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 1-5 of 5 results.

A088713 G.f. A(x) satisfies A(x/A(x)) = 1/(1-x).

Original entry on oeis.org

1, 1, 2, 6, 24, 118, 674, 4308, 30062, 225266, 1791964, 15009118, 131566314, 1201452248, 11389283418, 111761444078, 1132680800640, 11834071103246, 127261591139010, 1406778021294220, 15967144849210158, 185897394076705298
Offset: 0

Views

Author

Paul D. Hanna, Oct 12 2003

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 24*x^4 + 118*x^5 + 674*x^6 +...
Illustration of logarithmic derivation.
If we form an array of coefficients of x^k in A(x)^n, n>=1, like so:
A^1: [1],1,  2,   6,   24,   118,   674,    4308, ...;
A^2: [1, 2], 5,  16,   64,   308,  1716,   10724, ...;
A^3: [1, 3,  9], 31,  126,   600,  3278,   20070, ...;
A^4: [1, 4, 14,  52], 217,  1032,  5560,   33440, ...;
A^5: [1, 5, 20,  80,  345], 1651,  8820,   52270, ...;
A^6: [1, 6, 27, 116,  519,  2514],13385,   78420, ...;
A^7: [1, 7, 35, 161,  749,  3689, 19663], 114269, ...; ...
then the sums of the coefficients of x^k, k=0..n-1, in A(x)^n (shown above in brackets) begin:
1 = 1;
1 + 2 = 3;
1 + 3 +  9 = 13;
1 + 4 + 14 +  52 = 71;
1 + 5 + 20 +  80 +  345 = 451;
1 + 6 + 27 + 116 +  519 +  2514 = 3183;
1 + 7 + 35 + 161 +  749 +  3689 + 19663 = 24305; ...
and equal the coefficients in log(A(x)):
log(A(x)) = x + 3*x^2/2 + 13*x^3/3 + 71*x^4/4 + 451*x^5/5 + 3183*x^6/6 + 24305*x^7/7 + 197551*x^8/8 +...
The main diagonal in the above table forms the g.f. G(x) of A088714:
[1/1, 2/2, 9/3, 52/4, 345/5, 2514/6, 19663/7, ...]
where G(x) = 1 + x + 3*x^2 + 13*x^3 + 69*x^4 + 419*x^5 + 2809*x^6 +...
satisfies A'(x)/A(x) = (G(x) + x*G'(x)) / (1 - x*G(x)).

Links

Crossrefs

Cf. A088714.
Variants: A154677, A168448, A168449, A168478, A168479.

Programs

  • Mathematica
    terms = 22; A[] = 1; Do[A[x] = 1 + x*A[x]*A[1 - 1/A[x]] + O[x]^j // Normal, {j, terms}]; CoefficientList[A[x], x] (* Jean-François Alcover, Jan 15 2018 *)
  • PARI
    a(n)=local(A=1+x);for(i=1,n,A=(1+A*serreverse(x/(A+x*O(x^n))))^1);polcoeff(A,n)
for(n=0,30,print1(a(n),", ")) \\ Paul D. Hanna, Dec 06 2009
  • PARI
    {a(n)=local(A=1+x);if(n==0,1,for(i=1,n,
A=1+x*exp(sum(k=1,n-1,sum(j=0,k,polcoeff(A^k+x*O(x^j),j))*x^k/k)+x*O(x^n))));
polcoeff(A+x*O(x^n),n)}
for(n=0,30,print1(a(n),", ")) \\ Paul D. Hanna, Dec 09 2013

Formula

G.f. satisfies: A(x) = 1 + x*A(x)*A(1-1/A(x)).
G.f.: A(x*g(x)) = g(x) = (1-1/A(x))/x where g(x) is the g.f. of A088714.
From Paul D. Hanna, Dec 06 2009: (Start)
G.f. satisfies: A(x) = 1 + A(x)*Series_Reversion(x/A(x)).
G.f. satisfies: A( (x/(1+x)) / A(x/(1+x)) ) = 1 + x.
(End)
Logarithmic derivative: given g.f. A(x), let G(x) = A(x*G(x)) be the g.f. of A088714, then A'(x)/A(x) = (G(x) + x*G'(x)) / (1 - x*G(x)).

A154677 G.f. satisfies: A(x/A(x)) = G(x) where G(x) = 1 + x*G(x)^2 is the g.f. of A000108 (Catalan numbers).

Original entry on oeis.org

1, 1, 3, 13, 70, 440, 3116, 24274, 204407, 1836339, 17425275, 173329307, 1796783304, 19323703019, 214843877103, 2462522274426, 29032815570544, 351447240945518, 4361579736404011, 55424256247911490, 720399315622779670, 9569215299494074698, 129799982362958621827
Offset: 0

Views

Author

Paul D. Hanna, Jan 14 2009

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 70*x^4 + 440*x^5 + ... where
A(x/A(x)) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + ...
x/A(x) = x - x^2 - 2*x^3 - 8*x^4 - 43*x^5 - 277*x^6 - 2026*x^7 - ...

