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.

Previous Showing 31-39 of 39 results.

A319105 Expansion of e.g.f. Product_{k>=0} 1/(1 - x^(2^k))^(1/2^k).

Original entry on oeis.org

1, 1, 3, 9, 51, 255, 1845, 12915, 129465, 1165185, 13338675, 146725425, 2023126875, 26300649375, 405966485925, 6089497288875, 110674075136625, 1881459277322625, 36921598987147875, 701510380755809625, 15370603498046671875, 322782673458980109375, 7608990209632029343125, 175006774821536674891875
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 10 2018

Keywords

Crossrefs

Cf. A001511, A018819, A028342.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[Product[1/(1 - x^2^k)^(1/2^k), {k, 0, Ceiling[Log[2, nmax]]}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Exp[Sum[(IntegerExponent[k, 2] + 1) x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = (n - 1)! Sum[(IntegerExponent[k, 2] + 1) a[n - k]/(n - k)!, {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]

Formula

E.g.f.: exp(Sum_{k>=1} A001511(k)*x^k/k).

A322513 Expansion of e.g.f. log(1 + Sum_{k>=1} d(k) * x^k / k!), where d(k) = number of divisors of k (A000005).

Original entry on oeis.org

0, 1, 1, -2, 1, 11, -48, -6, 1241, -6431, -15320, 452970, -2317212, -17584137, 372119776, -1552313624, -31732274313, 565880016193, -1217992446564, -90197542736656, 1400682677566587, 1990004001731140, -384348195167184028, 5109122826021406702
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 03 2019

Keywords

Comments

Logarithmic transform of A000005.

Links

Crossrefs

Cf. A000005, A028342, A274805, A294363, A295739.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 0, (b-> b(n)-add(a(j)
     *binomial(n, j)*j*b(n-j), j=1..n-1)/n)(numtheory[tau]))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 06 2019
  • Mathematica
    nmax = 23; CoefficientList[Series[Log[1 + Sum[DivisorSigma[0, k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = DivisorSigma[0, n] - Sum[Binomial[n, k] DivisorSigma[0, n - k] k a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 0, 23}]

A327927 Expansion of e.g.f. exp(Sum_{i>=1} Sum_{j=1..i} x^(i*j) / (i*j)).

Original entry on oeis.org

1, 1, 2, 6, 30, 150, 1020, 7140, 63420, 611100, 6625080, 72875880, 977213160, 12876743880, 190951160400, 2975661889200, 51767677962000, 886225654314000, 17136230971860000, 329530590793404000, 7035395004749311200, 151961029211943151200
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 30 2019

Keywords

Crossrefs

Cf. A000005, A028260, A028342, A038548, A206303, A327940.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[Exp[Sum[Ceiling[DivisorSigma[0, k]/2] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Ceiling[DivisorSigma[0, k]/2] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 21}]
nmax = 20; CoefficientList[Series[Exp[Sum[-(x^(k*(1 + k))*LerchPhi[x^k, 1, 1 + k] + Log[1 - x^k])/k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 06 2019 *)

Formula

E.g.f.: exp(Sum_{k>=1} ceiling(A000005(k)/2) * x^k / k).
E.g.f.: exp(Sum_{k>=1} A038548(k) * x^k / k).
E.g.f.: Product_{k>=1} 1 / (1 - x^A028260(k))^(1/A028260(k)).

A327940 Expansion of e.g.f. exp(Sum_{i>=1} Sum_{j=1..i-1} x^(i*j) / (i*j)).

Original entry on oeis.org

1, 0, 1, 2, 9, 44, 385, 1854, 23233, 153656, 2151441, 18787130, 338487721, 3165541092, 60609811249, 835202858294, 14913805143105, 228441779869424, 5319673396479073, 81040768940877426, 2153026504862728201, 39759334398324543260, 988919906784578473761
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 30 2019

Keywords

Crossrefs

Cf. A000005, A026424, A028342, A056924, A206303, A327927.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Exp[Sum[Floor[DivisorSigma[0, k]/2] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Floor[DivisorSigma[0, k]/2] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 22}]

Formula

E.g.f.: exp(Sum_{k>=1} floor(A000005(k)/2) * x^k / k).
E.g.f.: exp(Sum_{k>=1} A056924(k) * x^k / k).
E.g.f.: Product_{k>=1} 1 / (1 - x^A026424(k))^(1/A026424(k)).

A345749 E.g.f.: Product_{k>=1} 1/(1 - (exp(x) - 1)^k)^(1/k).

Original entry on oeis.org

1, 1, 4, 21, 147, 1250, 12633, 147497, 1947676, 28699373, 466994003, 8309274754, 160368858609, 3336869582657, 74468098634660, 1773827462044421, 44905503103938915, 1203843692164105458, 34070243272290551113, 1015056385225183643721
Offset: 0

Views

Author

Seiichi Manyama, Jun 26 2021

Keywords

Comments

Stirling transform of A028342.

Links

Crossrefs

Cf. A000005, A028342, A048993, A167137, A294363, A305986, A345750, A345751.

