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 1-10 of 14 results. Next

A087906 a(n) = Sum_{d|n} (n-1)!/(d-1)!.

Original entry on oeis.org

1, 2, 3, 13, 25, 301, 721, 10921, 60481, 740881, 3628801, 106777441, 479001601, 12462690241, 134399865601, 2833553923201, 20922789888001, 892191453753601, 6402373705728001, 268633265290790401, 3652732042831872001, 102181898422712908801, 1124000727777607680001
Offset: 1

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Author

Vladeta Jovovic, Oct 15 2003

Keywords

Links

Crossrefs

Cf. A057625, A061095, A038041, A038048, A005225, A209902 (exp).

Programs

  • Mathematica
    Array[n \[Function] DivisorSum[n, (n - 1)!/(# - 1)! &], 25] (* J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010 *)
  • PARI
    a(n)=sumdiv(n,d,(n-1)!/(d-1)!); \\ Joerg Arndt, May 21 2013

Formula

E.g.f.: Sum_{k>0} (exp(x^k)-1)/k = -Sum_{k>0} log(1-x^k)/k!.

Extensions

More terms from J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010

A209903 E.g.f.: Product_{n>=1} B(x^n) where B(x) = exp(exp(x)-1) = e.g.f. of Bell numbers.

Original entry on oeis.org

1, 1, 4, 17, 111, 752, 6893, 64171, 733540, 8751579, 119847295, 1716294780, 27583937857, 460405876777, 8428298492136, 160944930254405, 3309210789416387, 70814345769448444, 1617322515279759301, 38322855872232745163, 960820910852189283072
Offset: 0

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Author

Paul D. Hanna, Mar 15 2012

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 17*x^3/3! + 111*x^4/4! + 752*x^5/5! +...
Let B(x) = exp(exp(x)-1) be the e.g.f. of Bell numbers:
B(x) = 1 + x + 2*x^2/2! + 5*x^3/3! + 15*x^4/4! + 52*x^5/5! + 203*x^6/6! +...
then the e.g.f. of this sequence equals the infinite product:
A(x) = B(x)*B(x^2)*B(x^3)*B(x^4)*B(x^5)*B(x^6)...
The logarithm of the e.g.f. A(x) begins:
log(A(x)) = x + 3*x^2/2! + 7*x^3/3! + 37*x^4/4! + 121*x^5/5! + 1201*x^6/6! +...+ A057625(n)*x^n/n! +...

Links

Crossrefs

Cf. A057625 (log), A209902, A330199.

Programs

  • PARI
    {a(n)=local(Bell=exp(exp(x+x*O(x^n))-1));n!*polcoeff(prod(m=1,n,subst(Bell,x,x^m+x*O(x^n))),n)}
  • PARI
    {a(n)=n!*polcoeff(exp(sum(m=1,n,x^m/m!/(1-x^m+x*O(x^n)))),n)}
for(n=0,25,print1(a(n),", "))
  • PARI
    a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, k*sumdiv(k, d, 1/d!)*a(n-k)/(n-k)!)); \\ Seiichi Manyama, Jul 02 2021

Formula

E.g.f.: exp( Sum_{n>=1} x^n/n! / (1-x^n) ).
E.g.f.: exp( Sum_{n>=1} A057625(n)*x^n/n! ).
E.g.f.: exp( Sum_{n>=1} exp(x^n)-1 ).
a(n) = (n-1)! * Sum_{k=1..n} k * (Sum_{d|k} 1/d!) * a(n-k)/(n-k)! for n > 0. - Seiichi Manyama, Jul 02 2021

A298906 Expansion of e.g.f. Product_{k>=1} (1 + x^k)^(1/k!).

Original entry on oeis.org

1, 1, 1, 4, 2, 1, 77, 29, -4289, -14836, 283812, 1316855, -16548717, -292820579, 911200565, 52594983250, 100157634380, -3444629077653, 7961210574683, -2170805244559295, -41176659971108705, 348776485253486302, 35663019455311634058, 513993485453689440281
Offset: 0

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Author

Ilya Gutkovskiy, Jun 18 2018

Keywords

Examples

			E.g.f.: A(x) = 1 + x + x^2/2! + 4*x^3/3! + 2*x^4/4! + x^5/5! + 77*x^6/6! + ... = (1 + x) * (1 + x^2)^(1/2!) * (1 + x^3)^(1/3!) * (1 + x^4)^(1/4!) * ...