Links

Crossrefs

Cf. A000108.
Cf. variants: A168448, A168478. - Paul D. Hanna, Dec 06 2009

Programs

  • PARI
    {a(n)=local(A=1+x,F=sum(k=0,n,binomial(2*k+1,k)/(2*k+1)*x^k)+x*O(x^n)); for(i=0,n,A=subst(F,x,serreverse(x/(A+x*O(x^n)))));polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+A^2*serreverse(x/(A+x*O(x^n))));polcoeff(A,n)} \\ Paul D. Hanna, Dec 06 2009

Formula

G.f. satisfies: A( (x-x^2)/A(x-x^2) ) = 1/(1-x).
G.f. satisfies: A( (x/(1+x)^2)/A(x/(1+x)^2) ) = 1 + x.
G.f. satisfies: A(x) = 1 + A(x)^2*Series_Reversion(x/A(x)). - Paul D. Hanna, Dec 06 2009

A168478 G.f. satisfies: A(x/A(x)^3) = G(x) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 6, 60, 803, 13071, 244917, 5101603, 115451307, 2794682082, 71579132742, 1924722618873, 54022011952266, 1575777019075715, 47606721776494443, 1485688929610479498, 47790055655273649449, 1581727833458617151379
Offset: 0

Views

Author

Paul D. Hanna, Dec 06 2009

Keywords

Examples

			G.f.: A(x) = 1 + x + 6*x^2 + 60*x^3 + 803*x^4 + 13071*x^5 +...
A(x/A(x)^3) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...+ A001764(n)*x^n +...

Crossrefs

Cf. A168479 (cube), A168448 (variant), A001764.

Programs

  • PARI
    {a(n)=local(A=1+x, F=sum(k=0, n, binomial(3*k+1, k)/(3*k+1)*x^k)+x*O(x^n)); for(i=0, n, A=subst(F, x, serreverse(x/(A+x*O(x^n))^3))); polcoeff(A, n)}
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+A^3*serreverse(x/(A+x*O(x^n))^3)); polcoeff(A,n)}

Formula

G.f. satisfies: A(x) = 1 + A(x)^3*Series_Reversion[x/A(x)^3].
G.f. satisfies: A( (x*(1-x)^2)/A(x*(1-x)^2)^3 ) = 1/(1-x).
G.f. satisfies: A( (x/(1+x)^3)/A(x/(1+x)^3)^3 ) = 1 + x.

A168449 G.f. satisfies: A(x/A(x)) = C(x)^2 where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

Original entry on oeis.org

1, 2, 9, 60, 520, 5450, 65830, 886466, 13005906, 204607622, 3412713687, 59858823020, 1097439583778, 20934702108924, 414042879930671, 8466407067384676, 178587080601453990, 3878812336463745962
Offset: 0

Views

Author

Paul D. Hanna, Dec 06 2009

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 9*x^2 + 60*x^3 + 520*x^4 + 5450*x^5 +...
A(x/A(x)) = 1 + 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 132*x^5 + 429*x^6 +...

Crossrefs

Cf. A154677, A168448, A168479 (variant).

Programs

  • PARI
    {a(n)=local(A=1+x, F=sum(k=0, n, binomial(2*k+1, k)/(2*k+1)*x^k)+x*O(x^n)); for(i=0, n, A=subst(F^2, x, serreverse(x/(A+x*O(x^n))))); polcoeff(A, n)}
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=(1+A*serreverse(x/(A+x*O(x^n))))^2); polcoeff(A,n)}

Formula

G.f. satisfies: A(x) = [1 + A(x)*Series_Reversion(x/A(x))]^2.
G.f. satisfies: A( (x-x^2)/A(x-x^2) ) = 1/(1-x)^2.
G.f. satisfies: A( (x/(1+x)^2)/A(x/(1+x)^2)^2 ) = (1 + x)^2.
Self-convolution of A168448.

A168653 G.f. satisfies: A(x*A(x)) = G(x) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 2, 6, 21, 82, 340, 1478, 6622, 30433, 142331, 676203, 3248579, 15776459, 77196573, 380849394, 1888606247, 9430534212, 47236684433, 238214461960, 1202007809362, 6116704517639, 30997312336216, 159384351652358
Offset: 0

Views

Author

Paul D. Hanna, Dec 06 2009

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 21*x^4 + 82*x^5 + 340*x^6 +...
A(x*A(x)) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...

Crossrefs

Cf. A154677, A168478, A168448, A001764.

Programs

  • PARI
    {a(n)=local(A=1+x, F=sum(k=0, n, binomial(3*k+1, k)/(3*k+1)*x^k)+x*O(x^n)); for(i=0, n, A=subst(F, x, serreverse(x*(A+x*O(x^n))^1))); polcoeff(A, n)}
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+A^3*serreverse(x*(A+x*O(x^n)))); polcoeff(A,n)}

Formula

G.f. satisfies: A(x) = 1 + A(x)^3*Series_Reversion(x*A(x)).
G.f. satisfies: A( x*(1-x)^2*A(x*(1-x)^2) ) = 1/(1-x).
G.f. satisfies: A( (x/(1+x)^3)*A(x/(1+x)^3) ) = 1 + x.
Showing 1-5 of 5 results.

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