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.

A346547 E.g.f.: Product_{k>=1} 1 / (1 - x^k)^exp(x).

Original entry on oeis.org

1, 1, 6, 36, 282, 2575, 28075, 340809, 4657996, 69874305, 1145441713, 20279904337, 386803154474, 7874727448757, 170678885319787, 3919163707551187, 95029714996046680, 2424604353738271201, 64940619086990938317, 1820746123923294245293, 53328181409328560026038
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 16 2021

Keywords

Comments

Exponential transform of A002745.

Crossrefs

Cf. A000203, A002745, A053529, A346545, A346546, A346548.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Product[1/(1 - x^k)^Exp[x], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[Exp[Exp[x] Sum[DivisorSigma[1, k] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
A002745[n_] := Sum[Binomial[n, k] DivisorSigma[1, k] (k - 1)!, {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A002745[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

Formula

E.g.f.: exp( exp(x) * Sum_{k>=1} sigma(k) * x^k / k ).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A002745(k) * a(n-k).

A346548 E.g.f.: Product_{k>=1} 1 / (1 - x^k)^exp(-x).

Original entry on oeis.org

1, 1, 2, 6, 42, 175, 2015, 10843, 157388, 1240377, 20118077, 172029231, 4052166250, 36360150385, 952965601471, 11194257455977, 316421367496344, 3722989943371217, 134504815853036649, 1641201826969536379, 67298415781492985366, 935342610632498431241, 40176825083871581430723
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 16 2021

Keywords

Comments

Exponential transform of A002743.
The first negative term is a(71) = -1.2234788... * 10^104.

Crossrefs

Cf. A000203, A002743, A053529, A346545, A346546, A346547.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Product[1/(1 - x^k)^Exp[-x], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Exp[-x] Sum[DivisorSigma[1, k] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
A002743[n_] := Sum[(-1)^(n - k) Binomial[n, k] DivisorSigma[1, k] (k - 1)!, {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A002743[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]

Formula

E.g.f.: exp( exp(-x) * Sum_{k>=1} sigma(k) * x^k / k ).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A002743(k) * a(n-k).

A346546 E.g.f.: Product_{k>=1} 1 / (1 - x^k)^(exp(-x)/k).

Original entry on oeis.org

1, 1, 1, 2, 15, 44, 485, 1854, 25781, 170288, 2477485, 12571140, 435748665, 2049818198, 64651106637, 628176476186, 18837010964105, 93248340364152, 6695745240354169, 33794005826851192, 2549048418922818525, 20209158430316698922, 1138228671555859916609
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 16 2021

Keywords

Comments

Exponential transform of A002744.
The first negative term is a(37) = -2641429247236224246927617458359165366254750.

Crossrefs

Cf. A000005, A002744, A028342, A346545, A346547, A346548.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Product[1/(1 - x^k)^(Exp[-x]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Exp[-x] Sum[DivisorSigma[0, k] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
A002744[n_] := Sum[(-1)^(n - k) Binomial[n, k] DivisorSigma[0, k] (k - 1)!, {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A002744[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]

Formula

E.g.f.: exp( exp(-x) * Sum_{k>=1} d(k) * x^k / k ).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A002744(k) * a(n-k).

A347895 E.g.f.: Product_{k>=1} 1 / (1 - x^k)^(sin(x)/k).

Original entry on oeis.org

1, 0, 2, 6, 24, 190, 1054, 10752, 81512, 964836, 10271986, 128768090, 1692130756, 25920268218, 373473594246, 6417995468640, 113429495582192, 2067370129203944, 40640473402056690, 853163776496470734, 18080227324013473596, 417591451359251698830, 9865838382742988603390
Offset: 0

Views

Author

Seiichi Manyama, Sep 18 2021

Keywords

Crossrefs

Cf. A000005 (d(n)), A346545, A346841, A347896, A347897.

Programs

  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(1/prod(k=1, N, (1-x^k)^(sin(x)/k))))
  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(exp(sin(x)*sum(k=1, N, numdiv(k)*x^k/k))))

Formula

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

A347896 E.g.f.: Product_{k>=1} 1 / (1 - x^k)^(cos(x)/k).

Original entry on oeis.org

1, 1, 3, 8, 35, 154, 1069, 6992, 63181, 581792, 6370661, 67452274, 951915997, 11969405746, 179011910505, 2779710261520, 48258699341241, 801399592338584, 16001984887453273, 296633739485713758, 6451644638913737817, 139116145404549492478, 3202263932381289406973, 72019768700608119636692
Offset: 0

Views

Author

Seiichi Manyama, Sep 18 2021

Keywords

Crossrefs

Cf. A000005 (d(n)), A346545, A347895, A347897.

Programs

  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(1/prod(k=1, N, (1-x^k)^(cos(x)/k))))
  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(exp(cos(x)*sum(k=1, N, numdiv(k)*x^k/k))))

Formula

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

A347897 E.g.f.: Product_{k>=1} 1 / (1 - x^k)^(tan(x)/k).

Original entry on oeis.org

1, 0, 2, 6, 36, 250, 1744, 18312, 158960, 2046672, 23152216, 332066240, 4628867680, 75851021376, 1225796994720, 22407297808560, 420285940934912, 8427749606274560, 177279678667864320, 3930905732908421376, 91016443490231306112, 2210008179756128156160, 55958663509700641300736
Offset: 0

Views

Author

Seiichi Manyama, Sep 18 2021

Keywords

Crossrefs

Cf. A000005 (d(n)), A346545, A347895, A347896.

Programs

  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(1/prod(k=1, N, (1-x^k)^(tan(x)/k))))
  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(exp(tan(x)*sum(k=1, N, numdiv(k)*x^k/k))))

Formula

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

A356590 Expansion of e.g.f. ( Product_{k>0} 1/(1 - (k * x)^k)^(1/k) )^exp(x).

Original entry on oeis.org

1, 1, 8, 96, 2382, 100035, 6995185, 699004551, 96910745876, 17476222963065, 4000562831147323, 1127335505294104887, 384099492016873956422, 155403154609857016567601, 73680868272553092728379865, 40444727351284600806487687057
Offset: 0

Views

Author

Seiichi Manyama, Aug 14 2022

Keywords

Crossrefs

Cf. A346545, A346547.
Cf. A023881, A356588, A356589.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1-(k*x)^k)^(1/k))^exp(x)))
  • PARI
    a356589(n) = n!*sum(k=1, n, sigma(k, k)/(k*(n-k)!));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356589(j)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A356589(k) * binomial(n-1,k-1) * a(n-k).

A356595 Expansion of e.g.f. ( Product_{k>0} 1/(1 - x^k)^k )^exp(x).

Original entry on oeis.org

1, 1, 8, 60, 582, 6555, 88585, 1333731, 22602020, 420261225, 8536210843, 187294058787, 4420961159582, 111409233290537, 2986570482052729, 84773698697674837, 2539347801355477960, 80003306259203052465, 2644032803825175398175, 91425359712959262036223
Offset: 0

Views

Author

Seiichi Manyama, Aug 15 2022

Keywords

Crossrefs

Cf. A346545, A346547.
Cf. A356337, A356600.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1-x^k)^k)^exp(x)))
  • PARI
    a356600(n) = n!*sum(k=1, n, sigma(k, 2)/(k*(n-k)!));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356600(j)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A356600(k) * binomial(n-1,k-1) * a(n-k).
Showing 1-8 of 8 results.

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

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