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.

A143441 Decimal expansion of the (negated) value of q at which the q-Pochhammer symbol reaches a maximum along [ -1, 1].

Original entry on oeis.org

4, 1, 1, 2, 4, 8, 4, 7, 9, 1, 7, 7, 9, 5, 4, 7, 7, 3, 4, 4, 4, 0, 2, 5, 6, 6, 2, 4, 3, 5, 5, 7, 2, 4, 3, 6, 9, 7, 2, 0, 4, 0, 5, 0, 3, 6, 3, 3, 6, 0, 1, 1, 0, 5, 5, 7, 0, 2, 1, 1, 0, 1, 7, 8, 3, 6, 4, 4, 2, 9, 1, 3, 4, 5, 3, 3, 8, 1, 4, 4, 7, 1, 5, 0, 7, 7, 2, 0, 9, 5, 0, 6, 3, 3, 3, 9, 2, 4, 1, 8, 5, 6, 7, 3, 1
Offset: 0

Views

Author

Eric W. Weisstein, Aug 14 2008

Keywords

Examples

			-0.41124847917795477344...

Links

Crossrefs

Cf. A143440, A242168, A258232, A002039.

Programs

  • Mathematica
    q0 = q /. FindRoot[ QPochhammer'[q] == 0, {q, -1/2}, WorkingPrecision -> 300]; RealDigits[q0, 10, 105] // First (* Jean-François Alcover, Dec 05 2013 *)

Formula

Equals -1/lim_{n->infinity} A002039(n)^(1/n). - Vaclav Kotesovec, Jun 02 2018

A002040 Related to partitions.

Original entry on oeis.org

1, 2, 4, 8, 21, 52, 131, 316, 765, 1846, 4494, 10944, 26654, 64798, 157502, 382868, 931028, 2264106, 5505777, 13387880, 32553601, 79156974, 192479838, 468039888, 1138098210, 2767421826, 6729311459, 16363118556, 39788886610, 96751470494
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 21*x^4 + 52*x^5 + 131*x^6 + 316*x^7 + ...

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002039, A000203, A010815.

Programs

  • Mathematica
    max = 29; f[q_] := Product[1 - (-q)^k, {k, 1, max + 1}]; CoefficientList[ Series[1/f'[q], {q, 0, max}], q] (* Jean-François Alcover, Jun 18 2012, after Michael Somos *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ 1 / D[ Normal @ Series[ QPochhammer[ -x], {x, 0, n + 1}], x], {x, 0, n}]]; (* Michael Somos, May 31 2016 *)
  • PARI
    {a(n) = polcoeff( 1 / eta( -x + x^2 * O(x^n))', n)};

Formula

G.f.: 1/(f(q)') where f(-q)=Product_{k>0} (1-q^k) is one of Ramanujan's theta functions. - Michael Somos, Apr 08 2003
a(n) = sum_{k=0..n} (-1)^k*A000041(k)*A002039(n-k). - Mircea Merca, Feb 27 2014
a(n) ~ c * d^n, where d = -1/A143441 = 2.431619934495323994754... and c = 0.623278923942755977756856780504941340332933121682037117752100... - Vaclav Kotesovec, Jun 02 2018

Extensions

Formula corrected and sequence extended by Michael Somos

A307241 a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1)*d(k+1)*a(n-k), where d() is the number of divisors (A000005).

Original entry on oeis.org

1, 2, 2, 3, 6, 12, 23, 42, 75, 135, 248, 460, 849, 1554, 2837, 5192, 9527, 17490, 32083, 58809, 107781, 197578, 362280, 664320, 1218069, 2233202, 4094289, 7506602, 13763219, 25234674, 46266927, 84828138, 155528132, 285154061, 522819002, 958568628, 1757496665, 3222295912
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 30 2019

Keywords

Crossrefs

Cf. A000005, A002039, A129921, A307242.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[(-1)^(k + 1) DivisorSigma[0, k + 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 37}]
nmax = 37; CoefficientList[Series[-x/Sum[(-x)^k/(1 - (-x)^k), {k, 1, nmax + 1}], {x, 0, nmax}], x]
nmax = 37; CoefficientList[Series[1/D[Log[Product[(1 - (-x)^k)^(1/k), {k, 1, nmax + 1}]], x], {x, 0, nmax}], x]

Formula

G.f.: -x / Sum_{k>=1} (-x)^k/(1 - (-x)^k).
G.f.: 1 / (d/dx) log(Product_{k>=1} (1 - (-x)^k)^(1/k)).

