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.

A155202 G.f.: A(x) = exp( Sum_{n>=1} (2^n - 1)^n * x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 1, 5, 119, 12783, 5739069, 10426379903, 76135573607705, 2234839096465512877, 263966776643953756165279, 125532809982533901346598445525, 240383033223427436734891985275952307
Offset: 0

Views

Author

Paul D. Hanna, Feb 04 2009

Keywords

Comments

More generally, for m integer, exp( Sum_{n>=1} (m^n + y)^n * x^n/n ) is a power series in x and y with integer coefficients.

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 119*x^3 + 12783*x^4 + 5739069*x^5 +...
log(A(x)) = x + 3^2*x^2/2 + 7^3*x^3/3 + 15^4*x^4/4 + 31^5*x^5/5 +...

Crossrefs

Cf. A055601, A155200, A155202, A155810 (triangle), variants: A155205, A155209.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n+1,(2^m-1)^m*x^m/m)+x*O(x^n)),n)}

A155205 G.f.: A(x) = exp( Sum_{n>=1} (3^n - 1)^n * x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 2, 34, 5924, 10252294, 166020197708, 24810918565918804, 34076399079565985138408, 428687477154543524080261047622, 49247086840315416213775472777558582540
Offset: 0

Views

Author

Paul D. Hanna, Feb 04 2009

Keywords

Comments

More generally, for m integer, exp( Sum_{n>=1} (m^n + y)^n * x^n/n ) is a power series in x and y with integer coefficients.

Examples

			G.f.: A(x) = 1 + 2*x + 34*x^2 + 5924*x^3 + 10252294*x^4 +...
log(A(x)) = 2*x + 8^2*x^2/2 + 26^3*x^3/3 + 80^4*x^4/4 + 242^5*x^5/5 +...

Crossrefs

Cf. A060613, A155203, A155204, A155206, A155812 (triangle), variants: A155202, A155209.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n+1,(3^m-1)^m*x^m/m)+x*O(x^n)),n)}

A155207 G.f.: A(x) = exp( Sum_{n>=1} 4^(n^2) * x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 4, 136, 87904, 1074100576, 225184288253824, 787061981348092400896, 45273238870711805132010916864, 42535296046210357883346895894694749696, 649556283428320264374891976653586736162144180224
Offset: 0

Views

Author

Paul D. Hanna, Feb 04 2009

Keywords

Comments

More generally, for m integer, exp( Sum_{n>=1} m^(n^2) * x^n/n ) is a power series in x with integer coefficients.

Examples

			G.f.: A(x) = 1 + 4*x + 136*x^2 + 87904*x^3 + 1074100576*x^4 +...
log(A(x)) = 4*x + 4^4*x^2/2 + 4^9*x^3/3 + 4^16*x^4/4 + 4^25*x^5/5 +...

Crossrefs

Cf. A060757, A155208, A155209, A155210, variants: A155200, A155203.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n+1,4^(m^2)*x^m/m)+x*O(x^n)),n)}

Formula

G.f. satisfies: A'(x)/A(x) = 4 + 64*x*A'(16*x)/A(16*x). - Paul D. Hanna, Nov 15 2022
a(n) ~ 4^(n^2)/n. - Vaclav Kotesovec, Oct 31 2024

A155208 G.f.: A(x) = exp( Sum_{n>=1} (4^n + 1)^n * x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 5, 157, 92285, 1091087581, 226287110093405, 788215837483128170845, 45292586018794926904179045725, 42540488665745908362239138191829777245, 649578584556365450465861374646071307864262693725
Offset: 0

Views

Author

Paul D. Hanna, Feb 04 2009

Keywords

Comments

More generally, for m integer, exp( Sum_{n>=1} (m^n + y)^n * x^n/n ) is a power series in x and y with integer coefficients.

Examples

			G.f.: A(x) = 1 + 5*x + 157*x^2 + 92285*x^3 + 1091087581*x^4 +...
log(A(x)) = 5*x + 17^2*x^2/2 + 65^3*x^3/3 + 257^4*x^4/4 + 1025^5*x^5/5 +...

Crossrefs

Cf. A155207, A155209, A155210, variants: A155201, A155204.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n+1,(4^m+1)^m*x^m/m)+x*O(x^n)),n)}

A155210 G.f.: A(x) = exp( Sum_{n>=1} (4^n - 1)^n/3^(n-1) * x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 3, 42, 9378, 39179127, 2766569881269, 3234201150559172040, 62076685218110095082936700, 19446778350632942283719042004313725, 98999235365955012033013202024947235500115415
Offset: 0

Views

Author

Paul D. Hanna, Feb 04 2009

Keywords

Comments

More generally, for m integer, exp( Sum_{n>=1} (m^n - 1)^n/(m-1)^(n-1) * x^n/n ) is a power series in x with integer coefficients.
Note that g.f. exp( Sum_{n>=1} (4^n - 1)^n/3^n * x^n/n ) has fractional coefficients as a power series in x.

Examples

			G.f.: A(x) = 1 + 3*x + 42*x^2 + 9378*x^3 + 39179127*x^4 +...
log(A(x)) = 3*x + 15^2/3*x^2/2 + 63^3/3^2*x^3/3 + 255^4/3^3*x^4/4 +...

Crossrefs

Cf. A155207, A155208, A155209, variant: A155206.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n+1,(4^m-1)^m/3^(m-1)*x^m/m)+x*O(x^n)),n)}
Showing 1-5 of 5 results.

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