Links

Crossrefs

Cf. A000110, A168243, A209902, A345762.

Programs

  • Maple
    a:=series(exp(add((-1)^(k+1)*(exp(x^k)-1)/k,k=1..100)),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    nmax = 23; CoefficientList[Series[Exp[Sum[(-1)^(k + 1) (Exp[x^k] - 1)/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1)/(d - 1)!, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 23}]

Formula

E.g.f.: exp(Sum_{k>=1} (-1)^(k+1)*(exp(x^k) - 1)/k).
E.g.f.: Product_{k>=1} B(x^k)^((-1)^(k+1)/k), where B(x) = exp(exp(x) - 1) = e.g.f. of Bell numbers (A000110).

A356025 Expansion of e.g.f. ( Product_{k>0} 1/(1-x^k)^(1/k!) )^(1/(1-x)).

Original entry on oeis.org

1, 1, 5, 28, 206, 1786, 18347, 212745, 2773927, 39901109, 628298992, 10725440221, 197349522471, 3888090474399, 81659016005387, 1820049574958950, 42895622543757084, 1065460090285463634, 27811791343693345811, 760920657403831436463
Offset: 0

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Author

Seiichi Manyama, Aug 04 2022

Keywords

Crossrefs

Cf. A209902, A356009, A356336.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((1/prod(k=1, N, (1-x^k)^(1/k!)))^(1/(1-x))))
  • PARI
    a356009(n) = n!*sum(k=1, n, sumdiv(k, d, 1/(d*(k/d)!)));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356009(j)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A356009(k) * binomial(n-1,k-1) * a(n-k).

A345762 E.g.f.: Product_{k>=1} (1 - x^k)^(1/k!).

Original entry on oeis.org

1, -1, -1, 2, 0, 29, -135, 727, -1967, -6074, 94510, 1548051, -41361089, 408842095, 213929807, -41951737904, 130060640466, 10569226878107, -229371598130229, 3327344803563111, -31418096993670379, -383829978086171112, 17799865170898698140, 220582224147105677385
Offset: 0

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Author

Seiichi Manyama, Jun 26 2021

Keywords

Links

Crossrefs

Cf. A028343, A087906, A185895, A209902, A330199, A345758, A346039, A346057.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1-x^k)^(1/k!))))
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(-sum(k=1, N, (exp(x^k)-1)/k))))
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(-sum(k=1, N, sumdiv(k, d, 1/(d-1)!)*x^k/k))))
  • PARI
    a(n) = if(n==0, 1, -(n-1)!*sum(k=1, n, sumdiv(k, d, 1/(d-1)!)*a(n-k)/(n-k)!));

Formula

E.g.f.: exp( -Sum_{k>=1} (exp(x^k) - 1)/k ).
E.g.f.: exp( -Sum_{k>=1} A087906(k)*x^k/k! ).
a(n) = -(n-1)! * Sum_{k=1..n} (Sum_{d|k} 1/(d-1)!) * a(n-k)/(n-k)! for n > 0.

A356487 Expansion of e.g.f. Product_{k>0} 1/(1 - (k * x)^k)^(1/k!).

Original entry on oeis.org

1, 1, 6, 45, 580, 7105, 170076, 2654575, 116426528, 2386183761, 209503380160, 3455683548691, 969334978024920, 15164681616944353, 6510178188269825720, 223847763757748796975, 81261936394687862700256, 1581790511799886415713825
Offset: 0

Views

Author

Seiichi Manyama, Aug 09 2022

Keywords

Crossrefs

Cf. A023882, A209902, A356486.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1-(k*x)^k)^(1/k!))))
  • PARI
    a356486(n) = (n-1)!*sumdiv(n, d, d^n/(d-1)!);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356486(j)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A356486(k) * binomial(n-1,k-1) * a(n-k).

A345756 E.g.f.: Product_{k>=1} 1/(1 - (exp(x) - 1)^k)^(1/k!).

