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.

A205477 L.g.f.: Sum_{n>=1} (x^n/n) * Product_{d|n} (1 + n*x^d/d).

Original entry on oeis.org

1, 3, 4, 7, 11, 12, 29, 15, 49, 43, 100, 100, 157, 45, 299, 159, 273, 795, 761, 307, 830, 2126, 1657, 3276, 1711, 965, 3505, 6405, 1509, 9967, 6976, 9375, 8188, 24483, 8089, 26299, 20795, 29871, 40408, 112475, 51497, 164022, 27650, 83398, 74639, 208015, 280074
Offset: 1

Views

Author

Paul D. Hanna, Jan 27 2012

Keywords

Examples

			L.g.f.: L(x) = x + 3*x^2/2 + 4*x^3/3 + 7*x^4/4 + 11*x^5/5 + 12*x^6/6 +...
By definition:
L(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 +...
Exponentiation yields the g.f. of A205476:
exp(L(x)) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 8*x^5 + 12*x^6 + 20*x^7 +...

Crossrefs

Cf. A205476 (exp), A205479, A205481, A205483, A205485, A205487, A205489, A205491.

Programs

  • Mathematica
    max = 50; s = Sum[(x^(n-1)/n)*Product[1+n*x^d/d, {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+m*x^d/d+x*O(x^n))))), n)}

Formula

Forms the logarithmic derivative of A205476.

A205478 G.f.: exp( Sum_{n>=1} (x^n/n) * Product_{d|n} (1 + d*x^n) ).

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 8, 8, 14, 15, 24, 25, 43, 45, 69, 74, 113, 120, 187, 198, 291, 314, 452, 483, 720, 770, 1089, 1182, 1657, 1784, 2530, 2724, 3764, 4102, 5593, 6053, 8361, 9049, 12183, 13304, 17831, 19378, 26097, 28355, 37548, 41107, 54031, 58894, 78008, 85052
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) ) does not yield an integer series.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 4*x^5 + 8*x^6 + 8*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^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 + ...
Explicitly,
log(A(x)) = x + 3*x^2/2 + x^3/3 + 7*x^4/4 + x^5/5 + 15*x^6/6 + x^7/7 + 15*x^8/8 + 10*x^9/9 + 13*x^10/10 + ... + A205479(n)*x^n/n + ...

Crossrefs

Cf. A205479, A198296, A205476, A205480, A205482, A205484, A205486, A205488, A205490.

Programs

  • Mathematica
    max = 50; s = Exp[Sum[(x^n/n)*Product[1+d*x^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, log(1+d*x^m+x*O(x^n)))))), n)}

Formula

Logarithmic derivative yields A205479.

A205481 L.g.f.: Sum_{n>=1} x^n/n * Product_{d|n} (1 + d*x^(n/d))^d.

Original entry on oeis.org

1, 3, 7, 23, 76, 249, 974, 4151, 16558, 70308, 342937, 1680725, 8012252, 40903572, 222539812, 1202060807, 6608077855, 38523427818, 228629565951, 1349303611408, 8257330774574, 53118486147015, 345693735519287, 2252515985849693, 15028013765653626, 102689873016938288
Offset: 1

Views

Author

Paul D. Hanna, Jan 27 2012

Keywords

Examples

			L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 23*x^4/4 + 76*x^5/5 + 249*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^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 +...
Exponentiation yields the g.f. of A205480:
exp(L(x)) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 27*x^5 + 76*x^6 + 242*x^7 +...

Crossrefs

Cf. A205480 (exp), A205477, A205479, A205483, A205485, A205487, A205489, A205491.

Programs

  • PARI
    {a(n)=n*polcoeff(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

Forms the logarithmic derivative of A205480.

A205483 L.g.f.: Sum_{n>=1} x^n/n * Product_{d|n} (1 + d*x^n)^d.

Original entry on oeis.org

1, 3, 1, 11, 1, 45, 1, 59, 109, 53, 1, 869, 1, 101, 961, 3643, 1, 3555, 1, 18101, 3235, 245, 1, 92645, 21876, 341, 11287, 74141, 1, 722045, 1, 324667, 20329, 581, 502076, 5280611, 1, 725, 40054, 7567509, 1, 27239663, 1, 906301, 7838224, 1061, 1, 181474021
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 + 11*x^4/4 + x^5/5 + 45*x^6/6 +...
By definition:
L(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 +...
Exponentiation yields the g.f. of A205482:
exp(L(x)) = 1 + x + 2*x^2 + 2*x^3 + 5*x^4 + 5*x^5 + 15*x^6 + 15*x^7 +...

