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.

A211895 G.f.: exp( Sum_{n>=1} 3 * Jacobsthal(n)^3 * x^n/n ), where Jacobsthal(n) = A001045(n).

Original entry on oeis.org

1, 3, 6, 36, 186, 1254, 8208, 57540, 404619, 2913705, 21146694, 155231256, 1147302756, 8538393900, 63879354096, 480212156664, 3624581868297, 27456690186507, 208644709097070, 1589982296208492, 12147079485362406, 93012131704072698, 713676733469348352
Offset: 0

Views

Author

Paul D. Hanna, Apr 25 2012

Keywords

Comments

Given g.f. A(x), note that A(x)^(1/3) is not an integer series.

Examples

			G.f.: A(x) = 1 + 3*x + 6*x^2 + 36*x^3 + 186*x^4 + 1254*x^5 + 8208*x^6 +...
such that
log(A(x))/3 = x + x^2/2 + 3^3*x^3/3 + 5^3*x^4/4 + 11^3*x^5/5 + 21^3*x^6/6 + 43^3*x^7/7 +...+ Jacobsthal(n)^3*x^n/n +...
Jacobsthal numbers begin:
A001045 = [1,1,3,5,11,21,43,85,171,341,683,1365,2731,5461,10923,...].

Links

Crossrefs

Cf. A211893, A211894, A211896, A207970, A001045 (Jacobsthal).

Programs

  • Mathematica
    CoefficientList[Series[((1+x)*(1+4*x)^3/((1-2*x)^3*(1-8*x)))^(1/9), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 24 2012 *)
  • PARI
    {Jacobsthal(n)=polcoeff(x/(1-x-2*x^2+x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(k=1, n, 3*Jacobsthal(k)^3*x^k/k)+x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=polcoeff(( (1+x)*(1+4*x)^3 / ((1-2*x)^3*(1-8*x)+x*O(x^n)) )^(1/9),n)}

Formula

G.f.: ( (1+x)*(1+4*x)^3 / ((1-2*x)^3*(1-8*x)) )^(1/9).
G.f.: exp( Sum_{n>=1} (2^n - (-1)^n)^3 / 9 * x^n/n ).
Recurrence: n*a(n) = (5*n-2)*a(n-1) + 6*(5*n-12)*a(n-2) - 8*(5*n-12)*a(n-3) - 64*(n-4)*a(n-4). - Vaclav Kotesovec, Oct 24 2012
a(n) ~ 3^(2/9)*8^n/(Gamma(1/9)*n^(8/9)). - Vaclav Kotesovec, Oct 24 2012

A211892 G.f.: exp( Sum_{n>=1} 3 * Jacobsthal(n^2) * x^n/n ), where Jacobsthal(n) = A001045(n).

Original entry on oeis.org

1, 3, 12, 198, 16962, 6762210, 11473594848, 80455865485692, 2306084412391039038, 268657100633050977422322, 126765866001055606588876061400, 241678197713843578271875740922972788, 1858396158245858742065123341776166504084452
Offset: 0

Views

Author

Paul D. Hanna, Apr 24 2012

Keywords

Comments

Given g.f. A(x), note that A(x)^(1/3) is not an integer series.

Examples

			G.f.: A(x) = 1 + 3*x + 12*x^2 + 198*x^3 + 16962*x^4 + 6762210*x^5 +...
such that
log(A(x))/3 = x + 5*x^2/2 + 171*x^3/3 + 21845*x^4/4 + 11184811*x^5/5 + 22906492245*x^6/6 + 187649984473771*x^7/7 +...+ Jacobsthal(n^2)*x^n/n +...
Jacobsthal numbers begin:
A001045 = [1,1,3,5,11,21,43,85,171,341,683,1365,2731,5461,10923,21845,...].

Crossrefs

Cf. A211893, A211894, A211895, A211896, A207972, A001045, A155200.
Cf. A231279 (Jacobsthal(n^2)).

Programs

  • PARI
    {Jacobsthal(n)=polcoeff(x/(1-x-2*x^2+x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(k=1, n, 3*Jacobsthal(k^2)*x^k/k)+x*O(x^n)), n)}
for(n=0, 16, print1(a(n), ", "))

Formula

G.f.: (1+x) * exp( Sum_{n>=1} 2^(n^2) * x^n/n ).
a(n) = A155200(n) + A155200(n-1).

A211894 G.f.: exp( Sum_{n>=1} 3 * Jacobsthal(n)^2 * x^n/n ), where Jacobsthal(n) = A001045(n).

Original entry on oeis.org

1, 3, 6, 18, 57, 195, 684, 2460, 8970, 33102, 123204, 461868, 1741410, 6597750, 25099584, 95822928, 366943881, 1408947675, 5422742910, 20915079258, 80820382425, 312839889219, 1212812010804, 4708415402772, 18302630040504, 71230126892088, 277514015733168
Offset: 0

Views

Author

Paul D. Hanna, Apr 25 2012

Keywords

Comments

Given g.f. A(x), note that A(x)^(1/3) is not an integer series.

