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.

A136516 a(n) = (2^n+1)^n.

Original entry on oeis.org

1, 3, 25, 729, 83521, 39135393, 75418890625, 594467302491009, 19031147999601100801, 2460686496619787545743873, 1280084544196357822418212890625, 2672769719437237714909813214827010049, 22366167213460480200139104627873703828439041
Offset: 0

Views

Author

Paul D. Hanna, Jan 02 2008

Keywords

Comments

More generally, 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.
Main diagonal of A264871. - Omar E. Pol, Nov 27 2015

Examples

			A(x) = 1 + 3x + 5^2*x^2/2! + 9^3*x^3/3! + 17^4*x^4/4! +... + (2^n+1)^n*x^n/n! +...
A(x) = exp(x) + 2*exp(2x) + 2^4*exp(4x)*x^2/2! + 2^9*exp(8x)*x^3/3! +...+ 2^(n^2)*exp(2^n*x)*x^n/n! +...
This is a special case of the more general statement:
Sum_{n>=0} m^n * F(q^n*x)^b * log( F(q^n*x) )^n / n! = Sum_{n>=0} x^n * [y^n] F(y)^(m*q^n + b) where F(x) = exp(x), q=2, m=1, b=1.

Links

Crossrefs

Cf. A014050, A055601, A243918, A202989, A326011, A326012.

Programs

  • Magma
    [(2^n+1)^n: n in [0..45]]; // Vincenzo Librandi, Apr 21 2011
  • Maple
    seq((2^n+1)^n, n=0..30); # Robert Israel, Nov 27 2015
  • Mathematica
    Table[(2^n+1)^n,{n,0,16}] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011*)
  • PARI
    a(n)=polcoeff(sum(k=0,n,2^(k^2)*exp(2^k*x)*x^k/k!),n)
  • PARI
    {a(n)=polcoeff(sum(k=0, n, 2^(k^2)*x^k/(1-2^k*x +x*O(x^n))^(k+1)), n)} \\ Paul D. Hanna, Sep 15 2009

Formula

E.g.f.: A(x) = Sum_{n>=0} 2^(n^2) * exp(2^n*x) * x^n/n!.
O.g.f.: Sum_{n>=0} 2^(n^2)*x^n/(1 - 2^n*x)^(n+1) = Sum_{n>=0} (2^n+1)^n*x^n. [Paul D. Hanna, Sep 15 2009]
a(n) = 2^(n^2) + n 2^(n^2-n) + O(n^2 2^(n^2-2n)). - Robert Israel, Nov 27 2015

A060613 Number of n X n {-1,0,1} matrices with no zero rows.

Original entry on oeis.org

2, 64, 17576, 40960000, 829997587232, 148863517207035904, 238534446168822298080896, 3429499272008000182681600000000, 443223773846454955204927262062339154432
Offset: 1

Views

Author

Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 25 2001

Keywords

Links

Crossrefs

Cf. A055601, A202989.

Programs

  • PARI
    a(n)={(3^n - 1)^n} \\ Harry J. Smith, Jul 08 2009
  • PARI
    {a(n, q=3, m=1, b=-1)=(m*q^n + b)^n} \\ Paul D. Hanna, Dec 26 2011
  • PARI
    /* E.g.f. series identity: */
{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)} \\ Paul D. Hanna, Dec 26 2011
  • PARI
    /* O.g.f. series identity: */
{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)} \\ Paul D. Hanna, Dec 26 2011

Formula

a(n) = (3^n - 1)^n.
E.g.f.: Sum_{n>=0} 3^(n^2) * exp(-3^n*x) * x^n/n!. - Paul D. Hanna, Dec 26 2011
O.g.f.: Sum_{n>=0} 3^(n^2) * x^n/(1+3^n*x)^(n+1). - Paul D. Hanna, Dec 26 2011

Extensions

More terms from Harry J. Smith, Jul 08 2009

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.

A244004 a(n) = Sum_{k=0..n} binomial(n,k) * (1 + 3^k)^k.

Original entry on oeis.org

1, 5, 109, 22265, 45300601, 865092893645, 151339416167296549, 240067372975786948895105, 3437874756032842865408439504241, 443629315988897178081546025182658298645, 515464811455682924553846499988261249729439172189
Offset: 0

