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 1-8 of 8 results.

A205479 L.g.f.: Sum_{n>=1} (x^n/n) * Product_{d|n} (1 + d*x^n).

Original entry on oeis.org

1, 3, 1, 7, 1, 15, 1, 15, 10, 13, 1, 67, 1, 17, 16, 63, 1, 168, 1, 37, 22, 25, 1, 555, 1, 29, 118, 49, 1, 520, 1, 543, 34, 37, 1, 1048, 1, 41, 40, 1125, 1, 554, 1, 73, 475, 49, 1, 6651, 1, 563, 52, 85, 1, 1680, 1, 1457, 58, 61, 1, 20632, 1, 65, 787, 5087, 1
Offset: 1

Views

Author

Paul D. Hanna, Jan 27 2012

Keywords

Examples

			L.g.f.: L(x) = x + 3*x^2/2 + x^3/3 + 7*x^4/4 + x^5/5 + 15*x^6/6 +...
By definition:
L(x) = x*(1+x) + x^2*(1+x^2)*(1+2*x^2)/2 + x^3*(1+x^3)*(1+3*x^3)/3 + x^4*(1+x^4)*(1+2*x^4)*(1+4*x^4)/4 + x^5*(1+x^5)*(1+5*x^5)/5 + x^6*(1+x^6)*(1+2*x^6)*(1+3*x^6)*(1+6*x^6)/6 +...
Exponentiation yields the g.f. of A205478:
exp(L(x)) = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 4*x^5 + 8*x^6 + 8*x^7 +...

Links

Crossrefs

Cf. A205478 (exp), A205477, A205481, A205483, A205485, A205487, A205489, A205491.

Programs

  • Mathematica
    max = 70; s = Sum[(x^(n-1)/n)*Product[1+d*x^n, {d, Divisors[n]}], {n, 1, max}] + O[x]^max; CoefficientList[s, x]*Range[max] (* Jean-François Alcover, Dec 23 2015 *)
  • PARI
    {a(n)=n*polcoeff(sum(m=1, n+1, x^m/m*exp(sumdiv(m, d, log(1+d*x^m+x*O(x^n))))), n)}

Formula

Forms the logarithmic derivative of A205478.

A205476 G.f.: exp( Sum_{n>=1} (x^n/n) * Product_{d|n} (1 + n*x^d/d) ).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 12, 20, 28, 45, 65, 101, 148, 221, 316, 469, 673, 969, 1420, 2025, 2892, 4100, 5905, 8314, 11860, 16645, 23399, 32838, 46071, 64274, 89761, 124977, 173231, 240492, 332978, 460015, 634271, 874464, 1200463, 1649499, 2263102, 3098661, 4239109
Offset: 0

Views

Author

Paul D. Hanna, Jan 27 2012

Keywords

Comments

Note: exp( Sum_{n>=1} (x^n/n) * Product_{d|n} (1 + x^d) ) does not yield an integer series.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 8*x^5 + 12*x^6 + 20*x^7 + ...
By definition:
log(A(x)) = x*(1+x) + x^2*(1+2*x)*(1+x^2)/2 + x^3*(1+3*x)*(1+x^3)/3 + x^4*(1+4*x)*(1+2*x^2)*(1+x^4)/4 + x^5*(1+5*x)*(1+x^5)/5 + x^6*(1+6*x)*(1+3*x^2)*(1+2*x^3)*(1+x^6)/6 + ...
Explicitly,
log(A(x)) = x + 3*x^2/2 + 4*x^3/3 + 7*x^4/4 + 11*x^5/5 + 12*x^6/6 + 29*x^7/7 + 15*x^8/8 + 49*x^9/9 + 43*x^10/10 + ... + A205477(n)*x^n/n + ...

Crossrefs

Cf. A205477 (log), A198304, A205478, A205480, A205482, A205484, A205486, A205488, A205490.

