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 1-3 of 3 results.

A144681 E.g.f. satisfies: A(x/A(x)) = exp(x).

Original entry on oeis.org

1, 1, 3, 22, 305, 6656, 204337, 8226436, 414585425, 25315924960, 1828704716801, 153433983789164, 14739472821255481, 1602471473448455104, 195300935112810494801, 26470100501608768436716
Offset: 0

Views

Author

Paul D. Hanna, Sep 19 2008

Keywords

Examples

			E.g.f. A(x) = 1 + x + 3*x^2/2! + 22*x^3/3! + 305*x^4/4! +...
A(x/A(x)) = 1 + x + x^2/2! + x^3/3! + x^4/4! + x^5/5! +...
1/A(x) = 1 + x - x^2/2! + 10*x^3/3! - 159*x^4/4! + 3816*x^5/5! -+...
A(log(A(x))) = 1 + x + 5*x^2/2! + 55*x^3/3! + 1005*x^4/4! + 26601*x^5/5! +...
ILLUSTRATE FORMULA a(n+1) = [x^n/n!] exp(x)*A(x)^(n+1) as follows.
Form a table of coefficients of x^k/k! in exp(x)*A(x)^n for n>=1, k>=0:
exp(x)*A(x)^1: [(1), 2, 6, 35, 416, 8437, 249340, ...];
exp(x)*A(x)^2: [1,(3), 13, 93, 1145, 22593, 645741, ...];
exp(x)*A(x)^3: [1, 4,(22), 181, 2320, 45199, 1257364, ...];
exp(x)*A(x)^4: [1, 5, 33,(305), 4097, 79825, 2177329, ...];
exp(x)*A(x)^5: [1, 6, 46, 471,(6656), 131001, 3529836, ...];
exp(x)*A(x)^6: [1, 7, 61, 685, 10201,(204337), 5477005, ...];
exp(x)*A(x)^7: [1, 8, 78, 953, 14960, 306643,(8226436), ...]; ...
then the terms along the main diagonal form this sequence shift left.

Links

Crossrefs

Cf. A144682, A144683, A144684, A087961, A140049.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(n=0,n,A=exp(serreverse(x/A)));n!*polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x+sum(k=2,n-1,a(k)*x^k/k!)+x*O(x^n));if(n==0,1,(n-1)!*polcoeff(exp(x+x*O(x^n))*A^n,n-1))}

Formula

E.g.f. satisfies: A(x) = exp( x*A(log A(x)) ).
E.g.f. satisfies: a(n+1) = [x^n/n!] exp(x)*A(x)^(n+1).
E.g.f: A(x) = G(2x)^(1/2) where G(x/G(x)^2) = exp(x) and G(x) is the e.g.f. of A144682.
E.g.f: A(x) = G(3x)^(1/3) where G(x/G(x)^3) = exp(x) and G(x) is the e.g.f. of A144683.
E.g.f: A(x) = G(4x)^(1/4) where G(x/G(x)^4) = exp(x) and G(x) is the e.g.f. of A144684.
E.g.f: A(x) = 1/G(-x) where G(x*G(x)) = exp(x) and G(x) is the e.g.f. of A087961.
E.g.f. A(log(A(x))) = log(A(x))/x = G(x) is the e.g.f of A140049 where G(x) satisfies G(x*exp(-x*G(x))) = exp(x*G(x)).

A144682 E.g.f. satisfies: A(x/A(x)^2) = exp(x).

Original entry on oeis.org

1, 1, 5, 73, 2073, 92481, 5775133, 471058953, 47961475313, 5904863932609, 858738633997941, 144899744540718729, 27970301202134146441, 6106540658691499524993, 1493749158085983126737165
Offset: 0

