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.

A168408 E.g.f.: Sum_{n>=0} (exp(3^n*x) - 1)^n/n!, an analog of the Bell numbers (A000110).

Original entry on oeis.org

1, 3, 90, 21897, 46281375, 882516062106, 153201395082609531, 241604140428719375972139, 3448358784659838019970862469260, 444238039567848645977924129826080612043
Offset: 0

Views

Author

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

Keywords

Examples

			E.g.f.: A(x) = 1 + 3*x + 90*x^2/2! + 21897*x^3/3! + 46281375*x^4/4! +...
A(x) = 1 + (exp(3*x) - 1) + (exp(9*x) - 1)^2/2! + (exp(27*x) - 1)^3/3! +...+ (exp(3^n*x) - 1)^n/n! +...
a(n) = coefficient of x^n/n! in Bell(x)^(3^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, A168407.
Cf. A008277.

Programs

  • Mathematica
    Table[BellB[n, 3^n], {n, 0, 15}] (* Vaclav Kotesovec, Jul 02 2025 *)
  • PARI
    {a(n)=local(infnty=n^5+10);round(exp(-3^n)*sum(k=0,infnty,(3^k*k)^n/k!))}
  • PARI
    {a(n)=n!*polcoeff(sum(k=0,n,(exp(3^k*x +x*O(x^n))-1)^k/k!),n)}
  • PARI
    {a(n)=n!*polcoeff(exp(3^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)*3^(n*k))} \\ Paul D. Hanna, Feb 15 2010
  • PARI
    {a(n)=polcoeff(sum(k=0,n,(3^k*x)^k/prod(j=1,k,1-j*3^k*x+x*O(x^n))),n)}

Formula

a(n) = exp(-3^n)*Sum_{k>=0} (3^k*k)^n/k!.
a(n) = [x^n/n! ] Bell(x)^(3^n) where Bell(x) = exp(exp(x) - 1) is the e.g.f. of the Bell numbers.
a(n) = Sum_{k=0..n} S2(n,k)*3^(n*k), where S2(n,k) = A008277(n,k) are the Stirling numbers of the second kind.
G.f.: A(x) = Sum_{n>=0} 3^(n^2)*x^n/[Product_{k=1..n} (1-k*3^n*x)].
a(n) ~ 3^(n^2). - Vaclav Kotesovec, Jul 02 2025

A277036 G.f.: Sum_{n>=0} exp(-n * 2^n * x) * [ Sum_{k>=1} k^n * 2^(n*k) * x^k / k! ]^n.

Original entry on oeis.org

1, 2, 16, 640, 102656, 63897600, 154597064704, 1463095704682496, 54479037904873062400, 8016231806154061580861440, 4675328432258454936484990418944, 10830326782491721013522399339743281152, 99782643106894570834269165391541758337220608, 3659836060539105945122413831815090863199825623515136, 534751190090057629985959636400471838795213939324687126364160
Offset: 0

Views

Author

Paul D. Hanna, Oct 08 2016

Keywords

Comments

More generally, for fixed integer q, G(x,q) = Sum_{n>=0} exp(-n * q^n * x) * [ Sum_{k>=1} k^n * q^(n*k) * x^k / k! ]^n is an integer series such that G(x,q) = Sum_{n>=0} q^(n^2) * [ Sum_{k=1..n} S2(n,k) * q^(n*k-n) * x^k ]^n.

Examples

			G.f.: A(x) = 1 + 2*x + 16*x^2 + 640*x^3 + 102656*x^4 + 63897600*x^5 + 154597064704*x^6 + 1463095704682496*x^7 +...
such that
A(x) = Sum_{n>=0} exp(-n*2^n*x) * (2^n*x + 2^n*2^(2*n)*x^2/2! + 3^n*2^(3*n)*x^3/3! +...+ k^n*2^(k*n)*x^k/k! +...)^n.
Explicitly,
A(x) = 1 + exp(-2*x) * (2*x + 2*2^2*x^2/2! + 3*2^3*x^3/3! + 4*2^4*x^4/4! +...) +
exp(-2*2^2*x) * (2^2*x + 4*2^4*x^2/2! + 9*2^6*x^3/3! + 16*2^8*x^4/4! +...)^2 +
exp(-3*2^3*x) * (2^3*x + 8*2^6*x^2/2! + 27*2^9*x^3/3! + 64*2^12*x^4/4! +...)^3 +
exp(-4*2^4*x) * (2^4*x + 16*2^8*x^2/2! + 81*2^12*x^3/3! + 256*2^16*x^4/4! +...)^4 +
exp(-5*2^5*x) * (2^5*x + 32*2^10*x^2/2! + 243*2^15*x^3/3! + 1024*2^20*x^4/4! +...)^5 +...
The g.f. can be written using the Stirling2 numbers like so:
A(x) = 1 + 2*x + (2^2*x + 2^4*x^2)^2 + (2^3*x + 3*2^6*x^2 + 2^9*x^3)^3 + (2^4*x + 7*2^8*x^2 + 6*2^12*x^3 + 2^16*x^4)^4 + (2^5*x + 15*2^10*x^2 + 25*2^15*x^3 + 10*2^20*x^4 + 2^25*x^5)^5 + (2^6*x + 31*2^12*x^2 + 90*2^18*x^3 + 65*2^24*x^4 + 15*2^30*x^5 + 2^36*x^6)^6 + (2^7*x + 63*2^14*x^2 + 301*2^21*x^3 + 350*2^28*x^4 + 140*2^35*x^5 + 21*2^42*x^6 + 2^49*x^7)^7 +...+ [ Sum_{k=1..n} S2(n,k) * 2^(n*k) * x^k ]^n +...

