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.

A202989 E.g.f: Sum_{n>=0} 3^(n^2) * exp(3^n*x) * x^n/n!.

Original entry on oeis.org

1, 4, 100, 21952, 45212176, 864866612224, 151334226289000000, 240066313618039143841792, 3437872835498096514323500400896, 443629285048033016198674962874808664064, 515464807019389919369209932597753906250000000000
Offset: 0

Views

Author

Paul D. Hanna, Dec 26 2011

Keywords

Comments

E.g.f. series identity: Sum_{n>=0} m^n * q^(n^2) * exp(b*q^n*x) * x^n/n! = Sum_{n>=0} (m*q^n + b)^n * x^n/n! for all q, m, b.
O.g.f. series identity: Sum_{n>=0} m^n * q^(n^2) * x^n/(1-b*q^n*x)^(n+1) = Sum_{n>=0} (m*q^n + b)^n * x^n for all q, m, b.

Examples

			E.g.f.: A(x) = 1 + 4*x + 100*x^2/2! + 21952*x^3/3! + 45212176*x^4/4! +..
By the series identity, the e.g.f.:
A(x) = exp(x) + 3*exp(3*x)*x + 3^4*exp(3^2*x)*x^2/2! + 3^9*exp(3^3*x)*x^3/3! +...
expands into:
A(x) = 1 + 4*x + 10^2*x^2/2! + 28^3*x^3/3! + 82^4*x^4/4! + 244^5*x^5/5! +...+ (3^n+1)^n*x^n/n! +...

Crossrefs

Cf. A165327, A060613, A244004, A136516, A326013.

Programs

  • PARI
    {a(n, q=3, m=1, b=1)=(m*q^n + b)^n}
  • PARI
    {a(n, q=3, m=1, b=1)=n!*polcoeff(sum(k=0, n, m^k*q^(k^2)*exp(b*q^k*x+x*O(x^n))*x^k/k!), n)}
  • PARI
    {a(n, q=3, m=1, b=1)=polcoeff(sum(k=0,n,m^k*q^(k^2)*x^k/(1-b*q^k*x +x*O(x^n))^(k+1)),n)}

Formula

a(n) = (3^n + 1)^n.
O.g.f.: Sum_{n>=0} 3^(n^2) * x^n/(1 - 3^n*x)^(n+1).

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)}

A241095 a(n) = (5^n - 1)^n.

Original entry on oeis.org

1, 4, 576, 1906624, 151613669376, 297546691796890624, 14546328186912540283109376, 17761976839391142146587243652890624, 542089984111981439129333008468155364987109376, 413588400456285638417956135979678948104381423950212890624
Offset: 0

Views

Author

Hannah Ko, Apr 15 2014

Keywords

Comments

Number of n X n {-2,-1,0,1,2} matrices with no zero rows.

Crossrefs

Cf. A055601, A060613, A240091.

Programs

  • Mathematica
    Join[{1},Table[(5^n-1)^n,{n,10}]] (* Harvey P. Dale, May 24 2023 *)

Extensions

More terms from Alois P. Heinz, Apr 23 2014

A241098 (4^n - 1)^n.

Original entry on oeis.org

1, 3, 225, 250047, 4228250625, 1120413075641343, 4715453174592516890625, 316777275155162685106909462527, 340240830764391036687105719527812890625, 5845805845679338940092384222659294205213252255743, 1606922719369349023077959288770050490334792256259918212890625
Offset: 0

Views

Author

Hannah Ko, Apr 15 2014

Keywords

Examples

			a(3) = 63^3 = 250047.

Crossrefs

Cf. A055601, A060613, A241095, A240091.

Programs

Formula

(4^n - 1)^n

A202990 E.g.f: Sum_{n>=0} 3^n * 2^(n^2) * exp(-2*2^n*x) * x^n/n!.

Original entry on oeis.org

1, 4, 100, 10648, 4477456, 7339040224, 47045881000000, 1186980379913527168, 118530511097526559703296, 47035767668340696232372862464, 74367598058372171073462490000000000, 469253945833810205185008441288962454059008
Offset: 0

Views

Author

Paul D. Hanna, Dec 26 2011

Keywords

Comments

E.g.f. series identity: Sum_{n>=0} m^n * q^(n^2) * exp(b*q^n*x) * x^n/n! = Sum_{n>=0} (m*q^n + b)^n * x^n/n! for all q, m, b.
O.g.f. series identity: Sum_{n>=0} m^n * q^(n^2) * x^n/(1-b*q^n*x)^(n+1) = Sum_{n>=0} (m*q^n + b)^n * x^n for all q, m, b.

