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 1-3 of 3 results.

A168407 E.g.f.: Sum_{n>=0} (exp(2^n*x) - 1)^n/n!, an analog of the Bell numbers (A000110).

Original entry on oeis.org

1, 2, 20, 712, 91920, 44874784, 85939843136, 660213878210688, 20540390859740217600, 2592165941692975372042752, 1324271564605167892188248409088, 2730585827960928853182474922961668096
Offset: 0

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Author

Paul D. Hanna, Nov 25 2009, Nov 25 2009, Feb 16 2010

Keywords

Examples

			E.g.f.: A(x) = 1 + 2*x + 20*x^2/2! + 712*x^3/3! + 91920*x^4/4! +...
A(x) = 1 + (exp(2*x) - 1) + (exp(4*x) - 1)^2/2! + (exp(8*x) - 1)^3/3! +...+ (exp(2^n*x) - 1)^n/n! +...
a(n) = coefficient of x^n/n! in Bell(x)^(2^n) where Bell(x) = exp(exp(x)-1):
Bell(x) = 1 + x + 2*x^2/2! + 5*x^3/3! + 15*x^4/4! + 52*x^5/5! + 203*x^6/6! +...+ A000110(n)*x^n/n! +...

Crossrefs

Cf. A000110, A168408, A277036, A008277.

Programs

  • Mathematica
    Table[BellB[n, 2^n], {n, 0, 15}] (* Vaclav Kotesovec, Jul 02 2025 *)
  • PARI
    {a(n)=local(infnty=n^4+10);round(exp(-2^n)*sum(k=0,infnty,(2^k*k)^n/k!))}
  • PARI
    {a(n)=n!*polcoeff(sum(k=0,n,(exp(2^k*x +x*O(x^n))-1)^k/k!),n)}
  • PARI
    {a(n)=n!*polcoeff(exp(2^n*(exp(x +x*O(x^n))-1)),n)}
  • PARI
    {S2(n,k)=(1/k!)*sum(i=0,k,(-1)^(k-i)*binomial(k,i)*i^n)}
{a(n)=sum(k=0,n,S2(n,k)*2^(n*k))}
  • PARI
    {a(n)=polcoeff(sum(k=0,n,(2^k*x)^k/prod(j=1,k,1-j*2^k*x+x*O(x^n))),n)}

Formula

a(n) = exp(-2^n) * Sum_{k>=0} (2^k*k)^n/k!.
a(n) = [x^n/n! ] B(x)^(2^n), where B(x) = exp(exp(x) - 1) is the e.g.f. of the Bell numbers.
a(n) = Sum_{k=0..n} S2(n,k) * 2^(n*k), where S2(n,k) = A008277(n,k) are the Stirling numbers of the second kind.
G.f.: A(x) = Sum_{n>=0} 2^(n^2) * x^n / [Product_{k=1..n} (1 - k*2^n*x)].
a(n) ~ 2^(n^2). - Vaclav Kotesovec, Jul 02 2025

A277037 G.f.: exp( Sum_{n>=1} [Sum_{k>=1} k^n * 2^(n*k) * x^k]^n / n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 2, 18, 484, 54630, 26638924, 53843811956, 442942117297000, 14725418961500037126, 1971239927985067569365772, 1060292226589575099894174194524, 2288290973515256950275126683431946552, 19795837218795604674370624304477542380054748, 685985356865646724678258830150265065104998427771576, 95174256167264272421248219248338459257647770713814222870312
Offset: 0

Views

Author

Paul D. Hanna, Oct 08 2016

Keywords

Comments

More generally, for fixed integer q, G(x,q) = exp( Sum_{n>=1} [Sum_{k>=1} k^n * q^(n*k) * x^k]^n / n ) is an integer series such that G(x,q) = exp( Sum_{n>=1} q^(n^2)/(1 - q^n*x)^(n^2+n) * [ Sum_{k=1..n} A008292(n,k) * q^(n*k-n) * x^k ]^n / n ), where A008292 are the Eulerian numbers.

Examples

			G.f.: A(x) = 1 + 2*x + 18*x^2 + 484*x^3 + 54630*x^4 + 26638924*x^5 + 53843811956*x^6 + 442942117297000*x^7 +...
such that the logarithm of g.f. A(x) equals the series:
log(A(x)) = Sum_{n>=1} (2^n*x + 2^n*2^(2*n)*x^2 + 3^n*2^(3*n)*x^3 +...+ k^n*2^(k*n)*x^k +...)^n/n.
This logarithmic series can be written using the Eulerian numbers like so:
log(A(x)) = 2*x/(1-2*x)^2 + 2^4*(x + 2^2*x^2)^2/(1-2^2*x)^6/2 + 2^9*(x + 4*2^3*x^2 + 2^6*x^3)^3/(1-2^3*x)^12/3 + 2^16*(x + 11*2^4*x^2 + 11*2^8*x^3 + 2^24*x^4)^4/(1-2^4*x)^20/4 + 2^25*(x + 26*2^5*x^2 + 66*2^10*x^3 + 26*2^15*x^4 + 2^20*x^5)^5/(1-2^5*x)^30/5 + 2^36*(x + 57*2^6*x^2 + 302*2^12*x^3 + 302*2^18*x^4 + 57*2^24*x^5 + 2^30*x^6)^6/(1-2^6*x)^42/6 +...+ [ Sum_{k=1..n} A008292(n,k) * 2^(n*k) * x^k ]^n / (1 - 2^n*x)^(n*(n+1))/n +...
Explicitly,
log(A(x)) = 2*x + 32*x^2/2 + 1352*x^3/3 + 214272*x^4/4 + 132616992*x^5/5 + 322738100480*x^6/6 + 3099838240135296*x^7/7 + 117796258487089512448*x^8/8 +...