A307242 a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1)*sigma_2(k+1)*a(n-k), where sigma_2() is the sum of squares of divisors (A001157).

Original entry on oeis.org

1, 5, 15, 46, 159, 570, 2036, 7208, 25400, 89456, 315335, 1112286, 3923867, 13841052, 48818892, 172186234, 607314043, 2142064478, 7555322206, 26648517536, 93992371863, 331521717928, 1169314641890, 4124305724658, 14546896171716, 51308559972146, 180971133233105, 638305788168090
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 30 2019

Keywords

Crossrefs

Cf. A001157, A002039, A307241, A320649.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[(-1)^(k + 1) DivisorSigma[2, k + 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 27}]
nmax = 27; CoefficientList[Series[-x/Sum[k^2 (-x)^k/(1 - (-x)^k), {k, 1, nmax + 1}], {x, 0, nmax}], x]
nmax = 27; CoefficientList[Series[1/D[Log[Product[(1 - (-x)^k)^k, {k, 1, nmax + 1}]], x], {x, 0, nmax}], x]

Formula

G.f.: -x / Sum_{k>=1} k^2*(-x)^k/(1 - (-x)^k).
G.f.: 1 / (d/dx) log(Product_{k>=1} (1 - (-x)^k)^k).

A335227 G.f.: x / (Sum_{k>=1} k * x^k / (1 + x^k)).

Original entry on oeis.org

1, -1, -3, 6, 1, -20, 24, 38, -132, 34, 411, -632, -601, 2914, -1664, -7822, 15649, 6802, -62082, 55672, 141109, -369310, -12036, 1275642, -1580834, -2343886, 8375349, -2648282, -25217490, 41097852, 33815048, -183252284, 117569579, 475949186, -1006346968, -344955964
Offset: 0

Views

Author

Ilya Gutkovskiy, May 27 2020

Keywords

Crossrefs

Cf. A000009, A000593, A002039, A180305, A320651, A335228.

Programs

  • Mathematica
    nmax = 35; CoefficientList[Series[x/Sum[k x^k/(1 + x^k), {k, 1, nmax + 1}], {x, 0, nmax}], x]
nmax = 35; CoefficientList[Series[1/D[Log[Product[(1 + x^k), {k, 1, nmax + 1}]], x], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -Sum[DivisorSum[k + 1, # &, OddQ[#] &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 35}]

Formula

G.f.: x / (Sum_{k>=1} (-1)^(k+1) * x^k / (1 - x^k)^2).
G.f.: 1 / log(g(x))', where g(x) = Product_{k>=1} (1 + x^k) is the g.f. for A000009.
G.f.: 1 / (Sum_{k>=0} A000593(k+1) * x^k).
a(0) = 1; a(n) = -Sum_{k=1..n} A000593(k+1) * a(n-k).

A335228 G.f.: x / (Sum_{k>=1} x^k / (1 + x^k)^2).

Original entry on oeis.org

1, 1, -3, -2, 9, 0, -32, 18, 108, -118, -333, 576, 911, -2466, -2040, 9702, 2529, -35622, 8254, 122436, -88275, -391882, 501660, 1148334, -2331810, -2949282, 9689949, 5791930, -37155906, -2645148, 133051344, -54698868, -445531893, 408566282, 1383325848, -2115234972
Offset: 0

Views

Author

Ilya Gutkovskiy, May 27 2020

Keywords

Crossrefs

Cf. A002039, A002129, A010054, A180305, A335227.

Programs

  • Mathematica
    nmax = 35; CoefficientList[Series[x/Sum[x^k/(1 + x^k)^2, {k, 1, nmax + 1}], {x, 0, nmax}], x]
nmax = 35; CoefficientList[Series[1/D[Log[Sum[x^(k (k + 1)/2), {k, 0, nmax}]], x], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -Sum[DivisorSum[k + 1, (-1)^(# + 1) # &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 35}]

Formula

G.f.: x / (Sum_{k>=1} (-1)^(k+1) * k * x^k / (1 - x^k)).
G.f.: 1 / log(g(x))', where g(x) = Sum_{k>=0} x^(k*(k + 1)/2) is the g.f. for A010054.
G.f.: 1 / (Sum_{k>=0} A002129(k+1) * x^k).
a(0) = 1; a(n) = -Sum_{k=1..n} A002129(k+1) * a(n-k).
Showing 1-6 of 6 results.

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

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