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 1-4 of 4 results.

A165941 G.f.: A(x) = exp( Sum_{n>=1} 2^n * x^n/(n*(1+x^n)) ).

Original entry on oeis.org

1, 2, 2, 6, 10, 18, 42, 78, 154, 314, 626, 1246, 2498, 4994, 9970, 19974, 39930, 79826, 159706, 319374, 638714, 1277530, 2554978, 5109854, 10219922, 20439714, 40879234, 81758854, 163517466, 327034514, 654069866, 1308139246, 2616277578
Offset: 0

Views

Author

Paul D. Hanna, Oct 20 2009

Keywords

Comments

Compare to: exp( Sum_{n>=1} x^n/(1+x^n)/n ) = Sum_{n>=0} x^(n*(n+1)/2).

Examples

			G.f.: A(x) = 1 + 2*x + 2*x^2 + 6*x^3 + 10*x^4 + 18*x^5 + 42*x^6 + 78*x^7 +...
such that
log(A(x)) = 2*x/(1+x) + 2^2*x^2/(2*(1+x^2)) + 2^3*x^3/(3*(1+x^3)) + 2^4*x^4/(4*(1+x^4)) + 2^5*x^5/(5*(1+x^5)) +...
Also, A(x) = (1 + x*B(x))/(1 - x*B(x)), where
B(x) = 1 + 2*x^2 + 2*x^4 + 4*x^5 + 2*x^6 + 8*x^7 + 6*x^8 + 20*x^9 + 18*x^10 + 36*x^11 + 54*x^12 + 76*x^13 + 150*x^14 + 172*x^15 +...
such that B(x) = (1 + x*C(x))/(1 - x*C(x)), where
C(x) = 1 + 2*x^3 + 2*x^6 + 4*x^7 + 2*x^9 + 8*x^10 + 4*x^11 + 10*x^12 + 12*x^13 + 16*x^14 + 22*x^15 + 32*x^16 + 44*x^17 + 66*x^18 +...
such that C(x) = (1 + x*D(x))/(1 - x*D(x)), where
D(x) = 1 + 2*x^4 + 2*x^8 + 4*x^9 + 2*x^12 + 8*x^13 + 4*x^14 + 8*x^15 + 2*x^16 + 12*x^17 + 16*x^18 + 20*x^19 + 18*x^20 + 24*x^21 +...
such that D(x) = (1 + x*E(x))/(1 - x*E(x)), where
E(x) = 1 + 2*x^5 + 2*x^10 + 4*x^11 + 2*x^15 + 8*x^16 + 4*x^17 + 8*x^18 + 2*x^20 + 12*x^21 + 16*x^22 + 20*x^23 + 16*x^24 + 10*x^25 +...
such that E(x) = (1 + x*F(x))/(1 - x*F(x)), where
F(x) = 1 + 2*x^6 + 2*x^12 + 4*x^13 + 2*x^18 + 8*x^19 + 4*x^20 + 8*x^21 + 2*x^24 + 12*x^25 + 16*x^26 + 20*x^27 + 16*x^28 + 8*x^29 + 18*x^30 + 16*x^31 + 36*x^32 +...
etc.
The coefficients in the above functions tend toward the terms in triangle A259192.

Crossrefs

Cf. A259192, A259273, A259274, A259275, A259276, A348902.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[Exp[Sum[2^k * x^k / (1 + x^k)/k, {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 18 2020 *)
  • PARI
    {a(n)=if(n==0, 1, polcoeff(exp(sum(m=1, n, 2^m*x^m/(1+x^m+x*O(x^n))/m)), n))}
for(n=0,30,print1(a(n),", "))
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=(1 + x^(n+1-i)*A)/(1 - x^(n+1-i)*A+ x*O(x^n)));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f.: -1 + 2/(1+x - 2*x/(1+x^2 - 2*x^2/(1+x^3 - 2*x^3/(1+x^4 - 2*x^4/(1+x^5 - 2*x^5/(1+x^6 - 2*x^6/(1+x^7 - 2*x^7/(1+x^8 - 2*x^8/(...))))))))), a continued fraction.
G.f.: A(x) = (1 + x*B(x))/(1 - x*B(x)), where B(x) = (1 + x^2*C(x))/(1 - x^2*C(x)), C(x) = (1 + x^3*D(x))/(1 - x^3*D(x)), D(x) = (1 + x^4*E(x))/(1 - x^4*E(x)), ... - Paul D. Hanna, Jun 14 2015
a(n) ~ c * 2^n, where c = 2^(7/8) / EllipticTheta(2, 0, 1/sqrt(2)) = 0.6091497110662286155211146043057245512950999410185846... - Vaclav Kotesovec, Oct 18 2020, updated Apr 18 2024

A259273 G.f.: A(x) = exp( Sum_{n>=1} 3^n * x^n/(n*(1+x^n)) ).

