeis.org

1, 2, 10, 90, 1240, 23800, 598788, 18932620, 729558240, 33475442400, 1796086010400, 111058345494624, 7820581741096320, 621007886404464000, 55143814204485434400, 5436629250445000648800, 591426542480093093242368
Offset: 0

Views

Author

Paul D. Hanna, Jul 28 2010

Keywords

Comments

Let F(x) be the e.g.f. of A179421, then x*F(x) equals the e.g.f. of column 0 in the matrix log of the Riordan array (F(x), x*F(x)).

Examples

			E.g.f.: A(x) = 1 + 2*x + 10*x^2/2! + 90*x^3/3! + 1240*x^4/4! +...
The e.g.f. of A179421 is:
F(x) = 1 + x + 4*x^2/2! + 33*x^3/3! + 440*x^4/4! + 8380*x^5/5! +...

Crossrefs

Cf. A179421.

Programs

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

Formula

a(n) = Sum_{k=0..n} C(n,k)*A179421(k)*A179421(n-k).

A179420 E.g.f. A(x) satisfies: A(A(x)) = x*A'(x) with A(0)=0, A'(0)=1.

Original entry on oeis.org

0, 1, 2, 12, 132, 2200, 50280, 1482768, 54171376, 2381590944, 123292821600, 7390709937600, 506182300962624, 39180896544097152, 3396777800819754624, 327323946734658720000, 34831825328790915321600
Offset: 0

Views

Author

Paul D. Hanna, Jul 13 2010

Keywords

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 132*x^4/4! + 2200*x^5/5! +...
E.g.f. 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! +...
Related expansions begin:
A*Dx(A)/2! = 2*x^2/2! + 15*x^3/3! + 180*x^4/4! + 3150*x^5/5! +...
A*Dx(A*Dx(A))/3! = 6*x^3/3! + 104*x^4/4! + 2140*x^5/5! +...
A*Dx(A*Dx(A*Dx(A)))/4! = 24*x^4/4! + 770*x^5/5! + 24600*x^6/6! +...
A*Dx(A*Dx(A*Dx(A*Dx(A))))/5! = 120*x^5/5! + 6264*x^6/6! +...
which generate iterations of A=A(x) as illustrated by:
A(A(x))/x = 1 + 2*A + 2^2*A*Dx(A)/2! + 2^3*A*Dx(A*Dx(A))/3! +...
A(A(A(x)))/x = 1 + 3*A + 3^2*A*Dx(A)/2! + 3^3*A*Dx(A*Dx(A))/3! +...
A_{-1}(x)/x = 1 - A + A*Dx(A)/2! - A*Dx(A*Dx(A))/3! +-...(inverse).
Illustrate a main property of the iterations A_n(x) of A(x) by:
A(x) = A(A(x)) * A(x)/[x*d/dx A(x)];
A(x) = A_3(x) * A_2(x)/[x*d/dx A_2(x)];
A(x) = A_4(x) * A_3(x)/[x*d/dx A_3(x)]; ...
which can be shown consistent by the chain rule of differentiation.
...
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)/x 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)*A(x) for k>=0.

Links

Crossrefs

Cf. A193202, A179421, A179422, A179423, A179424, variant: A179320.
a(n)/n! = A221019(n)/A221020(n).

Programs

  • Mathematica
    a[n_] := a[n] = Module[{A}, A[x_] = x+x^2+Sum[a[m]*x^m/m!, {m, 3, n-1}]; If[n<3, n!*Coefficient[A[x], x, n], n!*Coefficient[A[A[x]], x, n]/(n-2)] ]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Jan 15 2018, translated from PARI *)
  • Maxima
    Co(n, k, F):=if k=1  then F(n) else sum(F(i+1)*Co(n-i-1, k-1, F), i, 0, n-k);
a(n):=if n=0 then 0 else if n<3 then 1 else sum(Co(n,k,a)*a(k),k,2,n-1)/(n-2); /* Vladimir Kruchinin, Jun 29 2011 */
  • 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),n)/(n-2))}