Original entry on oeis.org

1, 1, 4, 20, 132, 1057, 10036, 110168, 1369395, 19009207, 291638340, 4898978911, 89387432140, 1760380295559, 37222139393757, 841009071062929, 20219172890524757, 515336552717107810, 13879978696592456136, 393920374851547833518, 11749388855614114735431
Offset: 0

Views

Author

Seiichi Manyama, Jun 26 2021

Keywords

Comments

Stirling transform of A209902.

Links

Crossrefs

Cf. A048993, A140585, A209902, A345757, A345758.

Programs

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

Formula

E.g.f.: exp( Sum_{k>=1} (exp((exp(x) - 1)^k) - 1)/k ).
a(n) = Sum_{k=0..n} Stirling2(n,k) * A209902(k).

A345870 Expansion of e.g.f. Product_{k>=1} ((1 + x^k)/(1 - x^k))^(1/k!).

Original entry on oeis.org

1, 2, 6, 26, 126, 742, 5166, 40462, 351742, 3458470, 37425406, 440788702, 5633316574, 77379974518, 1140707915262, 18053421105742, 302414295475134, 5364631473148614, 100769601500958078, 1988246969908681278, 41179474537324087454, 896909297854081874454
Offset: 0

Views

Author

Seiichi Manyama, Jun 27 2021

Keywords

Comments

Exponential convolution of the sequences A209902 and A298906.

Links

Crossrefs

Cf. A005408, A209902, A295792, A298906, A306041, A345871.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, ((1+x^k)/(1-x^k))^(1/k!))))
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(2*sum(k=0, N\2, (exp(x^(2*k+1))-1)/(2*k+1)))))

Formula

E.g.f.: exp( 2*Sum_{k>=0} (exp(x^(2*k+1)) - 1)/(2*k+1) ).

A345871 Expansion of e.g.f. Product_{k>=1} ((1 + x^k)/(1 - x^k))^(1/(2*k!)).

Original entry on oeis.org

1, 1, 2, 7, 23, 116, 707, 4775, 34092, 326723, 3255927, 35131570, 404387205, 5178352921, 72399248378, 1124473250815, 17051538263075, 287692287405292, 5225129815132463, 97469631563584567, 1891891950093538380, 40846072052629411027, 895935864764993940483
Offset: 0

Views

Author

Seiichi Manyama, Jun 27 2021

Keywords

Links

Crossrefs

Cf. A005408, A206303, A209902, A298906, A345870.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, ((1+x^k)/(1-x^k))^(1/(2*k!)))))
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=0, N\2, (exp(x^(2*k+1))-1)/(2*k+1)))))

Formula

E.g.f.: exp( Sum_{k>=0} (exp(x^(2*k+1)) - 1)/(2*k+1) ).

A346037 Expansion of e.g.f. Product_{k>=1} B(x^k)^(1/k!) where B(x) = exp(exp(x) - 1) = e.g.f. of Bell numbers.

Original entry on oeis.org

1, 1, 3, 9, 41, 183, 1145, 6835, 52043, 398441, 3577291, 32395905, 342875813, 3603992759, 42817702673, 518311440987, 6897155535843, 93092680608025, 1376879589495555, 20561329595474713, 333009853668160757, 5480574201430489831, 96322698607644959065
Offset: 0

Views

Author

Seiichi Manyama, Jul 02 2021

Keywords

Links

Crossrefs

Cf. A000110, A121860, A209902, A209903, A346039, A346056.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, exp(exp(x^k)-1)^(1/k!))))
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, (exp(x^k)-1)/k!))))
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, sumdiv(k, d, 1/(d!*(k/d)!))*x^k))))
  • PARI
    a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, k*sumdiv(k, d, 1/(d!*(k/d)!))*a(n-k)/(n-k)!));

Formula

E.g.f.: exp( Sum_{k>=1} (exp(x^k) - 1)/k! ).
E.g.f.: exp( Sum_{k>=1} A121860(k)*x^k/k! ).
a(n) = (n-1)! * Sum_{k=1..n} k * (Sum_{d|k} 1/(d! * (k/d)!)) * a(n-k)/(n-k)! for n > 0.
Showing 1-10 of 14 results. Next

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