Examples

			E.g.f.: A(x) = 1 + 4*x + 100*x^2/2! + 10648*x^3/3! + 4477456*x^4/4! +..
By the series identity, the e.g.f.:
A(x) = exp(-2*x) + 3*2*exp(-2*2*x)*x + 3^2*2^4*exp(-2*2^2*x)*x^2/2! + 3^3*2^9*exp(-2*2^3*x)*x^3/3! +...
expands into:
A(x) = 1 + 4*x + 10^2*x^2/2! + 22^3*x^3/3! + 46^4*x^4/4! + 94^5*x^5/5! +...+ (3*2^n-2)^n*x^n/n! +...

Crossrefs

Cf. A180602, A165327, A202989, A060613, A055601.

Programs

  • Mathematica
    Table[(3*2^n-2)^n,{n,0,12}] (* Harvey P. Dale, Jul 16 2023 *)
  • PARI
    {a(n, q=2, m=3, b=-2)=(m*q^n + b)^n}
  • PARI
    {a(n, q=2, m=3, b=-2)=n!*polcoeff(sum(k=0, n, m^k*q^(k^2)*exp(b*q^k*x+x*O(x^n))*x^k/k!), n)}
  • PARI
    {a(n, q=2, m=3, b=-2)=polcoeff(sum(k=0, n, m^k*q^(k^2)*x^k/(1-b*q^k*x+x*O(x^n))^(k+1)), n)}

Formula

a(n) = (3*2^n - 2)^n.
O.g.f.: Sum_{n>=0} 3^n * 2^(n^2) * x^n/(1 + 2*2^n*x)^(n+1).

A234811 (6^n - 1)^n.

Original entry on oeis.org

1, 5, 1225, 9938375, 2812412850625, 28412011938974609375, 10313098426045900054366890625, 134710177671603826682045331123397109375, 63339984974231689005132970549727976493235812890625, 1072138503990252055371856714088806945958716851785209488787109375
Offset: 0

Views

Author

Hannah Ko, Apr 19 2014

Keywords

Examples

			a(3) = 215^3 = 9938375.

Crossrefs

Cf. A055601, A060613, A240091, A241095, A241098.

Programs

A202991 E.g.f: Sum_{n>=0} 3^(n^2) * exp(-2*3^n*x) * x^n/n!.

Original entry on oeis.org

1, 1, 49, 15625, 38950081, 812990017201, 147640825624179889, 237771659632917369765625, 3425319186561140076700951192321, 443021141828981570872668681812345111521, 515202988063835984513918825523304657054713360049
Offset: 0

Views

Author

Paul D. Hanna, Dec 26 2011

Keywords

Comments

E.g.f. series identity: Sum_{n>=0} m^n * q^(n^2) * exp(b*q^n*x) * x^n/n! = Sum_{n>=0} (m*q^n + b)^n * x^n/n! for all q, m, b.
O.g.f. series identity: Sum_{n>=0} m^n * q^(n^2) * x^n/(1-b*q^n*x)^(n+1) = Sum_{n>=0} (m*q^n + b)^n * x^n for all q, m, b.

Examples

			E.g.f.: A(x) = 1 + x + 49*x^2/2! + 15625*x^3/3! + 38950081*x^4/4! +...
By the series identity, the g.f.:
A(x) = exp(-2*x) + 3*exp(-2*3*x)*x + 3^4*exp(-2*3^2*x)*x^2/2! + 3^9*exp(-2*3^3*x)*x^3/3! + 3^16*exp(-2*3^4*x)*x^4/4! +...
expands into:
A(x) = 1 + x + 7^2*x^2/2! + 25^3*x^3/3! + 79^4*x^4/4! + 241^5*x^5/5! +...+ (3^n-2)^n*x^n/n! +...

Crossrefs

Cf. A180602, A165327, A202990, A202989, A060613, A055601.

Programs

  • PARI
    {a(n, q=3, m=1, b=-2)=(m*q^n + b)^n}
  • PARI
    {a(n, q=3, m=1, b=-2)=n!*polcoeff(sum(k=0, n, m^k*q^(k^2)*exp(b*q^k*x+x*O(x^n))*x^k/k!), n)}
  • PARI
    {a(n, q=3, m=1, b=-2)=polcoeff(sum(k=0, n, m^k*q^(k^2)*x^k/(1-b*q^k*x+x*O(x^n))^(k+1)), n)}

Formula

a(n) = (3^n - 2)^n.
O.g.f.: Sum_{n>=0} 3^(n^2)*x^n/(1 + 2*3^n*x)^(n+1).
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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