Original entry on oeis.org

1, 3, 6, 21, 60, 174, 537, 1596, 4776, 14358, 43053, 129126, 387438, 1162272, 3486678, 10460307, 31380756, 94141830, 282426288, 847278282, 2541833808, 7625503749, 22876509444, 68629525032, 205888582014, 617665741140, 1852997213508, 5558991660912, 16676974967991, 50030924873862, 150092774683998
Offset: 0

Views

Author

Paul D. Hanna, Jun 23 2015

Keywords

Comments

Compare to: exp( Sum_{n>=1} x^n/(1+x^n)/n ) = Sum_{n>=0} x^(n*(n+1)/2).

Examples

			G.f.: A(x) = 1 + 3*x + 6*x^2 + 21*x^3 + 60*x^4 + 174*x^5 + 537*x^6 +...
such that
log(A(x)) = 3*x/(1+x) + 3^2*x^2/(2*(1+x^2)) + 3^3*x^3/(3*(1+x^3)) + 3^4*x^4/(4*(1+x^4)) + 3^5*x^5/(5*(1+x^5)) +...

Crossrefs

Cf. A165941, A259274, A259275, A259276.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[Exp[Sum[3^k * x^k / (1 + x^k)/k, {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 18 2020 *)
  • PARI
    {a(n)=if(n==0, 1, polcoeff(exp(sum(m=1, n, 3^m*x^m/(1+x^m+x*O(x^n))/m)), n))}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=(1 + x^(n+1-i)*A)/(1 - 2*x^(n+1-i)*A+ x*O(x^n))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f.: -1/2 + (3/2)/(1+x - 3*x/(1+x^2 - 3*x^2/(1+x^3 - 3*x^3/(1+x^4 - 3*x^4/(1+x^5 - 3*x^5/(1+x^6 - 3*x^6/(1+x^7 - 3*x^7/(1+x^8 - 3*x^8/(...))))))))), a continued fraction.
G.f.: A(x) = (1 + x*B(x))/(1 - 2*x*B(x)), where B(x) = (1 + x^2*C(x))/(1 - 2*x^2*C(x)), C(x) = (1 + x^3*D(x))/(1 - 2*x^3*D(x)), D(x) = (1 + x^4*E(x))/(1 - 2*x^4*E(x)), ...
a(n) ~ c * 3^n, where c = 2 / (3^(1/8) * EllipticTheta(2, 0, 1/sqrt(3))) = 0.7289909630241618243925302344904284400138198884186993... - Vaclav Kotesovec, Oct 18 2020, updated Apr 18 2024

A259274 G.f.: A(x) = exp( Sum_{n>=1} 4^n * x^n/(n*(1+x^n)) ).

Original entry on oeis.org

1, 4, 12, 52, 204, 804, 3244, 12948, 51756, 207108, 828364, 3313332, 13253580, 53014116, 212055852, 848224660, 3392897772, 13571588484, 54286358988, 217145432052, 868581718860, 3474326895460, 13897307565804, 55589230225428, 222356920980972, 889427683862724, 3557710735299660
Offset: 0

Views

Author

Paul D. Hanna, Jun 23 2015

Keywords

Comments

Compare to: exp( Sum_{n>=1} x^n/(1+x^n)/n ) = Sum_{n>=0} x^(n*(n+1)/2).

Examples

			G.f.: A(x) = 1 + 4*x + 12*x^2 + 52*x^3 + 204*x^4 + 804*x^5 + 3244*x^6 +...
such that
log(A(x)) = 4*x/(1+x) + 4^2*x^2/(2*(1+x^2)) + 4^3*x^3/(3*(1+x^3)) + 4^4*x^4/(4*(1+x^4)) + 4^5*x^5/(5*(1+x^5)) +...