Crossrefs

Cf. A277035, A156170, A277036, A008292.

Programs

  • PARI
    {a(n) = my(A=1,Oxn=x*O(x^n)); A = exp( sum(m=1,n+1, sum(k=1,n+1, k^m * 2^(m*k) * x^k +x*O(x^n) )^m / m )); polcoeff(A,n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {A008292(n, k) = sum(j=0, k, (-1)^j * (k-j)^n * binomial(n+1, j))}
{a(n) = my(A=1, Oxn=x*O(x^n)); A = exp( sum(m=1, n+1, sum(k=1, m, A008292(m, k) * 2^(m*k) * x^k / (1 - 2^m*x +Oxn)^(m+1) )^m / m ) ); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Formula

G.f.: exp( Sum_{n>=1} [ Sum_{k=1..n} A008292(n,k) * 2^(n*k) * x^k ]^n / (1 - 2^n*x)^(n*(n+1)) / n ), where A008292 are the Eulerian numbers.

A277035 G.f.: Sum_{n>=0} log(1 + Sum_{k>=1} k^n * 2^(n*k) * x^k )^n / n!, a power series in x with integer coefficients.

Original entry on oeis.org

1, 2, 14, 320, 21036, 4248736, 2753284608, 5889659239296, 42571528094271584, 1060699597956427433984, 92622122614950875482410496, 28732153964162783015337150191616, 32013867511269166370870196132112760832, 129287051721999031624124705228031781712207872, 1906782843976072893849368486957954962408685271556096
Offset: 0

Views

Author

Paul D. Hanna, Oct 08 2016

Keywords

Comments

More generally, for fixed integer q, G(x,q) = Sum_{n>=0} log(1 + Sum_{k>=1} k^n * q^(n*k) * x^k )^n / n! is an integer series such that G(x,q) = Sum_{n>=0} log(1 + [Sum_{k=1..n} A008292(n,k) * q^(n*k) * x^k]/(1 - q^n*x)^(n+1) )^n / n!, where A008292 are the Eulerian numbers.

Examples

			G.f.: A(x) = 1 + 2*x + 14*x^2 + 320*x^3 + 21036*x^4 + 4248736*x^5 + 2753284608*x^6 + 5889659239296*x^7 + 42571528094271584*x^8 +...
such that
A(x) = Sum_{n>=0} log(1 + 1^n*2^n*x + 2^n*2^(2*n)*x^2 + 3^n*2^(3*n)*x^3 +...+ k^n*2^(k*n)*x^k +...)^n/n!.
Equivalently,
A(x) = 1 + log(1 + 1*2*x + 2*2^2*x^2 + 3*2^3*x^3 + 4*2^4*x^4 +...) +
log(1 + 1^2*2^2*x + 2^2*2^4*x^2 + 3^2*2^6*x^3 + 4^2*2^8*x^4 +...)^2/2! +
log(1 + 1^3*2^3*x + 2^3*2^6*x^2 + 3^3*2^9*x^3 + 4^3*2^12*x^4 +...)^3/3! +
log(1 + 1^4*2^4*x + 2^4*2^8*x^2 + 3^4*2^12*x^3 + 4^4*2^16*x^4 +...)^4/4! +
...
The g.f. can be written using the Eulerian numbers like so:
A(x) = 1 + log(1 + 2*x/(1-2*x)^2) + log(1 + 2^2*(x + 2^2*x^2)/(1-2^2*x)^3)^2/2! + log(1 + 2^3*(x + 4*2^3*x^2 + 2^6*x^3)/(1-2^3*x)^4)^3/3! + log(1 + 2^4*(x + 11*2^4*x^2 + 11*2^8*x^3 + 2^24*x^4)/(1-2^4*x)^5)^4/4! + log(1 + 2^5*(x + 26*2^5*x^2 + 66*2^10*x^3 + 26*2^15*x^4 + 2^20*x^5)/(1-2^5*x)^6)^5/5! + log(1 + 2^6*(x + 57*2^6*x^2 + 302*2^12*x^3 + 302*2^18*x^4 + 57*2^24*x^5 + 2^30*x^6)/(1-2^6*x)^7)^6/6! +...+ log(1 + Sum_{k=1..n} A008292(n,k) * 2^(n*k) * x^k / (1 - 2^n*x)^(n+1) )^n/n! +...

Crossrefs

f. A277037, A277036, A008292.

Programs

  • PARI
    {a(n) = my(A=1, Oxn=x*O(x^n));
A = sum(m=0, n+1, log(1 + sum(k=1, n+1, k^m * 2^(m*k) * x^k +x*O(x^n)) )^m / m! );
polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {A008292(n, k) = sum(j=0, k, (-1)^j * (k-j)^n * binomial(n+1, j))}
{a(n) = my(A=1, Oxn=x*O(x^n));
A = sum(m=0, n+1, log(1 + sum(k=1, m+1, A008292(m,k) * 2^(m*k) * x^k) / (1 - 2^m*x +Oxn)^(m+1) )^m / m! );
polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Formula

G.f.: Sum_{n>=0} log(1 + [Sum_{k=1..n} A008292(n,k) * 2^(n*k) * x^k]/(1 - 2^n*x)^(n+1) )^n / n!, where A008292 are the Eulerian numbers.
Showing 1-3 of 3 results.

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