Programs

  • Mathematica
    max = 50; s = Exp[Sum[(x^n/n)*Product[1+n*x^d/d, {d, Divisors[n]}], {n, 1, max}]] + O[x]^max; CoefficientList[s , x] (* Jean-François Alcover, Dec 23 2015 *)
  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*exp(sumdiv(m, d, log(1+m*x^d/d+x*O(x^n)))))), n)}

Formula

Logarithmic derivative yields A205477.

A205480 G.f: exp( Sum_{n>=1} x^n/n * Product_{d|n} (1 + d*x^(n/d))^d ).

Original entry on oeis.org

1, 1, 2, 4, 10, 27, 76, 242, 852, 3016, 11262, 47004, 204761, 894673, 4134909, 20370101, 101904474, 521459464, 2813783214, 15616060213, 87143803196, 502477538546, 3039137586808, 18763942581733, 116737580008529, 742909490860950, 4846956807516551
Offset: 0

Views

Author

Paul D. Hanna, Jan 27 2012

Keywords

Comments

Note: exp( Sum_{n>=1} x^n/n * Product_{d|n} (1 + x^(n/d))^d ) does not yield an integer series.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 27*x^5 + 76*x^6 + 242*x^7 + ...
By definition:
log(A(x)) = x*(1+x) + x^2*(1+x^2)*(1+2*x)^2/2 + x^3*(1+x^3)*(1+3*x)^3/3 + x^4*(1+x^4)*(1+2*x^2)^2*(1+4*x)^4/4 + x^5*(1+x^5)*(1+5*x)^5/5 + x^6*(1+x^6)*(1+2*x^3)^2*(1+3*x^2)^3*(1+6*x)^6/6 + ...
Explicitly,
log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 23*x^4/4 + 76*x^5/5 + 249*x^6/6 + 974*x^7/7 + 4151*x^8/8 + 16558*x^9/9 + ... + A205481(n)*x^n/n + ...

Crossrefs

Cf. A205481 (log), A205476, A205478, A205482, A205484, A205486, A205488, A205490.

Programs

  • Mathematica
    max = 30; s = Exp[Sum[(x^n/n)*Product[(1+d*x^(n/d))^d, {d, Divisors[n]}], {n, 1, max}]] + O[x]^max; CoefficientList[s , x] (* Jean-François Alcover, Dec 23 2015 *)
  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*exp(sumdiv(m, d, d*log(1+d*x^(m/d)+x*O(x^n)))))), n)}

Formula

Logarithmic derivative yields A205481.

A205482 G.f.: exp( Sum_{n>=1} x^n/n * Product_{d|n} (1 + d*x^n)^d ).

Original entry on oeis.org

1, 1, 2, 2, 5, 5, 15, 15, 34, 46, 84, 96, 246, 282, 512, 696, 1421, 1713, 3436, 4084, 8227, 10821, 19128, 23258, 48474, 60943, 106780, 139313, 252748, 322577, 600660, 760872, 1365570, 1807979, 3064882, 3951491, 7358684, 9476993, 15962935, 21243381
Offset: 0

Views

Author

Paul D. Hanna, Jan 27 2012

Keywords

Comments

Note: exp( Sum_{n>=1} x^n/n * Product_{d|n} (1 + x^n)^d ) does not yield an integer series.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 2*x^3 + 5*x^4 + 5*x^5 + 15*x^6 + 15*x^7 + ...
By definition:
log(A(x)) = x*(1+x) + x^2*(1+x^2)*(1+2*x^2)^2/2 + x^3*(1+x^3)*(1+3*x^3)^3/3 + x^4*(1+x^4)*(1+2*x^4)^2*(1+4*x^4)^4/4 + x^5*(1+x^5)*(1+5*x^5)^5/5 + x^6*(1+x^6)*(1+2*x^6)^2*(1+3*x^6)^3*(1+6*x^6)^6/6 + ...
Explicitly,
log(A(x)) = x + 3*x^2/2 + x^3/3 + 11*x^4/4 + x^5/5 + 45*x^6/6 + x^7/7 + 59*x^8/8 + 109*x^9/9 + 53*x^10/10 + ... + A205483(n)*x^n/n + ...

