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.

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

A155204 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, 4, 58, 7528, 11333974, 173018964568, 25223063625377572, 34295288559321731710864, 429734241619476967064512081894, 49292144502053186639397817183561560472
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 + 4*x + 58*x^2 + 7528*x^3 + 11333974*x^4 + 173018964568*x^5 +...
log(A(x)) = 4*x + 10^2*x^2/2 + 28*x^3/3 + 82^4*x^4/4 + 244^5*x^5/5 +...
		

Crossrefs

Cf. A155203, A155205, A155206, A155812 (triangle), A202989; variants: A155201, A155208.

Programs

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

Formula

Equals row sums of triangle A155812.

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

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

A155209 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, 3, 117, 83691, 1057319541, 224085796087563, 785909534807110163445, 45253898808490419883694669835, 42530103981310660908750359650219091445, 649533982980850199063905669772208004250784346635
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 + 3*x + 117*x^2 + 83691*x^3 + 1057319541*x^4 +...
log(A(x)) = 3*x + 15^2*x^2/2 + 63^3*x^3/3 + 255^4*x^4/4 + 1023^5*x^5/5 +...
		

Crossrefs

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.