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 1-6 of 6 results.

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

A259276 G.f.: A(x) = exp( Sum_{n>=1} 6^n * x^n/(n*(1+x^n)) ).

Original entry on oeis.org

1, 6, 30, 186, 1110, 6630, 39846, 239010, 1433910, 8603790, 51622446, 309733890, 1858404990, 11150428470, 66902565630, 401415404586, 2408492418870, 14450954480790, 86705726950470, 520234361647890, 3121406169699270, 18728437018590366, 112370622111206670, 674223732666113010
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).
In general, for m > 1, if g.f. = exp(Sum_{k>=1} m^k * x^k/(k*(1+x^k))) then a(n) ~ 2 * m^(n - 1/8) / EllipticTheta(2, 0, 1/sqrt(m)). - Vaclav Kotesovec, Apr 18 2024

Examples

			G.f.: A(x) = 1 + 6*x + 30*x^2 + 186*x^3 + 1110*x^4 + 6630*x^5 +...
such that
log(A(x)) = 6*x/(1+x) + 6^2*x^2/(2*(1+x^2)) + 6^3*x^3/(3*(1+x^3)) + 6^4*x^4/(4*(1+x^4)) + 6^5*x^5/(5*(1+x^5)) +...

Crossrefs

Cf. A165941, A259273, A259274, A259275.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Exp[Sum[6^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, 6^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 - 5*x^(n+1-i)*A+ x*O(x^n))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f.: -1/5 + (6/5)/(1+x - 6*x/(1+x^2 - 6*x^2/(1+x^3 - 6*x^3/(1+x^4 - 6*x^4/(1+x^5 - 6*x^5/(1+x^6 - 6*x^6/(1+x^7 - 6*x^7/(1+x^8 - 6*x^8/(...))))))))), a continued fraction.
G.f.: A(x) = (1 + x*B(x))/(1 - 5*x*B(x)), where B(x) = (1 + x^2*C(x))/(1 - 5*x^2*C(x)), C(x) = (1 + x^3*D(x))/(1 - 5*x^3*D(x)), D(x) = (1 + x^4*E(x))/(1 - 5*x^4*E(x)), ...
a(n) ~ c * 6^n, where c = 2^(7/8) / (3^(1/8) * EllipticTheta(2, 0, 1/sqrt(6))) = 0.8537393061385536395511385858124987202486847622399194... - Vaclav Kotesovec, Oct 18 2020, updated Apr 18 2024

A348902 G.f. A(x) satisfies: A(x) = 1 / (1 + x - 2 * x * A(4*x)).

Original entry on oeis.org

1, 1, 9, 305, 39705, 20412737, 41846783913, 342892875489361, 11236600170415809849, 1472826135905484728387681, 772188014962631262957890704329, 1619397184353040716422147490531778929, 13584491414647344530078887450781292845554521
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 03 2021

Keywords

Crossrefs

Cf. A001003, A015085, A165941, A348188, A348901.

Programs

  • Mathematica
    nmax = 12; A[] = 0; Do[A[x] = 1/(1 + x - 2 x A[4 x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = -a[n - 1] + Sum[2^(2 k + 1) a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 12}]

Formula

a(0) = 1; a(n) = -a(n-1) + Sum_{k=0..n-1} 2^(2*k+1) * a(k) * a(n-k-1).
a(n) ~ c * 2^(n^2), where c = 2^(7/8) / EllipticTheta(2, 0, 1/sqrt(2)) = 0.6091497110662286155211146043057245512950999410185846745870491125003511... (same constant as in A165941). - Vaclav Kotesovec, Nov 03 2021, updated Apr 21 2024

A259192 Triangle, such that the g.f. satisfies: A(x,y) = (1 + x*A(x*y,y)) / (1 - x*A(x*y,y)).