Crossrefs

Cf. A205483 (log), A205476, A205478, A205480, A205484, A205486, A205488, A205490.

Programs

  • Mathematica
    max = 40; s = Exp[Sum[(x^n/n)*Product[(1 + d*x^n)^d, {d, Divisors[n]}], {n, 1, max}]] + O[x]^max; CoefficientList[s , x] (* Jean-François Alcover, Dec 23 2015 *)
  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*exp(sumdiv(m, d, d*log(1+d*x^m+x*O(x^n)))))), n)}

Formula

Logarithmic derivative yields A205483.

A205484 G.f.: exp( Sum_{n>=1} x^n/n * Product_{d|n} (1 + d*x^d)^n ).

Original entry on oeis.org

1, 1, 2, 3, 7, 14, 30, 65, 132, 280, 632, 1439, 3299, 7569, 17450, 40313, 92889, 212801, 483590, 1092649, 2467078, 5581232, 12690828, 29123728, 67648617, 159370347, 381080620, 923803158, 2264970530, 5599185887, 13909201590, 34612152762, 86049014990
Offset: 0

Views

Author

Paul D. Hanna, Jan 27 2012

Keywords

Comments

Note: exp( Sum_{n>=1} x^n/n * Product_{d|n} (1 + x^d)^n ) does not yield an integer series.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 7*x^4 + 14*x^5 + 30*x^6 + 65*x^7 + ...
By definition:
log(A(x)) = x*(1+x) + x^2*(1+x)^2*(1+2*x^2)^2/2 + x^3*(1+x)^3*(1+3*x^3)^3/3 + x^4*(1+x)^4*(1+2*x^2)^4*(1+4*x^4)^4/4 + x^5*(1+x)^5*(1+5*x^5)^5/5 + x^6*(1+x)^6*(1+2*x^2)^6*(1+3*x^3)^6*(1+6*x^6)^6/6 + ...
Explicitly,
log(A(x)) = x + 3*x^2/2 + 4*x^3/3 + 15*x^4/4 + 31*x^5/5 + 72*x^6/6 + 176*x^7/7 + 327*x^8/8 + 751*x^9/9 + ... + A205485(n)*x^n/n + ...

Crossrefs

Cf. A205485 (log), A205476, A205478, A205480, A205482, A205486, A205488, A205490.

Programs

  • Mathematica
    max = 40; s = Exp[Sum[(x^n/n)*Product[(1 + d*x^d)^n, {d, Divisors[n]}], {n, 1, max}]] + O[x]^max; CoefficientList[s , x] (* Jean-François Alcover, Dec 23 2015 *)
  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*exp(sumdiv(m, d, m*log(1+d*x^d+x*O(x^n)))))), n)}

Formula

Logarithmic derivative yields A205485.

A205486 G.f.: exp( Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - d*x^(n/d))^d ).

Original entry on oeis.org

1, 1, 2, 5, 16, 60, 259, 1273, 7048, 43241, 289685, 2097912, 16317134, 135574160, 1196898329, 11168544771, 109647222799, 1128440311914, 12139734936953, 136195813530558, 1590028534430967, 19277087785530470, 242235954813757132, 3149491477171141810
Offset: 0

Views

Author

Paul D. Hanna, Jan 27 2012

Keywords

Comments

Note: exp( Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - x^(n/d))^d ) does not yield an integer series.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 16*x^4 + 60*x^5 + 259*x^6 + 1273*x^7 +...
By definition:
log(A(x)) = x/(1-x) + (x^2/2)/((1-x^2)*(1-2*x)^2) + (x^3/3)/((1-x^3)*(1-3*x)^3) + (x^4/4)/((1-x^4)*(1-2*x^2)^2*(1-4*x)^4) + (x^5/5)/((1-x^5)*(1-5*x)^5) + (x^6/6)/((1-x^6)*(1-2*x^3)^2*(1-3*x^2)^3*(1-6*x)^6) +...
Explicitly,
log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 43*x^4/4 + 206*x^5/5 + 1104*x^6/6 + 6581*x^7/7 + 43227*x^8/8 +...+ A205487(n)*x^n/n +...

