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.

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A321190 a(n) = [x^n] 1/(1 - Sum_{k>=1} k^n*x^k/(1 - x^k)).

Original entry on oeis.org

1, 1, 6, 47, 778, 25476, 1752936, 242632397, 70015221566, 41446777283255, 49999934258165654, 125272856707074638221, 641938223803783115191706, 6731818441446626626586172740, 146378489075644780343627471981694, 6505906463580477520696075719916583118
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 29 2018

Keywords

Crossrefs

Cf. A023887, A129921, A180305, A319647, A320649, A321042.

Programs

  • Maple
    seq(coeff(series((1-add(k^n*x^k/(1-x^k),k=1..n))^(-1),x,n+1), x, n), n = 0 .. 25); # Muniru A Asiru, Oct 29 2018
  • Mathematica
    Table[SeriesCoefficient[1/(1 - Sum[k^n x^k/(1 - x^k), {k, 1, n}]), {x, 0, n}], {n, 0, 15}]
Table[SeriesCoefficient[1/(1 - Sum[DivisorSigma[n, k] x^k, {k, 1, n}]), {x, 0, n}], {n, 0, 15}]
Table[SeriesCoefficient[1/(1 - Sum[Sum[j^n x^(i j), {j, 1, n}], {i, 1, n}]), {x, 0, n}], {n, 0, 15}]

Formula

a(n) = [x^n] 1/(1 - Sum_{k>=1} sigma_n(k)*x^k).
a(n) = [x^n] 1/(1 - Sum_{i>=1, j>=1} j^n*x^(i*j)).
a(n) = [x^n] 1/(1 + x * (d/dx) log(Product_{k>=1} (1 - x^k)^(k^(n-1)))).

A321262 Expansion of 1/(1 - Sum_{k>=1} k*x^(2*k)/(1 - x^k)).

Original entry on oeis.org

1, 0, 1, 1, 4, 3, 14, 12, 43, 50, 140, 177, 474, 643, 1560, 2325, 5246, 8194, 17763, 28838, 60190, 101063, 204935, 352227, 700037, 1224816, 2394971, 4250616, 8209174, 14724570, 28175997, 50949079, 96797183, 176131780, 332804667, 608449008, 1144920041, 2100793404
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 01 2018

Keywords

Comments

Invert transform of A001065.

Links

Crossrefs

Cf. A001065, A180305, A318784, A319107, A320651, A321263.

Programs

  • Maple
    a:=series(1/(1-add(k*x^(2*k)/(1-x^k),k=1..100)),x=0,38): seq(coeff(a,x,n),n=0..37); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 37; CoefficientList[Series[1/(1 - Sum[k x^(2 k)/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 37; CoefficientList[Series[1/(1 - Sum[(k - EulerPhi[k]) x^k/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[(DivisorSigma[1, k] - k) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 37}]

Formula

G.f.: 1/(1 - Sum_{k>=1} (sigma(k) - k)*x^k).
G.f.: 1/(1 - Sum_{k>=1} (k - phi(k))*x^k/(1 - x^k)).
a(0) = 1; a(n) = Sum_{k=1..n} A001065(k)*a(n-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).

A352839 Expansion of g.f. 1/(1 - Sum_{k>=1} sigma_k(k) * x^k).

Original entry on oeis.org

1, 1, 6, 39, 370, 4132, 59288, 990705, 19577018, 439550259, 11142216938, 313147651821, 9680830606850, 325944181383936, 11875777329091878, 465292113335910106, 19507503314546762246, 871248546067010133794, 41295079536653463057146
Offset: 0

Views

Author

Seiichi Manyama, Apr 05 2022

Keywords

Crossrefs

Cf. A129921, A180305, A320649.
Cf. A023887, A352841, A352842.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(1/(1-sum(k=1, N, sigma(k, k)*x^k)))
  • PARI
    a(n) = if(n==0, 1, sum(k=1, n, sigma(k, k)*a(n-k)));

Formula

a(0) = 1; a(n) = Sum_{k=1..n} sigma_k(k) * a(n-k).

A318493 Expansion of 1/(1 - Sum_{i>=1, j>=1} i*j*x^(i*j)).

Original entry on oeis.org

1, 1, 5, 15, 53, 165, 561, 1807, 5993, 19586, 64491, 211466, 695101, 2281614, 7494995, 24610588, 80829373, 265437828, 871738976, 2862815763, 9401768055, 30875971366, 101399191222, 333001988025, 1093603789613, 3591473940515, 11794667169894, 38734550365835, 127207121681103, 417757532953031
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 27 2018

Keywords

Crossrefs

Cf. A000005, A038040, A088305, A129921, A180305, A280540, A280541.

Programs

  • Maple
    a:=series(1/(1-add(add(i*j*x^(i*j),j=1..100),i=1..100)),x=0,30): seq(coeff(a,x,n),n=0..29); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 29; CoefficientList[Series[1/(1 - Sum[Sum[i j x^(i j), {i, 1, nmax}], {j, 1, nmax}]), {x, 0, nmax}], x]
nmax = 29; CoefficientList[Series[1/(1 - Sum[k x^k/(1 - x^k)^2, {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 29; CoefficientList[Series[1/(1 - Sum[k DivisorSigma[0, k] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[k DivisorSigma[0, k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 29}]

Formula

G.f.: 1/(1 - Sum_{k>=1} k*x^k/(1 - x^k)^2).
G.f.: 1/(1 - Sum_{k>=1} k*d(k)*x^k), where d(k) = number of divisors of k (A000005).
a(0) = 1; a(n) = Sum_{k=1..n} A038040(k)*a(n-k).
a(n) ~ c / r^n, where r = 0.304499876501217750838861744045680232405337905509126... is the root of the equation Sum_{k>=1} k*r^k/(1 - r^k)^2 = 1 and c = 0.44152042515136849968144466258954953693306684400261343177792428746297872748... - Vaclav Kotesovec, Aug 28 2018
Previous Showing 11-16 of 16 results.

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