Crossrefs

Cf. A276746, A277037, A168407.

Programs

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

Formula

G.f.: Sum_{n>=0} [ Sum_{k=1..n} S2(n,k) * 2^(n*k) * x^k ]^n, where S2(n,k) = A008277(n,k) are the Stirling numbers of the second kind.

A168404 E.g.f.: Sum_{n>=0} tan(2^n*x)^n/n!.

Original entry on oeis.org

1, 2, 16, 528, 67584, 34210304, 69391122432, 565356426987520, 18478277930015260672, 2419401354886413876592640, 1267940756758206239694099841024, 2658665157828553829995392867121496064
Offset: 0

Views

Author

Paul D. Hanna, Nov 25 2009

Keywords

Examples

			E.g.f.: A(x) = 1 + 2*x + 16*x^2/2! + 528*x^3/3! + 67584*x^4/4! +...
A(x) = 1 + tan(2*x) + tan(4*x)^2/2! + tan(8*x)^3/3! + tan(16*x)^4/4! +...+ tan(2^n*x)^n/n! +...
a(n) = coefficient of x^n/n! in G(x)^(2^n) where G(x) = exp(tan(x)):
G(x) = 1 + x + x^2/2! + 3*x^3/3! + 9*x^4/4! + 37*x^5/5! + 177*x^6/6! +...+ A006229(n)*x^n/n! +...

Crossrefs

Cf. A006229 (exp(tan x)), variants: A136632, A168402, A168403, A168405, A168406, A168407, A168408.

Programs

  • PARI
    {a(n)=n!*polcoeff(sum(k=0,n,tan(2^k*x +x*O(x^n))^k/k!),n)}
  • PARI
    {a(n)=n!*polcoeff(exp(2^n*tan(x +x*O(x^n))),n)}

Formula

a(n) = [x^n/n! ] exp(2^n*tan(x)) for n>=0.

A168405 E.g.f.: Sum_{n>=0} arcsin(2^n*x)^n/n!.

Original entry on oeis.org

1, 2, 16, 520, 66560, 33882400, 69055283200, 564153087455360, 18462510039810703360, 2418626471936038215754240, 1267795676362601991645220044800, 2658560574070850656450883768752998400
Offset: 0

Views

Author

Paul D. Hanna, Nov 25 2009

Keywords

Examples

			E.g.f.: A(x) = 1 + 2*x + 16*x^2/2! + 520*x^3/3! + 66560*x^4/4! + ...
A(x) = 1 + arcsin(2*x) + arcsin(4*x)^2/2! + arcsin(8*x)^3/3! + arcsin(16*x)^4/4! + ... + arcsin(2^n*x)^n/n! + ...
a(n) = coefficient of x^n/n! in G(x)^(2^n) where G(x) = exp(arcsin(x)):
G(x) = 1 + x + x^2/2! + 2*x^3/3! + 5*x^4/4! + 20*x^5/5! + 85*x^6/6! + ... + A006228(n)*x^n/n! + ...

Crossrefs

Cf. A006228 (exp(arcsin x)), variants: A136632, A168402, A168403, A168404, A168406, A168407, A168408.

Programs

  • PARI
    {a(n)=n!*polcoeff(sum(k=0,n,asin(2^k*x +x*O(x^n))^k/k!),n)}
  • PARI
    {a(n)=n!*polcoeff(exp(2^n*asin(x +x*O(x^n))),n)}

Formula

a(n) = [x^n/n!] exp(2^n*arcsin(x)) for n >= 0.

A168406 E.g.f.: Sum_{n>=0} arctan(2^n*x)^n/n!.

Original entry on oeis.org

1, 2, 16, 496, 63488, 32899840, 68049141760, 560546415810560, 18415229458563727360, 2416302337337071616327680, 1267360474688679165942982246400, 2658246833688954938616062542151680000
Offset: 0

Views

Author

Paul D. Hanna, Nov 25 2009

Keywords

Examples

			E.g.f.: A(x) = 1 + 2*x + 16*x^2/2! + 496*x^3/3! + 63488*x^4/4! + ...
A(x) = 1 + arctan(2*x) + arctan(4*x)^2/2! + arctan(8*x)^3/3! + arctan(16*x)^4/4! + ... + arctan(2^n*x)^n/n! + ...
a(n) = coefficient of x^n/n! in G(x)^(2^n) where G(x) = exp(arctan(x)):
G(x) = 1 + x + x^2/2! - x^3/3! - 7*x^4/4! + 5*x^5/5! + 145*x^6/6! + ... + A002019(n)*x^n/n! + ...

Crossrefs

Cf. A002019 (exp(arctan x)), variants: A136632, A168402, A168403, A168404, A168405, A168407, A168408.

Programs

  • PARI
    {a(n)=n!*polcoeff(sum(k=0,n,atan(2^k*x +x*O(x^n))^k/k!),n)}
  • PARI
    {a(n)=n!*polcoeff(exp(2^n*atan(x +x*O(x^n))),n)}

Formula

a(n) = [x^n/n!] exp(2^n*arctan(x)) for n >= 0.
Showing 1-5 of 5 results.

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

Content is available under The OEIS End-User License Agreement.