Programs

  • Mathematica
    max = 19; Range[0, max]! * CoefficientList[Series[Product[1/(1 - (Exp[x] - 1)^k)^(1/k), {k, 1, max}], {x, 0, max}], x] (* Amiram Eldar, Jun 26 2021 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1-(exp(x)-1)^k)^(1/k))))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N,numdiv(k)*(exp(x)-1)^k/k))))

Formula

E.g.f.: exp( Sum_{k>=1} d(k) * (exp(x) - 1)^k / k ), where d(n) is the number of divisors of n.
a(n) = Sum_{k=0..n} Stirling2(n,k) * A028342(k).

A296048 Expansion of e.g.f. Product_{k>=1} ((1 - x^k)/(1 + x^k))^(1/k).

Original entry on oeis.org

1, -2, 2, -4, 32, -128, 496, -2336, 29312, -395776, 3194624, -21951488, 277270528, -4027191296, 38850203648, -739834458112, 19460560584704, -299971773661184, 3169121209090048, -51853341314514944, 1234704403684130816, -30653318499154788352, 658369600764729884672, -10809496145754051313664
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 03 2017

Keywords

Crossrefs

Cf. A001227, A028342, A028343, A054844, A156616, A168243, A285675, A294356, A295792.

Programs

  • Maple
    a:=series(mul(((1-x^k)/(1+x^k))^(1/k),k=1..100),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 27 2019
  • Mathematica
    nmax = 23; CoefficientList[Series[Product[((1 - x^k)/(1 + x^k))^(1/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Exp[-2 Sum[Total[Mod[Divisors[k], 2] x^k]/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: exp(-2*Sum_{k>=1} A001227(k)*x^k/k).
E.g.f.: exp(-Sum_{k>=1} A054844(k)*x^k/k).

A305128 Expansion of e.g.f. Product_{k>=1} 1/(1 - x^k)^AH(k), where AH(k) is the k-th alternating harmonic number.

Original entry on oeis.org

1, 1, 3, 14, 79, 539, 4663, 42468, 457945, 5433281, 71036231, 994289658, 15544425103, 253283689619, 4489180389835, 84521336758904, 1687130833152561, 35365641206048129, 790065486354237643, 18340253632236738022, 449655289227002010351, 11492300073384698090795, 306803167368168113022271
Offset: 0

Views

Author

Ilya Gutkovskiy, May 26 2018

Keywords

Comments

a(n)/n! is the Euler transform of [1, 1 - 1/2, 1 - 1/2 + 1/3, 1 - 1/2 + 1/3 - 1/4, 1 - 1/2 + 1/3 - 1/4 + 1/5, ...].

Links

Crossrefs

Cf. A028342, A058312, A058313, A168243, A303970.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Product[1/(1 - x^k)^Sum[(-1)^(j + 1)/j, {j, 1, k}], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d ((-1)^(d + 1) LerchPhi[-1, 1, d + 1] + Log[2]), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 22}]

Formula

E.g.f.: Product_{k>=1} 1/(1 - x^k)^(A058313(k)/A058312(k)).

A327941 Expansion of e.g.f. exp(Sum_{i>=1} Sum_{j>=2} x^(i*j) / (i*j)).

Original entry on oeis.org

1, 0, 1, 2, 15, 44, 595, 2274, 36673, 247400, 3660921, 29194010, 632617711, 5289743172, 117393123835, 1525153361354, 32315717350785, 433901475732944, 11698737221494513, 168831340268759730, 4894554062081828431, 87212857278031619420, 2398463635663863045411
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 30 2019

Keywords

Crossrefs

Cf. A000005, A028342, A032741, A206303.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Exp[Sum[(DivisorSigma[0, k] - 1) x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[(DivisorSigma[0, k] - 1) a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 22}]

Formula

E.g.f.: exp(Sum_{k>=1} (A000005(k) - 1) * x^k / k).
E.g.f.: exp(Sum_{k>=1} A032741(k) * x^k / k).
E.g.f.: Product_{k>=2} 1 / (1 - x^k)^(1/k).

A353186 Expansion of e.g.f. 1/(1 - Sum_{k>=1} d(k) * x^k / k), where d(n) = number of divisors of n (A000005).

Original entry on oeis.org

1, 1, 4, 22, 170, 1588, 18236, 240840, 3662424, 62456136, 1185150768, 24714979584, 562659843984, 13870798275072, 368324715871680, 10478253239415552, 317975367247809408, 10252138622419702656, 349999438215928660992, 12612365665457524786944, 478414908509124826439424
Offset: 0

Views

Author

Seiichi Manyama, Apr 29 2022

Keywords

Crossrefs

Cf. A000005, A028342, A129921, A305305, A318249, A340903.

Programs

  • Mathematica
    d[k_] := d[k] = DivisorSigma[0, k]; a[0] = 1; a[n_] := a[n] = Sum[(k - 1)! * d[k] * Binomial[n, k] * a[n - k], {k, 1, n}]; Array[a, 21, 0] (* Amiram Eldar, Apr 30 2022 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=1, N, numdiv(k)*x^k/k))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, (j-1)!*numdiv(j)*binomial(i, j)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A318249(k) * binomial(n,k) * a(n-k).
Previous Showing 31-39 of 39 results.

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