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.

Previous Showing 11-16 of 16 results.

A261885 E.g.f. satisfies: A(x) = A( x/(1-x)^3 ) * (1-x)/(1+2*x) with A(0)=0.

Original entry on oeis.org

0, 3, -6, 33, -288, 3240, -41328, 538776, -6772608, 108502848, -3985200000, 160423053120, -1094935449600, -334098799856640, 6171528273039360, 2583214608931891200, -150276838897808179200, -27910708844766496358400, 3428432556793417713254400, 462153194693572097337753600, -107869378361311784037187584000
Offset: 0

Views

Author

Paul D. Hanna, Sep 04 2015

Keywords

Examples

			E.g.f.: A(x) = 3*x - 6*x^2/2! + 33*x^3/3! - 288*x^4/4! + 3240*x^5/5! - 41328*x^6/6! + 538776*x^7/7! - 6772608*x^8/8! + 108502848*x^9/9! - 3985200000*x^10/10! +...
such that
A(x/(1-x)^3) = 3*x + 12*x^2/2! + 33*x^3/3! + 108*x^4/4! + 900*x^5/5! + 2952*x^6/6! - 19152*x^7/7! + 1694592*x^8/8! + 1847232*x^9/9! - 1796670720*x^10/10! +...
where
A(x/(1-x)^3) = (1+2*x)/(1-x)*A(x).

Crossrefs

Cf. A261886, A179320, A179420, A179330.

Programs

  • PARI
    /* E.g.f. satisfies: A(x) = (1-x)/(1+2*x)*A(x/(1-x)^3): */
{a(n)=local(A=3*x, B); for(m=2, n, B=(1-x)/(1+2*x+O(x^(n+3)))*subst(A, x, x/(1-x+O(x^(n+3)))^3); A=A-polcoeff(B, m+1)*x^m/(m-1)/3); n!*polcoeff(A, n)}
for(n=0,20,print1(a(n),", "))
  • PARI
    /* 1/(1-x)^3 = 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/(1-x+x*O(x^n))^3+1+sum(m=1, n, (D=A*deriv(x*D+x*O(x^n)))/m!); -n!*polcoeff(R, n)}
for(n=0,20,print1(a(n),", "))

Formula

E.g.f. A = A(x) satisfies:
(1) 1/(1-x)^3 = 1 + A + A*Dx(A)/2! + A*Dx(A*Dx(A))/3! + A*Dx(A*Dx(A*Dx(A)))/4! +...
(2) G(-x)^3 = 1 - A + A*Dx(A)/2! - A*Dx(A*Dx(A))/3! + A*Dx(A*Dx(A*Dx(A)))/4! -+..., where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.
(3) (1-x)^6/((1-x)^3 - x)^3 = 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! +...
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

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

Views

Author

Paul D. Hanna, Jul 15 2011

Keywords

Examples

			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! +...

Crossrefs

Cf. A001028, A193099, A193100, A179420.

Programs

  • PARI
    {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

Views

Author

Paul D. Hanna, Jul 15 2011

Keywords

Examples

			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! +...

Crossrefs

Cf. A001028, A193098, A193100, A179420.

Programs

  • PARI
    {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)}

A186632 E.g.f. satisfies: A(A(A(x))) = d/dx A(x)^2/2 with A(0)=0, A'(0)=1.

Original entry on oeis.org

1, 2, 24, 588, 22560, 1201320, 82875744, 7085373456, 727827579648, 87810122446560, 12227912455651200, 1938610503725962176, 346053263639068417536, 68919212804888587849344
Offset: 1

Views

Author

Paul D. Hanna, Feb 24 2011

Keywords

Examples

			E.g.f: A(x) = x + 2*x^2/2! + 24*x^3/3! + 588*x^4/4! + 22560*x^5/5! +...
A(x)^2/2 = x^2/2! + 6*x^3/3! + 108*x^4/4! + 3420*x^5/5! + 158760*x^6/6! +...
Iterations of the e.g.f. begin:
A(A(x)) = x + 4*x^2/2! + 60*x^3/3! + 1680*x^4/4! + 70920*x^5/5! +...
A(A(A(x))) = x + 6*x^2/2! + 108*x^3/3! + 3420*x^4/4! + 158760*x^5/5! +...
A(A(A(A(x)))) = x + 8*x^2/2! + 168*x^3/3! + 5952*x^4/4! + 302640*x^5/5! +...
A(A(A(A(A(x))))) = x + 10*x^2/2! + 240*x^3/3! + 9420*x^4/4! + 522000*x^5/5! +...
Related expansions:
A'(A(x)) = 1 + 2*x + 28*x^2/2! + 780*x^3/3! + 33384*x^4/4! + 1956120*x^5/5! +...
A'(A(A(x))) = 1 + 2*x + 32*x^2/2! + 996*x^3/3! + 46944*x^4/4! + 2998680*x^5/5! +...

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,subst(A,x,A))-deriv(A^2/2), n)/(n-2))}

Formula

E.g.f. satisfies: A(A(A(x))) = A'(x)*A(x).
E.g.f. satisfies: A(A(A(A(x)))) = A'(A(x))*A(A(x)) = A( A'(x)*A(x) ).
E.g.f. satisfies: A(A(A(A(A(x))))) = A'(A(A(x)))*A'(x)*A(x).
n(n-1) divides a(n) for n>=2.

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))).
Previous Showing 11-16 of 16 results.

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