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.

A158093 a(n) = 3^(n^2+n)*C(1/3^n, n) = [x^n] (1 + 3^(n+1)*x)^(1/3^n).

Original entry on oeis.org

1, 3, -36, 6201, -10519740, 168009075234, -24937507748845692, 34147337933260567913832, -429040882807948915054596365580, 49262806958277650055073574841789707655
Offset: 0

Views

Author

Paul D. Hanna, Apr 21 2009

Keywords

Comments

A(1) = Sum_{n>=0} C(1/3^n,n) = Sum_{n>=0} log(1+1/3^n)^n/n! = 1.293240509200709604261070...

Examples

			G.f.: A(x) = 1 +3*x/3^2 -36*x^2/3^6 +6201*x^3/3^12 -10519740*x^4/3^20 +...
A(x) = 1 + log(1+x/3) + log(1+x/9)^2/2! + log(1+x/27)^3/3! +...+ log(1+x/3^n)^n/n! +...
Illustrate a(n) = [x^n] (1 + 3^(n+1)*x)^(1/3^n):
(1+9*x)^(1/3) = 1 + (3)*x - 9*x^2 + 45*x^3 - 270*x^4 +...
(1+27*x)^(1/9) = 1 + 3*x - (36)*x^2 + 612*x^3 - 11934*x^4 +...
(1+81*x)^(1/27) = 1 + 3*x - 117*x^2 + (6201)*x^3 - 372060*x^4 +...
(1+243*x)^(1/81) = 1 + 3*x - 360*x^2 + 57960*x^3 - (10519740)*x^4 +...
Special values of A(x).
A(1) = 1 + log(4/3) + log(10/9)^2/2! + log(28/27)^3/3! +...
A(3) = 1 + log(2) + log(4/3)^2/2! + log(10/9)^3/3! +...
A(9) = 1 + log(4) + log(2)^2/2! + log(4/3)^3/3! + log(10/9)^4/4! +...
A(r) = 2 at r=4.50548200106313905...
A(r) = 3 at r=12.21509538023664538...
A(r) = 4 at r=22.9609516534592247304...

Links

Crossrefs

Cf. A159478, A159558, A183131.

Programs

  • PARI
    a(n)=3^(n^2+n)*binomial(1/3^n,n)

Formula

G.f.: A(x) = Sum_{n>=0} a(n)*x^n/3^(n^2+n) = Sum_{n>=0} log(1+x/3^n)^n/n!.

A159318 a(n) = 2^(n^2+n) * binomial(2*n-1 + 1/2^n, n) / (n*2^n + 1).

Original entry on oeis.org

1, 2, 26, 1804, 591894, 860081340, 5338683113364, 138637536961147800, 14872932935424544987110, 6538678365573711555851779180, 11717380780236748297970244719026812
Offset: 0

Views

Author

Paul D. Hanna, Apr 22 2009

Keywords

Examples

			G.f.: A(x) = 1 + 2*x/2^2 + 26*x^2/2^6 + 1804*x^3/2^12 + 591894*x^4/2^20 + ...
G.f.: A(x) = Sum_{n>=0} log( 2^n*(1-sqrt(1 - 4*x/2^n))/(2*x) )^n/n!.
A(x) = 1 + log(F(x/2)) + log(F(x/4))^2/2! + log(F(x/8))^3/3! + ... where F(x) = (1-sqrt(1 - 4*x))/(2*x).
Special values.
A(1/2) = 1 + log(2) + log(4-4*sqrt(1/2))^2/2! + log(8-8*sqrt(3/4))^3/3! + log(16-16*sqrt(7/8))^4/4! + ...
A(1/2) = 1.70573970062357248928512380703308976974285275...
A(-1/2) = 1 + log(2*sqrt(2)-2) + log(4*sqrt(3/2)-4)^2/2! + log(8*sqrt(5/4)-8)^3/3! + log(16*sqrt(9/8)-16)^4/4! + ...
A(-1/2) = 0.81741280310249092844743171863299249334671633...
Illustrate a(n) = [x^n] {(1-sqrt(1-2^(n+3)*x))/(2^(n+2)*x)}^(1/2^n):
n=0: (1) + 2*x + 8*x^2 + 40*x^3 + 224*x^4 + 1344*x^5 + ...
n=1: 1 + (2)*x + 14*x^2 + 132*x^3 + 1430*x^4 + 16796*x^5 + ...
n=2: 1 + 2*x + (26)*x^2 + 476*x^3 + 10150*x^4 + 236060*x^5 + ...
n=3: 1 + 2*x + 50*x^2 + (1804)*x^3 + 76342*x^4 + 3534076*x^5 + ...
n=4: 1 + 2*x + 98*x^2 + 7020*x^3 + (591894)*x^4 + 54673468*x^5 + ...
n=5: 1 + 2*x + 194*x^2 + 27692*x^3 + 4660950*x^4 + (860081340)*x^5 + ...
coefficients in parenthesis form the initial terms of this sequence.