Crossrefs

Cf. A205482 (exp), A205477, A205479, A205481, A205485, A205487, A205489, A205491.

Programs

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

Formula

Forms the logarithmic derivative of A205482.

A205485 L.g.f.: Sum_{n>=1} x^n/n * Product_{d|n} (1 + d*x^d)^n.

Original entry on oeis.org

1, 3, 4, 15, 31, 72, 176, 327, 751, 2063, 5138, 12708, 30993, 75386, 182644, 433255, 1004854, 2279349, 5115960, 11580835, 26533616, 62024966, 149683357, 373141332, 957942931, 2516465279, 6694846987, 17883365774, 47644695777, 125952933062, 329364348277
Offset: 1

Views

Author

Paul D. Hanna, Jan 27 2012

Keywords

Examples

			L.g.f.: L(x) = x + 3*x^2/2 + 4*x^3/3 + 15*x^4/4 + 31*x^5/5 + 72*x^6/6 +...
By definition:
L(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 +...
Exponentiation yields the g.f. of A205484:
exp(L(x)) = 1 + x + 2*x^2 + 3*x^3 + 7*x^4 + 14*x^5 + 30*x^6 + 65*x^7 +...

Crossrefs

Cf. A205484 (exp), A205477, A205479, A205481, A205483, A205487, A205489, A205491.

Programs

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

Formula

Forms the logarithmic derivative of A205484.

A205487 L.g.f.: Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - d*x^(n/d))^d.

Original entry on oeis.org

1, 3, 10, 43, 206, 1104, 6581, 43227, 307927, 2351288, 19124238, 165102052, 1507907818, 14512524085, 146581677005, 1548261405595, 17054944088112, 195518380169283, 2328512358930925, 28759349826041248, 367752208054445945, 4860792910118985370
Offset: 1

Views

Author

Paul D. Hanna, Jan 27 2012

Keywords

Examples

			L.g.f.: L(x) = x + 3*x^2/2 + 10*x^3/3 + 43*x^4/4 + 206*x^5/5 + 1104*x^6/6 +...
By definition:
L(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) +...
Exponentiation yields the g.f. of A205486:
exp(L(x)) = 1 + x + 2*x^2 + 5*x^3 + 16*x^4 + 60*x^5 + 259*x^6 + 1273*x^7 +...

Crossrefs

Cf. A205486 (exp), A205477, A205479, A205481, A205483, A205485, A205489, A205491.

Programs

  • PARI
    {a(n)=n*polcoeff(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

Forms the logarithmic derivative of A205486.

A205489 L.g.f.: Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - d*x^n)^d.

Original entry on oeis.org

1, 3, 4, 15, 6, 78, 8, 247, 202, 708, 12, 4146, 14, 5498, 8964, 24135, 18, 81114, 20, 206520, 193736, 225558, 24, 2314378, 242656, 1278332, 3622954, 9209950, 30, 26654118, 32, 58890983, 59213598, 35652216, 28736938, 628796418, 38, 179307278, 878319368
Offset: 1

Views

Author

Paul D. Hanna, Jan 27 2012

Keywords

Examples

			L.g.f.: L(x) = x + 3*x^2/2 + 4*x^3/3 + 15*x^4/4 + 6*x^5/5 + 78*x^6/6 +...
By definition:
L(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) +...
Exponentiation yields the g.f. of A205488:
exp(L(x)) = 1 + x + 2*x^2 + 3*x^3 + 7*x^4 + 9*x^5 + 26*x^6 + 32*x^7 +...

Crossrefs

Cf. A205488 (exp), A205477, A205479, A205481, A205483, A205485, A205487, A205491.

Programs

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

Formula

Forms the logarithmic derivative of A205488.

A205491 L.g.f.: Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - d*x^d)^n.

Original entry on oeis.org

1, 3, 7, 23, 51, 165, 386, 1039, 2554, 6963, 17260, 47825, 124840, 340658, 911037, 2484687, 6614616, 17735646, 46647167, 122536323, 318125129, 825153684, 2130076369, 5522611009, 14375957026, 37817347272, 100579846732, 271246531726, 740731197176
Offset: 1

Views

Author

Paul D. Hanna, Jan 27 2012

Keywords

Examples

			 L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 23*x^4/4 + 51*x^5/5 + 165*x^6/6 +...
By definition:
L(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) +...
Exponentiation yields the g.f. of A205490:
exp(L(x)) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 22*x^5 + 57*x^6 + 134*x^7 +...

Crossrefs

Cf. A205490 (exp), A205477, A205479, A205481, A205483, A205485, A205487, A205489.

Formula

Forms the logarithmic derivative of A205490.
Showing 1-8 of 8 results.

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