Views

Author

Paul D. Hanna, Jun 17 2014

Keywords

Examples

			O.g.f.: A(x) = 1 + 5*x + 109*x^2 + 22265*x^3 + 45300601*x^4 +...
where the g.f. may be expressed by the series identity:
A(x) = 1/(1-x) + 4*x/(1-x)^2 + 10^2*x^2/(1-x)^3 + 28^3*x^3/(1-x)^4 + 82^4*x^4/(1-x)^5 + 244^5*x^5/(1-x)^6 + 730^6*x^6/(1-x)^7 +...
A(x) = 1/(1-2*x) + 3*x/(1-4*x)^2 + 3^4*x^2/(1-10*x)^3 + 3^9*x^3/(1-28*x)^4 + 3^16*x^4/(1-82*x)^5 + 3^25*x^5/(1-244*x)^6 + 3^36*x^6/(1-730*x)^7 +...
Illustration of initial terms:
a(0) = 1;
a(1) = 1 + (1+3);
a(2) = 1 + 2*(1+3) + (1+3^2)^2;
a(3) = 1 + 3*(1+3) + 3*(1+3^2)^2 + (1+3^3)^3;
a(4) = 1 + 4*(1+3) + 6*(1+3^2)^2 + 4*(1+3^3)^3 + (1+3^4)^4;
a(5) = 1 + 5*(1+3) + 10*(1+3^2)^2 + 10*(1+3^3)^3 + 5*(1+3^4)^4 + (1+3^5)^5; ...
Also, by a binomial identity we have
a(0) = 1;
a(1) = 2 + 3;
a(2) = 2^2 + 2*(1+3)*3 + 3^4;
a(3) = 2^3 + 3*(1+3)^2*3 + 3*(1+3^2)*3^4 + 3^9;
a(4) = 2^4 + 4*(1+3)^3*3 + 6*(1+3^2)^2*3^4 + 4*(1+3^3)*3^9 + 3^16;
a(5) = 2^5 + 5*(1+3)^4*3 + 10*(1+3^2)^3*3^4 + 10*(1+3^3)^2*3^9 + 5*(1+3^4)*3^16 + 3^25; ...

Crossrefs

Cf. A243918, A202989.

Programs

  • Mathematica
    Table[Sum[Binomial[n,k]*(1+3^k)^k,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jun 18 2014 *)
  • PARI
    {a(n)=sum(k=0, n, binomial(n, k)*(1+3^k)^k)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n)=sum(k=0, n, binomial(n, k)*(1+3^k)^(n-k)*3^(k^2))}
for(n=0, 20, print1(a(n), ", "))

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * (1 + 3^k)^(n-k) * 3^(k^2).
O.g.f.: Sum_{n>=0} (1 + 3^n)^n * x^n / (1-x)^(n+1).
O.g.f.: Sum_{n>=0} 3^(n^2) * x^n / (1 - (1+3^n)*x)^(n+1).
E.g.f.: exp(x) * Sum_{n>=0} (1 + 3^n)^n * x^n / n!.
a(n) ~ 3^(n^2). - Vaclav Kotesovec, Jun 18 2014

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).

A326013 a(n) = (n+1) * (3^n + 1)^n.

Original entry on oeis.org

1, 8, 300, 87808, 226060880, 5189199673344, 1059339584023000000, 1920530508944313150734336, 30940855519482868628911503608064, 4436292850480330161986749628748086640640, 5670112877213289113061309258575292968750000000000, 64696388003897763308476146732369244462492297181932629786624
Offset: 0

Views

Author

Paul D. Hanna, Jun 05 2019

Keywords

Comments

More generally, the following sums are equal:
(1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n,
(2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - p*q^n*r)^(n+k),
for any fixed integer k; here, k = 2 and p = 1, q = 3, r = x.

Examples

			O.g.f.: A(x) = 1 + 8*x + 300*x^2 + 87808*x^3 + 226060880*x^4 + 5189199673344*x^5 + 1059339584023000000*x^6 + 1920530508944313150734336*x^7 + ... + (n+1) * (3^n + 1)^n*x^n + ...
such that
A(x) = 1/(1 - x)^2 + 2*3*x/(1 - 3*x)^3 + 3*3^4*x^2/(1 - 3^2*x)^4 + 4*3^9*x^3/(1 - 3^3*x)^5 + 5*3^16*x^4/(1 - 3^4*x)^6 + 6*3^25*x^5/(1 - 3^5*x)^7 + 7*3^36*x^6/(1 - 3^6*x)^8 + ... (n+1)*3^(n^2)*x^n/(1 - 3^n*x)^(n+2) + ...

Crossrefs

Cf. A202989.

Programs

  • PARI
    {a(n) = (n+1) * (3^n + 1)^n}
for(n=0,15, print1(a(n),", "))
  • PARI
    /* O.g.f. */
{a(n) = my(A = sum(m=0,n, (m+1) * 3^(m^2) * x^m / (1 - 3^m*x +x*O(x^n))^(m+2) )); polcoeff(A,n)}
for(n=0,15, print1(a(n),", "))
  • PARI
    /* E.g.f. */
{a(n) = my(A = sum(m=0,n, (m+1 + 3^m*x) * 3^(m^2) * exp(3^m*x +x*O(x^n)) * x^m/m! )); n!*polcoeff(A,n)}
for(n=0,15, print1(a(n),", "))

Formula

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

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.

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