Links

Crossrefs

Cf. A159558, A159478, A158093, A000108.

Programs

  • Magma
    [2^(n^2 +n)*Binomial(2*n -1 +1/2^n, n)/(n*2^n +1): n in [0..50]]; // G. C. Greubel, Jun 26 2018
  • Mathematica
    Table[2^(n^2 +n)*Binomial[2*n -1 +1/2^n, n]/(n*2^n +1), {n, 0, 50}] (* G. C. Greubel, Jun 26 2018 *)
  • PARI
    a(n)=2^(n^2+n)*binomial(2*n-1+1/2^n, n)/(n*2^n + 1)
  • PARI
    a(n)=polcoeff(((1-sqrt(1 - 2^(n+3)*x))/2^(n+2)/x)^(1/2^n),n)
  • PARI
    {a(n)=polcoeff(1/(1-2^(n+1)*x+x*O(x^n))^(n+1/2^n),n)/(n*2^n+1)} \\ Paul D. Hanna, Jun 15 2010

Formula

a(n) = [x^n] {(1-sqrt(1 - 2^(n+3)*x))/(2^(n+2)*x)}^(1/2^n).
G.f.: A(x) = Sum_{n>=0} a(n)*x^n/2^(n^2+n).
G.f.: A(x) = Sum_{n>=0} log(F(x/2^n))^n/n! where F(x) = (1-sqrt(1-4*x))/(2*x) is the Catalan function (A000108).
Radius of convergence of A(x) is |x| <= 1/2.
a(n) = [x^n] (1/(1 - 2^(n+1)*x)^(n + 1/2^n))/(n*2^n + 1). - Paul D. Hanna, Jun 15 2010

A183131 a(n) = 3^(n^2+n) * C(n-1 + 1/3^n, n).

Original entry on oeis.org

1, 3, 45, 6930, 11006901, 170914738743, 25094213868636459, 34223931232325024404500, 429380124224154112727394224229, 49276413396437070036301200874674619650, 51540221234153816900126405027724655190954838780
Offset: 0

Views

Author

Paul D. Hanna, Dec 26 2010

Keywords

Examples

			G.f.: A(x) = 1 + 3*x/3^2 + 45*x^2/3^6 + 6930*x^3/3^12 + 11006901*x^4/3^20 + 170914738743*x^5/3^30 +...
A(x) = 1 - log(1-x/3) + log(1-x/9)^2/2! - log(1-x/27)^3/3! + log(1-x/81)^4/4! +...+ (-1)^n*log(1-x/3^n)^n/n! +...
Illustrate a(n) = [x^n] 1/(1 - 3^(n+1)*x)^(1/3^n):
(1-9*x)^(-1/3) = 1 + (3)*x + 18*x^2 + 126*x^3 + 945*x^4 +...
(1-27*x)^(-1/9) = 1 + 3*x + (45)*x^2 + 855*x^3 + 17955*x^4 +...
(1-81*x)^(-1/27) = 1 + 3*x + 126*x^2 + (6930)*x^3 + 426195*x^4 +...
(1-243*x)^(-1/81) = 1 + 3*x + 369*x^2 + 60147*x^3 + (11006901)*x^4 +...
(1-729*x)^(-1/243) = 1 + 3*x + 1098*x^2 + 534726*x^3 + 292762485*x^4 + (170914738743)*x^5 +...
Special values.
A(1) = 1 + log(3/2) + log(9/8)^2/2! + log(27/26)^3/3! + log(81/80)^4/4! +...
A(-1) = 1 + log(3/4) + log(9/10)^2/2! + log(27/28)^3/3! + log(81/82)^4/4! +...
A(1/3) = 1 + log(9/8) + log(27/26)^2/2! + log(81/80)^3/3! + log(243/242)^4/4! +...
A(4/3) = 1 + log(9/5) + log(27/23)^2/2! + log(81/77)^3/3! + log(243/239)^4/4! +...
A(1) = 1.412410489973035808125672257400880...
A(-1) = 0.7178603309478784469203322438498398552...
A(-3) = 0.3480384480558263511525077084408616142...
A(3) is indeterminate.