Views

Author

Paul D. Hanna, Sep 19 2008

Keywords

Examples

			E.g.f. A(x) = 1 + x + 5*x^2/2! + 73*x^3/3! + 2073*x^4/4! +...
ILLUSTRATE FORMULA a(n+1) = [x^n/n!] exp(x)*A(x)^(2*n+2) as follows.
Form a table of coefficients of x^k/k! in exp(x)*A(x)^(2*n) for n>=1, k>=0:
exp(x)*A(x)^2: [(1), 3, 17, 219, 5665, 239283, 14432433, ...];
exp(x)*A(x)^4: [1,(5), 41, 605, 15633, 638325, 37250233, ...];
exp(x)*A(x)^6: [1, 7,(73), 1207, 31825, 1274407, 72322201, ...];
exp(x)*A(x)^8: [1, 9, 113,(2073), 56545, 2249769, 124959057, ...];
exp(x)*A(x)^10:[1, 11, 161, 3251,(92481), 3695451, 202282081, ...];
exp(x)*A(x)^12:[1, 13, 217, 4789, 142705,(5775133), 313637833, ...];
exp(x)*A(x)^14:[1, 15, 281, 6735, 210673, 8688975,(471058953), ...]; ...
then the terms along the main diagonal form this sequence shift left.

Crossrefs

Cf. A144681, A144683, A144684.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(n=0,n,A=exp(serreverse(x/A^2)));n!*polcoeff(A,n)}
{a(n)=local(A=1+x+sum(k=2, n-1, a(k)*x^k/k!)+x*O(x^n)); if(n==0, 1, (n-1)!*polcoeff(exp(x+x*O(x^n))*A^(2*n), n-1))}

Formula

E.g.f. satisfies: A(x) = exp( x*A(log A(x))^2 ).
E.g.f. satisfies: a(n+1) = [x^n/n!] exp(x)*A(x)^(2*n+2).
E.g.f. A(x) = G(x/2)^2 where G(x/G(x)) = exp(x) and G(x) is the g.f. of A144681.

A144683 E.g.f. satisfies: A(x/A(x)^3) = exp(x).

Original entry on oeis.org

1, 1, 7, 154, 6625, 446776, 42088609, 5171653432, 792466370161, 146737621112464, 32079442602647521, 8134165156479090136, 2358873194743497162889, 773523517692799444058632, 284154419348084944647780289
Offset: 0

Views

Author

Paul D. Hanna, Sep 19 2008

Keywords

Examples

			E.g.f. A(x) = 1 + x + 7*x^2/2! + 154*x^3/3! + 6625*x^4/4! +...
ILLUSTRATE FORMULA a(n+1) = [x^n/n!] exp(x)*A(x)^(3*n+3) as follows.
Form a table of coefficients of x^k/k! in exp(x)*A(x)^(3*n) for n>=1, k>=0:
exp(x)*A(x)^3: [(1), 4, 34, 685, 27256, 1747159, 159049000, ...];
exp(x)*A(x)^6: [1, (7), 85, 1909, 75193, 4654417, 410053357, ...];
exp(x)*A(x)^9: [1, 10,(154), 3835, 153208, 9284725, 795356632, ...];
exp(x)*A(x)^12:[1, 13, 241, (6625), 272641, 16385713, 1373165425, ...];
exp(x)*A(x)^15:[1, 16, 346, 10441,(446776), 26918851, 2221660936, ...];
exp(x)*A(x)^18:[1, 19, 469, 15445, 690841, (42088609), 3443635405, ...];
exp(x)*A(x)^21:[1, 22, 610, 21799, 1022008, 63371617,(5171653432), ...]; ...
then the terms along the main diagonal form this sequence shift left.

Crossrefs

Cf. A144681, A144682, A144684.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(n=0,n,A=exp(serreverse(x/A^3)));n!*polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x+sum(k=2, n-1, a(k)*x^k/k!)+x*O(x^n)); if(n==0, 1, (n-1)!*polcoeff(exp(x+x*O(x^n))*A^(3*n), n-1))}

Formula

E.g.f. satisfies: A(x) = exp( x*A(log A(x))^3 ).
E.g.f. satisfies: a(n+1) = [x^n/n!] exp(x)*A(x)^(3*n+3).
E.g.f. A(x) = G(3x)^(1/3) where G(x/G(x)) = exp(x) and G(x) is the g.f. of A144681.
Showing 1-3 of 3 results.

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