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.

A155203 G.f.: A(x) = exp( Sum_{n>=1} 3^(n^2) * x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 3, 45, 6687, 10782369, 169490304819, 25016281429306077, 34185693516532070487615, 429210580094546346191627404353, 49269611092414945570325157106493868771
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 + 3*x + 45*x^2 + 6687*x^3 + 10782369*x^4 + 169490304819*x^5 +...
log(A(x)) = 3*x + 3^4*x^2/2 + 3^9*x^3/3 + 3^16*x^4/4 + 3^25*x^5/5 +...

Crossrefs

Cf. A060722, A155204, A155205, A155206, A155812 (triangle), variants: A155200, A155207.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n+1,3^(m^2)*x^m/m)+x*O(x^n)),n)}

Formula

Equals column 0 of triangle A155812.
G.f. satisfies: A'(x)/A(x) = 3 + 27*x*A'(9*x)/A(9*x). - Paul D. Hanna, Nov 15 2022
a(n) ~ 3^(n^2)/n. - Vaclav Kotesovec, Oct 31 2024

A155201 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, 3, 17, 285, 21747, 7894143, 12593691755, 84961748935779, 2379148487805445513, 273416748863491468927893, 128009274688933686165252807225, 242979449433397149030644307317592609, 1863847996727745781866688849374488247858333, 57652096246331953203644653244501049018464175026133
Offset: 0

Views

Author

Paul D. Hanna, Feb 04 2009

Keywords

Comments

More generally, it appears that 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 + 3*x + 17*x^2 + 285*x^3 + 21747*x^4 + 7894143*x^5 +...
log(A(x)) = 3*x + 5^2*x^2/2 + 9^3*x^3/3 + 17^4*x^4/4 + 33^5*x^5/5 +...

Crossrefs

Cf. A136516, A155200, A155202, A155810 (triangle), variants: A155204, A155208.

Programs

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

Formula

Equals row sums of triangle A155810.
a(n) = (1/n)*Sum_{k=1..n} (2^k + 1)^k * a(n-k) for n>0, with a(0)=1.
a(n) = B_n( 0!*(2^1+1)^1, 1!*(2^2+1)^2, 2!*(2^3+1)^3, ..., (n-1)!*(2^n+1)^n ) / n!, where B_n() is the n-th complete Bell polynomial. - Max Alekseyev, Oct 10 2014

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)}

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)}

A155812 Triangle, read by rows, where g.f.: A(x,y) = exp( Sum_{n>=1} (3^n + y)^n * x^n/n ) is a power series in x and y with integer coefficients.

Original entry on oeis.org

1, 3, 1, 45, 12, 1, 6687, 801, 39, 1, 10782369, 540720, 10764, 120, 1, 169490304819, 3499254081, 29275956, 129348, 363, 1, 25016281429306077, 206071208583660, 709664882337, 1321144632, 1459773, 1092, 1, 34185693516532070487615
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,y) = 1 + (3 + y)x + (45 + 12y + y^2)x^2 + (6687 + 801y + 39y^2 + y^3)x^3 +...
Triangle begins:
1;
3, 1;
45, 12, 1;
6687, 801, 39, 1;
10782369, 540720, 10764, 120, 1;
169490304819, 3499254081, 29275956, 129348, 363, 1;
25016281429306077, 206071208583660, 709664882337, 1321144632, 1459773, 1092, 1;
34185693516532070487615, 109444624780070083617, 150302858159634327, 115097787387369, 53628299415, 15815241, 3279, 1; ...

Crossrefs

Cf. A155203 (column 0), A155204 (row sums), A155813 (column 1).

Programs

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

Formula

G.f.: A(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k)*x^n*y^k.

A155206 G.f.: A(x) = exp( Sum_{n>=1} (3^n - 1)^n/2^(n-1) * x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 2, 18, 1498, 1283090, 10377556482, 775351592888722, 532444511048570910746, 3349121447720205394546014978, 192371436319107536207473420480152034, 100642626897912335112447860229547933463000450
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} (3^n - 1)^n/2^n * x^n/n ) has fractional coefficients as a power series in x.

Examples

			G.f.: A(x) = 1 + 2*x + 18*x^2 + 1498*x^3 + 1283090*x^4 + 10377556482*x^5 +...
log(A(x)) = 2*x + 8^2/2*x^2/2 + 26^3/2^2*x^3/3 + 80^4/2^3*x^4/4 + 242^5/2^4*x^5/5 +...

Crossrefs

Cf. A155203, A155204, A155205, A155812 (triangle), variant: A155210.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n+1,(3^m-1)^m/2^(m-1)*x^m/m)+x*O(x^n)),n)}

A156911 G.f.: A(x) = exp( Sum_{n>=1} 3^(n^2)/(1 - 3^n*x)^n * x^n/n ).

Original entry on oeis.org

1, 3, 54, 7470, 11326446, 173007630594, 25222890606413004, 34295263336258106333292, 429734207324188407742780371030, 49292144072318945019920850119049478578
Offset: 0

Views

Author

Paul D. Hanna, Feb 17 2009

Keywords

Comments

An example of this logarithmic identity at q=3:
Sum_{n>=1} [q^(n^2)/(1 - q^n*x)^n]*x^n/n = Sum_{n>=1} [(1 + q^n)^n - 1]*x^n/n.

Examples

			G.f.: A(x) = 1 + 3*x + 54*x^2 + 7470*x^3 + 11326446*x^4 +...
Log(A(x)) = 3/(1-3*x)*x + 3^4/(1-3^2*x)^2*x^2/2 + 3^9/(1-3^3*x)^3*x^3/3 +...
Log(A(x)) = (4-1)*x + (10^2-1)*x^2/2 + (28^3-1)*x^3/3 + (82^4-1)*x^4/4 +...

Crossrefs

Cf. A155204, A155203, A155205, A156910.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n+1, 3^(m^2)/(1-3^m*x)^m*x^m/m)+x*O(x^n)), n)}
  • PARI
    /* As First Differences of A155204: */
{a(n)=polcoeff((1-x)*exp(sum(m=1, n+1, (3^m+1)^m*x^m/m)+x*O(x^n)), n)}

Formula

G.f.: A(x) = (1-x)*exp( Sum_{n>=1} (1 + 3^n)^n * x^n/n );
Equals the first differences of A155204.
Showing 1-7 of 7 results.

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