Crossrefs

Cf. A205487 (log), A205476, A205478, A205480, A205482, A205484, A205488, A205490.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*exp(sumdiv(m, d, -d*log(1-d*x^(m/d)+x*O(x^n)))))), n)}

Formula

Logarithmic derivative yields A205487.

A205488 G.f.: exp( Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - d*x^n)^d ).

Original entry on oeis.org

1, 1, 2, 3, 7, 9, 26, 32, 85, 129, 293, 389, 1105, 1426, 3242, 5121, 11405, 15790, 37704, 52289, 118470, 181770, 373257, 543137, 1250814, 1796474, 3713201, 5790866, 11919993, 17663189, 37445802, 55499537, 113653306, 177038255, 347919239, 529686395, 1091858128
Offset: 0

Views

Author

Paul D. Hanna, Jan 27 2012

Keywords

Comments

Note: exp( Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - x^n)^d ) does not yield an integer series.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 7*x^4 + 9*x^5 + 26*x^6 + 32*x^7 +...
By definition:
log(A(x)) = x/(1-x) + (x^2/2)/((1-x^2)*(1-2*x^2)^2) + (x^3/3)/((1-x^3)*(1-3*x^3)^3) + (x^4/4)/((1-x^4)*(1-2*x^4)^2*(1-4*x^4)^4) + (x^5/5)/((1-x^5)*(1-5*x^5)^5) + (x^6/6)/((1-x^6)*(1-2*x^6)^2*(1-3*x^6)^3*(1-6*x^6)^6) +...
Explicitly,
log(A(x)) = x + 3*x^2/2 + 4*x^3/3 + 15*x^4/4 + 6*x^5/5 + 78*x^6/6 + 8*x^7/7 + 247*x^8/8 + 202*x^9/9 + 708*x^10/10 +...+ A205489(n)*x^n/n +...

Crossrefs

Cf. A205489 (log), A205476, A205478, A205480, A205482, A205484, A205486, A205490.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*exp(sumdiv(m, d, -d*log(1-d*x^m+x*O(x^n)))))), n)}

Formula

Logarithmic derivative yields A205489.

A205490 G.f.: exp( Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - d*x^d)^n ).

Original entry on oeis.org

1, 1, 2, 4, 10, 22, 57, 134, 331, 797, 1995, 4879, 12367, 31056, 79315, 202370, 521575, 1339934, 3456778, 8885907, 22848211, 58576714, 150117209, 384135566, 983789032, 2522109065, 6485104365, 16736092434, 43408268497, 113201300205, 296975753940, 783578962587
Offset: 0

Views

Author

Paul D. Hanna, Jan 27 2012

Keywords

Comments

Note: exp( Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - x^d)^n ) does not yield an integer series.

Examples

			 G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 22*x^5 + 57*x^6 + 134*x^7 +...
By definition:
log(A(x)) = x/(1-x) + (x^2/2)/((1-x)^2*(1-2*x^2)^2) + (x^3/3)/((1-x)^3*(1-3*x^3)^3) + (x^4/4)/((1-x)^4*(1-2*x^2)^4*(1-4*x^4)^4) + (x^5/5)/((1-x)^5*(1-5*x^5)^5) + (x^6/6)/((1-x)^6*(1-2*x^2)^6*(1-3*x^3)^6*(1-6*x^6)^6) +...
Explicitly,
log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 23*x^4/4 + 51*x^5/5 + 165*x^6/6 + 386*x^7/7 + 1039*x^8/8 + 2554*x^9/9 +...+ A205491(n)*x^n/n +...

Crossrefs

Cf. A205491 (log), A205476, A205478, A205480, A205482, A205484, A205486, A205488.

Formula

Logarithmic derivative yields A205491.
Showing 1-8 of 8 results.

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