Links

Crossrefs

Cf. A159558, A158093.

Programs

  • PARI
    a(n)=3^(n^2+n)*binomial(n-1+1/3^n, n)
  • PARI
    {a(n)=3^(n^2+n)*polcoeff(1+sum(m=1,n,(-log(1 - x/3^m +x*O(x^n)))^m/m!),n)}

Formula

a(n) = [x^n] 1/(1 - 3^(n+1)*x)^(1/3^n).
G.f.: Sum_{n>=0} (-log(1 - x/3^n))^n/n! = Sum_{n>=0} a(n)*x^n/3^(n^2+n).

A224883 a(n) = 2^(n^2) * binomial(n-1 + 1/2^(n-1), n).

Original entry on oeis.org

1, 2, 6, 60, 2550, 476476, 384115732, 1305385229720, 18382187112952806, 1060603038396055882860, 248959068848694059131153020, 236689359381076468102847994171880, 908758498534088142521911865612937786108, 14063550492706544341683006937639901739122886616
Offset: 0

Views

Author

Paul D. Hanna, Jul 23 2013

Keywords

Examples

			G.f.: A(x) = 1 + 2*x/2 + 6*x^2/2^4 + 60*x^3/2^9 + 2550*x^4/2^16 + 476476*x^5/2^25 +...+ a(n)*x^n/2^(n^2) +...
where
A(x) = 1 - 2*log(1-x/2) + 4*log(1-x/4)^2/2! - 8*log(1-x/8)^3/3! + 16*log(1-x/16)^4/4! +...+ (-2)^n*log(1-x/2^n)^n/n! +...
Illustrate a(n) = [x^n] 1/(1 - 2^n*x)^(2/2^n):
(1-x)^(-2/1) = (1) + 2*x + 3*x^2 + 4*x^3 + 5*x^4 + 6*x^5 +...
(1-2*x)^(-2/2) = 1 + (2)*x + 4*x^2 + 8*x^3 + 16*x^4 + 32*x^5 +...
(1-4*x)^(-2/4) = 1 + 2*x + (6)*x^2 + 20*x^3 + 70*x^4 + 252*x^5 +...
(1-8*x)^(-2/8) = 1 + 2*x + 10*x^2 + (60)*x^3 + 390*x^4 + 2652*x^5 +...
(1-16*x)^(-2/16) = 1 + 2*x + 18*x^2 + 204*x^3 + (2550)*x^4 + 33660*x^5 +...
(1-32*x)^(-2/32) = 1 + 2*x + 34*x^2 + 748*x^3 + 18326*x^4 + (476476)*x^5 +...
where the coefficients in parenthesis form the initial terms of this sequence.
Particular values.
A(1) = 1 + 2*log(2) + 4*log(4/3)^2/2! + 8*log(8/7)^3/3! + 16*log(16/15)^4/4! +...
A(1/2) = 1 + 2*log(4/3) + 4*log(8/7)^2/2! + 8*log(16/15)^3/3! +...
A(1/4) = 1 + 2*log(8/7) + 4*log(16/15)^2/2! + 8*log(32/31)^3/3! +...
A(3/2) = 1 + 2*log(4) + 4*log(8/5)^2/2! + 8*log(16/13)^3/3! + 16*log(32/29)^4/4! +...
Explicitly,
A(1) = 2.55500248436101360804704969796239525102504151...
A(1/2) = 1.61138451105646219391156983544059555709337920...
A(1/4) = 1.27543593708175757392940597050033002345086132...
A(3/2) = 4.22639446385430649517540615961613624264078875...

Links

Crossrefs

Cf. A159558, A246900.

Programs

  • Mathematica
    Table[2^(n^2) Binomial[n-1+1/2^(n-1),n],{n,0,20}] (* Harvey P. Dale, Feb 01 2017 *)
  • PARI
    {a(n)=2^(n^2)*binomial(n-1+1/2^(n-1), n)}
for(n=0,20,print1(a(n),", "))
  • PARI
    {a(n)=(2^n/n!)*prod(k=0, n-1, 2^(n-1)*k + 1)}
for(n=0,20,print1(a(n),", "))
  • PARI
    {a(n)=2^(n^2)*polcoef(sum(k=0, n, (-2)^k*log(1-x/2^k +x*O(x^n))^k/k!),n)}
for(n=0,20,print1(a(n),", "))

Formula

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

A159319 a(n) = 3^(n^2+n) * C(2*n-1 + 1/3^n, n) / (n*3^n + 1).