Formula

E.g.f. A(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 A=A(x) satisfies:
A(x)/x = 1 + A + A*Dx(A)/2! + A*Dx(A*Dx(A))/3! + A*Dx(A*Dx(A*Dx(A)))/4! +...
A_{-1}(x)/x = 1 - A + A*Dx(A)/2! - A*Dx(A*Dx(A))/3! + A*Dx(A*Dx(A*Dx(A)))/4! -+...
A_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).
Further, we have: A(x) = A_{n+1}(x) * A_n(x)/[x*d/dx A_n(x)] which holds for all n.
a(n)=sum(k=2..n-1, R(n-1,k-1)*a(k))/(n-2), n>2, a(1)=1, a(2)=1, where R is the Riordan array (A(x)/x, A(x)). [Vladimir Kruchinin, Jun 29 2011]
E.g.f. satisfies: A(x) = Series_Reversion(-G(-x)) where G(x) is the e.g.f. of A193202 and satisfies: G(G(x)) = x*G'(G(x)). [Paul D. Hanna, Jul 22 2011]

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.

A179498 E.g.f. satisfies: A(x) = A(x*A(x))^2 - x*A'(x).

Original entry on oeis.org

1, 1, 6, 78, 1648, 49500, 1957968, 97097336, 5834581632, 414370221696, 34127635732800, 3211425586911168, 341164552018811904, 40517022329819203584, 5335290940894955228160, 773591071307555130451200
Offset: 0

Views

Author

Paul D. Hanna, Jul 31 2010

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 6*x^2/2! + 78*x^3/3! + 1648*x^4/4! + 49500*x^5/5! +...
Related expansions:
. x*A(x) = x + 2*x^2/2! + 18*x^3/3! + 312*x^4/4! + 8240*x^5/5! +...
. x*A(x)^2 = x + 4*x^2/2! + 42*x^3/3! + 768*x^4/4! + 20680*x^5/5! +..
. x*A'(x) = x + 12*x^2/2! + 234*x^3/3! + 6592*x^4/4! + 247500*x^5/5! +...
. A(x*A(x)) = 1 + x + 8*x^2/2! + 132*x^3/3! + 3400*x^4/4! + 120940*x^5/5! +...
. A(x*A(x))^2 = 1 + 2*x + 18*x^2/2! + 312*x^3/3! + 8240*x^4/4! + 297000*x^5/5! +...
Illustrate the iterations G_n(x) of G(x) = x*A(x) by:
. [G_3(x)/x]^2 = A(x)^2 * G_2'(x);
. [G_4(x)/x]^2 = A(x)^2 * G_3'(x);
. [G_5(x)/x]^2 = A(x)^2 * G_4'(x); ...
which can be shown 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 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;
. 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 g.f. of column k = (k+1)*x*A(x)^2 for k>=0.

Crossrefs

Cf. A179497, A179499, A179421.

Programs

  • PARI
    {a(n)=local(A=1+x+sum(m=2,n-1,a(m)*x^m/m!)+x*O(x^(n+5)));if(n<2,n!*polcoeff(A,n),n!*polcoeff(subst(A,x,x*A)^2,n)/(n-1))}

Formula

E.g.f. satisfies: x*A(x)^2 equals the g.f. of column 0 in the matrix log of the Riordan array (A(x), x*A(x)).
E.g.f.: A(x) = G(x)/x where G(x) = e.g.f. of A179497.
Let G_n(x) denote the n-th iteration of x*A(x) with G_0(x)=x, then
. [G_{n+1}(x)/x]^2 = A(x)^2*G_n'(x) for all n,
and L=x*A(x)^2 satisfies the series:
. A(x) = 1 + L + L*Dx(L)/2! + L*Dx(L*Dx(L))/3! + L*Dx(L*Dx(L*Dx(L)))/4! +...
. G_{-1}(x)/x = 1 - L + L*Dx(L)/2! - L*Dx(L*Dx(L))/3! + L*Dx(L*Dx(L*Dx(L)))/4! -+...
. G_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).

A179493 E.g.f. A(x) satisfies: L(x) = A(x)/(x*A'(x)) * L(A(x)) where L(x) = x + x*A(x).

