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 1-9 of 9 results.

A346545 E.g.f.: Product_{k>=1} 1 / (1 - x^k)^(exp(x)/k).

Original entry on oeis.org

1, 1, 5, 26, 175, 1384, 12933, 135050, 1582901, 20380208, 286577757, 4352682256, 71247772121, 1244923243966, 23166410620637, 456940648889070, 9521696033968393, 208851154175983608, 4812156417656806393, 116112764199821653284, 2928658457243240595901, 77042063713731887400418
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 16 2021

Keywords

Comments

Exponential transform of A002746.

Crossrefs

Cf. A000005, A002746, A028342, A346546, A346547, A346548.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[Product[1/(1 - x^k)^(Exp[x]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Exp[Exp[x] Sum[DivisorSigma[0, k] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
A002746[n_] := Sum[Binomial[n, k] DivisorSigma[0, k] (k - 1)!, {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A002746[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]

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) * A002746(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).

A346841 E.g.f.: Product_{k>=1} 1 / (1 - x^k)^sin(x).

Original entry on oeis.org

1, 0, 2, 9, 40, 360, 2480, 28833, 266936, 3562920, 45634258, 659631225, 10231705196, 176661237948, 3080315922294, 59430009554685, 1217593208993232, 25766943601055184, 583245289316927058, 13861911731632256457, 343615639889119016556, 8925102256331257339140, 242399591002192962709230
Offset: 0

Views

Author

Seiichi Manyama, Sep 18 2021

Keywords

Crossrefs

Cf. A000203, A053529, A335626, A346547, A346941, A347774.

Programs

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

Formula

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

A346941 E.g.f.: Product_{k>=1} 1 / (1 - x^k)^cos(x).

Original entry on oeis.org

1, 1, 4, 15, 90, 555, 4815, 41034, 443268, 4977381, 64274655, 857332366, 13328296014, 207666642131, 3620701556017, 65845797790798, 1294049887432888, 26168756518235801, 576107273399556987, 12940593913711504118, 311924384689270232770, 7752903433736003497447, 203126367130952306670541
Offset: 0

Views

Author

Seiichi Manyama, Sep 18 2021

Keywords

Crossrefs

Cf. A000203, A346547, A346841, A347774.

Programs

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

Formula

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

A347774 E.g.f.: Product_{k>=1} 1 / (1 - x^k)^tan(x).

Original entry on oeis.org

1, 0, 2, 9, 52, 450, 3410, 41748, 415952, 5985144, 79468648, 1263309960, 20581146056, 375092849040, 7053697259856, 144054799315560, 3108398855786496, 70281839877041088, 1687564595412611520, 42264952015652902656, 1114043035100431983744, 30552235678578565203840
Offset: 0

Views

Author

Seiichi Manyama, Sep 18 2021

Keywords

Crossrefs

Cf. A000203, A346547, A346841, A346941.

Programs

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

Formula

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

A347915 Expansion of e.g.f. Product_{k>=1} (1 + x^k)^exp(x).

Original entry on oeis.org

1, 1, 4, 24, 150, 1235, 11725, 126987, 1512084, 20313897, 296921623, 4700713787, 80221988726, 1468879687145, 28661345212981, 594457831566757, 13027193829914920, 301079987772726257, 7318797530268562203, 186496088631167771143, 4971371842655844396298, 138384071439982000722737
Offset: 0

Views

Author

Seiichi Manyama, Sep 18 2021

Keywords

Crossrefs

Cf. A000593, A088311, A346547, A347916, A354507.
Cf. A354503, A354504.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Product[(1 + x^k)^Exp[x], {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Aug 17 2022 *)
  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, (1+x^k)^exp(x))))
  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(exp(exp(x)*sum(k=1, N, sigma(k>>valuation(k, 2))*x^k/k))))
  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(exp(exp(x)*sum(k=1, N, x^k/(k*(1-x^(2*k)))))))
  • PARI
    a354507(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*d)/(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, a354507(j)*binomial(i-1, j-1)*v[i-j+1])); v; \\ Seiichi Manyama, Aug 16 2022

Formula

E.g.f.: exp( exp(x) * Sum_{k>=1} A000593(k)*x^k/k ).
E.g.f.: exp( exp(x) * Sum_{k>=1} x^k/(k*(1 - x^(2*k))) ).
a(0) = 1; a(n) = Sum_{k=1..n} A354507(k) * binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Aug 16 2022

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 1-9 of 9 results.

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

Content is available under The OEIS End-User License Agreement.