Original entry on oeis.org

1, 3, 126, 66708, 379033074, 21399656315607, 11566324342205917416, 58678275719834357303044728, 2762222169999029718435709903699050, 1197781369953334546750963984948238943438411
Offset: 0

Views

Author

Paul D. Hanna, Apr 23 2009

Keywords

Examples

			G.f.: A(x) = 1 + 3*x/3^2 + 126*x^2/3^6 + 66708*x^3/3^12 + 379033074*x^4/3^20 +...
A(x) = Sum_{n>=0} log( (1-sqrt(1-4*x/3^n))/(2*x/3^n) )^n/n!.
A(x) = 1 + log(F(x/3)) + log(F(x/9))^2/2! + log(F(x/27))^3/3! +... where F(x) = (1-sqrt(1-4*x))/(2*x).
Special values.
A(3/4) = 1 + log(2) + log(6-6*sqrt(2/3))^2/2! + log(18-18*sqrt(8/9))^3/3! + log(54-54*sqrt(26/27))^4/4! +...
A(3/4) = 1.6977820781412737038286578011417848301231627494589650...
A(-3/4) = 1 + log(2*sqrt(2)-2) + log(6*sqrt(4/3)-6)^2/2! + log(18*sqrt(10/9)-18)^3/3! + log(54*sqrt(28/27)-54)^4/4! +...
A(-3/4) = 0.8145458917316632938137444904602229430460096517471900...
Illustrate (3^n)-th root formula:
a(n)/3^(n^2+n) = [x^n] F(x)^(1/3^n) or, equivalently,
a(n) = [x^n] F(3^(n+1)*x)^(1/3^n) where F(x)=Catalan(x):
F(3*x) = (1) + 3*x + 18*x^2 + 135*x^3 + 1134*x^4 + 10206*x^5 +...
F(9*x)^(1/3) = 1 + (3)*x + 45*x^2 + 936*x^3 + 22572*x^4 +...
F(27*x)^(1/9) = 1 + 3*x + (126)*x^2 + 7659*x^3 + 546480*x^4 +...
F(81*x)^(1/27) = 1 + 3*x + 369*x^2 + (66708)*x^3 + 14215230*x^4 +...
F(243*x)^(1/81) = 1 + 3*x + 1098*x^2 + 593775*x^3 + (379033074)*x^4 +...
coefficients in parenthesis form the initial terms of this sequence.

Links

Crossrefs

Cf. A159318, A158093, A159558, A159478, A000108.

Programs

  • Magma
    [3^(n^2 +n)*Binomial(2*n -1 +1/3^n, n)/(n*3^n +1): n in [0..40]]; // G. C. Greubel, Jun 26 2018
  • Mathematica
    Table[3^(n^2 +n)*Binomial[2*n -1 +1/3^n, n]/(n*3^n +1), {n, 0, 50}] (* G. C. Greubel, Jun 26 2018 *)
  • PARI
    {a(n)=3^(n^2+n)*binomial(2*n-1+1/3^n, n)/(n*3^n + 1)}
  • PARI
    {a(n)=3^(n^2+n)*polcoeff(1/(1-x+x*O(x^n))^(n+1/3^n)/(n*3^n + 1),n)}
  • PARI
    {a(n)=3^(n^2+n)*polcoeff(((1-sqrt(1-4*x+x^2*O(x^n)))/(2*x))^(1/3^n),n)}
  • PARI
    {a(n)=3^(n^2+n)*polcoeff(sum(k=0,n,log((1-sqrt(1-4*x/3^k+x^2*O(x^n)))/(2*x/3^k))^k/k!),n)}

Formula

G.f.: A(x) = Sum_{n>=0} a(n)*x^n/3^(n^2+n).
G.f.: A(x) = Sum_{n>=0} log(F(x/3^n))^n/n! and
a(n)/3^(n^2+n) = [x^n] F(x)^(1/3^n) where
F(x) = (1-sqrt(1-4*x))/(2*x) is the Catalan function (A000108).
a(n)/3^(n^2+n) = [x^n] 1/(1-x)^(n + 1/3^n)/(n*3^n + 1).
Radius of convergence of series A(x) is |x| <= 3/4.
Showing 1-5 of 5 results.

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