Original entry on oeis.org

0, 1, 2, 12, 108, 1420, 24660, 541968, 14547792, 465228720, 17385553440, 747776581200, 36566808933600, 2012537262763872, 123612631608883872, 8412289268206662720, 630378349868153698560, 51733701375836221013760
Offset: 0

Views

Author

Paul D. Hanna, Jul 23 2010

Keywords

Examples

			E.g.f: A(x) = x + 2*x^2/2! + 12*x^3/3! + 108*x^4/4! + 1420*x^5/5! +...
Related expansions:
. L(x) = x + 2*x^2/2! + 6*x^3/3! + 48*x^4/4! + 540*x^5/5! +...
. L(A(x)) = x + 4*x^2/2! + 30*x^3/3! + 348*x^4/4! + 5560*x^5/5! +...
. x*A'(x) = x + 4*x^2/2! + 36*x^3/3! + 432*x^4/4! + 7100*x^5/5! +...
. A(x)/x = 1 + x + 4*x^2/2! + 27*x^3/3! + 284*x^4/4! + 4110*x^5/5! +...
where L(x) = x + x*A(x) = A(x)/(x*A'(x)) * L(A(x)).
...
The RIORDAN ARRAY (A(x)/x, A(x)) begins:
1;
1, 1;
4/2!, 2, 1;
27/3!, 10/2!, 3, 1;
284/4!, 78/3!, 18/2!, 4, 1;
4110/5!, 880/4!, 159/3!, 28/2!, 5, 1;
77424/6!, 13220/5!, 1932/4!, 276/3!, 40/2!, 6, 1;
1818474/7!, 252828/6!, 30390/5!, 3608/4!, 435/3!, 54/2!, 7, 1; ...
where the g.f. of column k = A(x)^(k+1)/x^k for k>=0.
...
The MATRIX LOG of the above Riordan array (A(x)/x, A(x)) begins:
0;
1, 0;
2/2!, 2, 0;
6/3!, 4/2!, 3, 0;
48/4!, 12/3!, 6/2!, 4, 0;
540/5!, 96/4!, 18/3!, 8/2!, 5, 0;
8520/6!, 1080/5!, 144/4!, 24/3!, 10/2!, 6, 0;
172620/7!, 17040/6!, 1620/5!, 192/4!, 30/3!, 12/2!, 7, 0; ...
where the g.f. of column k = (k+1)*(x + x*A(x)) for k>=0.
...
To illustrate the inversion series, let L=L(x)=x + x*A(x), then:
. A(A(x)) = x + 4*x^2/2! + 36*x^3/3! + 480*x^4/4! + 8720*x^5/5! +...
. A(A(x))/x = 1 + 2*L + 2^2*L*Dx(L)/2! + 2^3*L*Dx(L*Dx(L))/3! +...
. A_3(x) = x + 6*x^2/2! + 72*x^3/3! + 1260*x^4/4! + 29340*x^5/5! +...
. A_3(x)/x = 1 + 3*L + 3^2*L*Dx(L)/2! + 3^3*L*Dx(L*Dx(L))/3! +...
where Dx(F) = d/dx(x*F).

Crossrefs

Cf. A179494, A179420, A179421.

Programs

  • PARI
    {a(n)=local(A=[1,1]);for(i=2,n, A=concat(A,0);G=x*Ser(A);A[ #A]=polcoeff(1+subst(G,x,G)+O(x^#A)-(1+G)*deriv(G)*x^2/G^2,#A-1)/(#A-2));if(n<1,0,n!*A[n])}

Formula

E.g.f. satisfies: A(A(x)) = -1 + (1 + A(x))*A'(x)*x^2/A(x)^2.
Let A_n(x) denote the n-th iteration of e.g.f. A(x), then
. A_{n+1}(x) = -1 + (1 + A(x))*A_n'(x)*x^2/A_n(x)^2.
. L(x) = A_n(x)/(x*A_n'(x)) * L(A_n(x)) where L(x) = x + x*A(x).
...
Let L = L(x) = x + x*A(x), then:
. A(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).

Extensions

Typos in formula and example corrected by Paul D. Hanna, Jul 28 2010

A179494 E.g.f. A(x) = G(x)/x where G(x) is the e.g.f. of A179493.

Original entry on oeis.org

1, 1, 4, 27, 284, 4110, 77424, 1818474, 51692080, 1738555344, 67979689200, 3047234077800, 154810558674144, 8829473686348848, 560819284547110848, 39398646866759606160, 3043158904460954177280, 257091879144869492997120
Offset: 0

Views

Author

Paul D. Hanna, Jul 23 2010

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 27*x^3/3! + 284*x^4/4! +...
x + x^2*A(x) = x + 2*x^2/2! + 6*x^3/3! + 48*x^4/4! + 540*x^5/5! +...