Original entry on oeis.org

1, 2, 2, 4, 2, 8, 4, 8, 2, 12, 16, 20, 16, 8, 16, 2, 16, 36, 48, 68, 40, 64, 40, 32, 16, 32, 2, 20, 64, 108, 176, 172, 208, 216, 160, 168, 144, 128, 80, 64, 32, 64, 2, 24, 100, 216, 388, 528, 612, 784, 704, 792, 672, 728, 576, 560, 384, 464, 288, 256, 160, 128, 64, 128, 2, 28, 144, 388, 784, 1300, 1696, 2316, 2544, 2864, 2976, 3000, 3024, 2856, 2560, 2400, 2416, 1856, 1776, 1408, 1248, 1024, 928, 576, 512, 320, 256, 128, 256
Offset: 0

Views

Author

Paul D. Hanna, Jun 21 2015

Keywords

Comments

Row sums = A006318, the large Schröder numbers.
Antidiagonal sums = A165941; g.f.: exp( Sum_{n>=1} 2^n*x^n/(n*(1+x^n)) ).
G.f. evaluated at y=1/2: A(x,1/2) = 1/(1-2*x).

Examples

			G.f.: A(x,y) = Sum_{n>=0} Sum_{k=0..n*(n-1)/2} T(n,k) * x^n*y^k.
G.f.: A(x,y) = 1 + x*(2) + x^2*(2 + 4*y) +
x^3*(2 + 8*y + 4*y^2 + 8*y^3) +
x^4*(2 + 12*y + 16*y^2 + 20*y^3 + 16*y^4 + 8*y^5 + 16*y^6) +
x^5*(2 + 16*y + 36*y^2 + 48*y^3 + 68*y^4 + 40*y^5 + 64*y^6 + 40*y^7 + 32*y^8 + 16*y^9 + 32*y^10) +
x^6*(2 + 20*y + 64*y^2 + 108*y^3 + 176*y^4 + 172*y^5 + 208*y^6 + 216*y^7 + 160*y^8 + 168*y^9 + 144*y^10 + 128*y^11 + 80*y^12 + 64*y^13 + 32*y^14 + 64*y^15) +...
such that
A(x,y) = (1 + x*A(x*y,y)) / (1 - x*A(x*y,y)).
This triangle of coefficients begins:
1;
2;
2, 4;
2, 8, 4, 8;
2, 12, 16, 20, 16, 8, 16;
2, 16, 36, 48, 68, 40, 64, 40, 32, 16, 32;
2, 20, 64, 108, 176, 172, 208, 216, 160, 168, 144, 128, 80, 64, 32, 64;
2, 24, 100, 216, 388, 528, 612, 784, 704, 792, 672, 728, 576, 560, 384, 464, 288, 256, 160, 128, 64, 128;
2, 28, 144, 388, 784, 1300, 1696, 2316, 2544, 2864, 2976, 3000, 3024, 2856, 2560, 2400, 2416, 1856, 1776, 1408, 1248, 1024, 928, 576, 512, 320, 256, 128, 256;
2, 32, 196, 640, 1476, 2808, 4260, 6104, 7844, 9216, 10816, 11264, 12512, 12424, 12608, 11784, 12384, 10848, 10880, 9328, 8992, 7888, 7488, 5952, 5856, 4352, 4064, 3072, 3008, 2048, 1856, 1152, 1024, 640, 512, 256, 512; ...

Links

Crossrefs

Cf. A006318, A165941.

Programs

  • PARI
    {T(n,k) = local(A=1+2*x); for(i=1,n, A = (1 + x*subst(A,x,x*y))/(1 - x*subst(A,x,x*y +x*O(x^n))) ); polcoeff( polcoeff(A,n,x) ,k,y) }
for(n=0,10, for(k=0,n*(n-1)/2, print1( T(n,k),", "));print(""))

Formula

G.f.: A(x,y) = -1 + 2/(1+x - 2*x/(1+x*y - 2*x*y/(1+x*y^2 - 2*x*y^2/(1+x*y^3 - 2*x*y^3/(1+x*y^4 - 2*x*y^4/(1+x*y^5 - 2*x*y^5/(1+x*y^6 - 2*x*y^6/(1+x*y^7 -...)))))))), a continued fraction.
Showing 1-6 of 6 results.

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