Crossrefs

Cf. A165941, A259273, A259275, A259276.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Exp[Sum[4^k * x^k / (1 + x^k)/k, {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 18 2020 *)
  • PARI
    {a(n)=if(n==0, 1, polcoeff(exp(sum(m=1, n, 4^m*x^m/(1+x^m+x*O(x^n))/m)), n))}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=(1 + x^(n+1-i)*A)/(1 - 3*x^(n+1-i)*A+ x*O(x^n))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f.: -1/3 + (4/3)/(1+x - 4*x/(1+x^2 - 4*x^2/(1+x^3 - 4*x^3/(1+x^4 - 4*x^4/(1+x^5 - 4*x^5/(1+x^6 - 4*x^6/(1+x^7 - 4*x^7/(1+x^8 - 4*x^8/(...))))))))), a continued fraction.
G.f.: A(x) = (1 + x*B(x))/(1 - 3*x*B(x)), where B(x) = (1 + x^2*C(x))/(1 - 3*x^2*C(x)), C(x) = (1 + x^3*D(x))/(1 - 3*x^3*D(x)), D(x) = (1 + x^4*E(x))/(1 - 3*x^4*E(x)), ...
a(n) ~ c * 4^n, where c = 2^(3/4) / EllipticTheta[2, 0, 1/2] = 0.789970474669932371974378022396788915338046391238667... - Vaclav Kotesovec, Oct 18 2020, updated Mar 17 2024

A259275 G.f.: A(x) = exp( Sum_{n>=1} 5^n * x^n/(n*(1+x^n)) ).

Original entry on oeis.org

1, 5, 20, 105, 520, 2580, 12945, 64680, 323320, 1616780, 8083745, 40418380, 202092620, 1010462480, 5052310420, 25261556205, 126307777920, 631538879180, 3157694416720, 15788472066780, 78942360284720, 394711801527505, 1973559007551520, 9867795037511480, 49338975188073020
Offset: 0

Views

Author

Paul D. Hanna, Jun 23 2015

Keywords

Comments

Compare to: exp( Sum_{n>=1} x^n/(1+x^n)/n ) = Sum_{n>=0} x^(n*(n+1)/2).

Examples

			G.f.: A(x) = 1 + 5*x + 20*x^2 + 105*x^3 + 520*x^4 + 2580*x^5 +...
such that
log(A(x)) = 5*x/(1+x) + 5^2*x^2/(2*(1+x^2)) + 5^3*x^3/(3*(1+x^3)) + 5^4*x^4/(4*(1+x^4)) + 5^5*x^5/(5*(1+x^5)) +...

Crossrefs

Cf. A165941, A259273, A259274, A259276.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Exp[Sum[5^k * x^k / (1 + x^k)/k, {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 18 2020 *)
  • PARI
    {a(n)=if(n==0, 1, polcoeff(exp(sum(m=1, n, 5^m*x^m/(1+x^m+x*O(x^n))/m)), n))}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=(1 + x^(n+1-i)*A)/(1 - 4*x^(n+1-i)*A+ x*O(x^n))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f.: -1/4 + (5/4)/(1+x - 5*x/(1+x^2 - 5*x^2/(1+x^3 - 5*x^3/(1+x^4 - 5*x^4/(1+x^5 - 5*x^5/(1+x^6 - 5*x^6/(1+x^7 - 5*x^7/(1+x^8 - 5*x^8/(...))))))))), a continued fraction.
G.f.: A(x) = (1 + x*B(x))/(1 - 4*x*B(x)), where B(x) = (1 + x^2*C(x))/(1 - 4*x^2*C(x)), C(x) = (1 + x^3*D(x))/(1 - 4*x^3*D(x)), D(x) = (1 + x^4*E(x))/(1 - 4*x^4*E(x)), ...
a(n) ~ c * 5^n, where c = 2 / (5^(1/8) * EllipticTheta(2, 0, 1/sqrt(5))) = 0.8277706439469762656495798472679610454060848013727259... - Vaclav Kotesovec, Oct 18 2020, updated Apr 18 2024
Showing 1-4 of 4 results.

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