Crossrefs

Cf. A179493, A179421.

Programs

  • PARI
    {a(n)=local(A=[1,1]);for(i=2,n, A=concat(A,0);G=x*Ser(A);A[ #A]=polcoeff(1+subst(G,x,G)+O(x^#A)-(1+G)*deriv(G)*x^2/G^2,#A-1)/(#A-2));if(n<0,0,n!*A[n+1])}

Formula

Let G(x) denote the e.g.f. of A179493, then G(x) satisfies:
. L(x) = G(x)/(x*G'(x)) * L(G(x)) where L(x) = x + x*G(x); see A179493 for more formulas.
Let R = the Riordan array (A(x), x*A(x)), then the e.g.f. of column k in the matrix log of R equals (k+1)*(x + x^2*A(x)).

A351920 E.g.f. A(x) satisfies: A(x/A(x)) = A(x) / (A(x) - x*A(x)').

Original entry on oeis.org

1, 1, -2, 15, -200, 3920, -102924, 3424946, -139217280, 6733296720, -379945682400, 24634909864752, -1813131434947392, 149981854962931680, -13828816882622028000, 1411324560147609680400
Offset: 0

Views

Author

Paul D. Hanna, Feb 25 2022

Keywords

Comments

Given e.g.f. A(x) of this sequence, 1/A(x) equals the e.g.f. of A179421 - see entry A179421 for further formulas.

Examples

			E.g.f.: A(x) = 1 + x - 2*x^2/2! + 15*x^3/3! - 200*x^4/4! + 3920*x^5/5! - 102924*x^6/6! + 3424946*x^7/7! - 139217280*x^8/8! + ...
where A(x/A(x)) = A(x) / (A(x) - x*A(x)').
Related table.
Here we illustrate the related formula
a(n) = [x^n/n!] A(x) = [x^n/n!] A(x)^n (n >= 0).
The table of coefficients of x^k/k! in A(x)^n begins:
n=0: [1, 0,  0,   0,    0,    0,       0,       0, ...];
n=1: [1, 1, -2,  15, -200, 3920, -102924, 3424946, ...];
n=2: [1, 2, -2,  18, -256, 5240, -142308, 4869676, ...];
n=3: [1, 3,  0,  15, -240, 5220, -147672, 5212410, ...];
n=4: [1, 4,  4,  12, -200, 4640, -137016, 4992008, ...];
n=5: [1, 5, 10,  15, -160, 3920, -120420, 4521370, ...];
n=6: [1, 6, 18,  30, -120, 3240, -102924, 3970596, ...];
n=7: [1, 7, 28,  63,  -56, 2660,  -86688, 3424946, ...]; ...
in which the main diagonal equals this sequence and is found in row n = 1.
Related series.
Notice the relation to A179421, given by
1/A(x) = 1 - x + 4*x^2/2! - 33*x^3/3! + 440*x^4/4! - 8380*x^5/5! + 211824*x^6/6! - 6771422*x^7/7! + ... + A179421(n)*x^n/n! + ...
Also, the following composition of functions
A(x/A(x)) = 1 + x - 4*x^2/2! + 39*x^3/3! - 632*x^4/4! + 14620*x^5/5! - 445104*x^6/6! + 16958522*x^7/7! - 781426848*x^8/8! + ...
equals A(x) / (A(x) - x*A(x)'), as specified in the definition.

Crossrefs

Cf. A179421, A179420.

Programs

  • PARI
    {a(n) = my(A=[1,1]); for(m=1,n, A=concat(A,0);
A[#A] = -Vec(Ser(A)^(#A-1))[#A]/(#A-2) ); n!*A[n+1]}
for(n=0,12,print1(a(n),", "))

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies:
(1) A(x/A(x)) = A(x) / (A(x) - x*A(x)').
(2) a(n) = [x^n/n!] A(x) = [x^n/n!] A(x)^n, for n >= 0.
(3) A(x) = x*A'(x) / (1 - 1/A(x/A(x))).
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