Examples

			G.f.: A(x) = 1 + 3*x + 6*x^2 + 18*x^3 + 57*x^4 + 195*x^5 + 684*x^6 +...
such that
log(A(x))/3 = x + x^2/2 + 3^2*x^3/3 + 5^2*x^4/4 + 11^2*x^5/5 + 21^2*x^6/6 + 43^2*x^7/7 +...+ Jacobsthal(n)^2*x^n/n +...
Jacobsthal numbers begin:
A001045 = [1,1,3,5,11,21,43,85,171,341,683,1365,2731,5461,10923,...].

Crossrefs

Cf. A211893, A211895, A211896, A054888, A207969, A001045 (Jacobsthal).

Programs

  • Mathematica
    CoefficientList[Series[(1+2*x)^(2/3) / ((1-x)*(1-4*x))^(1/3), {x, 0, 30}], x] (* Vaclav Kotesovec, Oct 18 2020 *)
  • PARI
    {Jacobsthal(n)=polcoeff(x/(1-x-2*x^2+x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(k=1, n, 3*Jacobsthal(k)^2*x^k/k)+x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=polcoeff(((1+2*x)^2/((1-x)*(1-4*x) +x*O(x^n)))^(1/3),n)}

Formula

G.f.: (1+2*x)^(2/3) / ((1-x)*(1-4*x))^(1/3).
G.f.: exp( Sum_{n>=1} (2^n - (-1)^n)^2 / 3 * x^n/n ).
a(n) ~ 3^(1/3) * 2^(2*n) / (n^(2/3) * Gamma(1/3)). - Vaclav Kotesovec, Oct 18 2020

A211896 G.f.: exp( Sum_{n>=1} 3 * Jacobsthal(n)^4 * x^n/n ), where Jacobsthal(n) = A001045(n).

Original entry on oeis.org

1, 3, 6, 90, 723, 10689, 130428, 1862580, 25594611, 368313993, 5289203262, 77279744418, 1134460916361, 16798605635235, 249994099311288, 3740771822960664, 56208829313956998, 847934859174601650, 12834366187138678836, 194855374723972622988, 2966358133685609559042
Offset: 0

Views

Author

Paul D. Hanna, Apr 25 2012

Keywords

Comments

Given g.f. A(x), note that A(x)^(1/3) is not an integer series.

Examples

			G.f.: A(x) = 1 + 3*x + 6*x^2 + 90*x^3 + 723*x^4 + 10689*x^5 + 130428*x^6 +...
such that
log(A(x))/3 = x + x^2/2 + 3^4*x^3/3 + 5^4*x^4/4 + 11^4*x^5/5 + 21^4*x^6/6 + 43^4*x^7/7 +...+ Jacobsthal(n)^4*x^n/n +...
Jacobsthal numbers begin:
A001045 = [1,1,3,5,11,21,43,85,171,341,683,1365,2731,5461,10923,...].

Crossrefs

Cf. A211893, A211894, A211895, A207969, A001045 (Jacobsthal).

Programs

  • PARI
    {Jacobsthal(n)=polcoeff(x/(1-x-2*x^2+x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(k=1, n, 3*Jacobsthal(k)^4*x^k/k)+x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=polcoeff(((1+2*x)^4*(1+8*x)^4/((1-x)*(1-4*x)^6*(1-16*x))+x*O(x^n))^(1/27),n)}

Formula

G.f.: ( (1+2*x)^4*(1+8*x)^4 / ((1-x)*(1-4*x)^6*(1-16*x)) )^(1/27).
G.f.: exp( Sum_{n>=1} (2^n - (-1)^n)^4 / 27 * x^n/n ).
a(n) ~ 3^(5/27) * 2^(4*n) / (5^(1/27) * Gamma(1/27) * n^(26/27)). - Vaclav Kotesovec, Oct 18 2020

A231292 a(n) = Jacobsthal(n)^n, where Jacobsthal(n) = A001045(n), for n>=1.

Original entry on oeis.org

1, 1, 27, 625, 161051, 85766121, 271818611107, 2724905250390625, 125015825667824393931, 21259046894411315872085401, 15087863296794400779633937999667, 41840013551409555494294964922119140625, 470091178834036922915254196307625156782873691
Offset: 1

Views

Author

Paul D. Hanna, Nov 06 2013

Keywords

Crossrefs

Cf. A211893, A231279.

Programs

  • Mathematica
    Module[{nn=20},#[[1]]^#[[2]]&/@Thread[{Rest[LinearRecurrence[{1,2},{0,1},nn+1]],Range[nn]}]] (* Harvey P. Dale, Jan 17 2022 *)
  • PARI
    {a(n)=(2^n-(-1)^n)^n/3^n}
for(n=1, 15, print1(a(n), ", "))

Formula

a(n) = (2^n - (-1)^n)^n / 3^n.
One-third the logarithmic derivative of A211893.
Showing 1-5 of 5 results.

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

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