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 10 results.

A057625 a(n) = n! * sum 1/k! where the sum is over all positive integers k that divide n.

Original entry on oeis.org

1, 3, 7, 37, 121, 1201, 5041, 62161, 423361, 5473441, 39916801, 818959681, 6227020801, 130784734081, 1536517382401, 32256486662401, 355687428096001, 10679532671808001, 121645100408832001, 3770998783116364801, 59616236292028416001, 1686001119824999577601
Offset: 1

Views

Author

Leroy Quet, Oct 09 2000

Keywords

Comments

Sets of lists of equal size, cf. A000262. - Vladeta Jovovic, Nov 02 2003
From Gus Wiseman, Jan 10 2019: (Start)
Number of matrices whose entries are 1,...,n, up to column permutations. For example, inequivalent representatives of the a(4) = 37 matrices are:
One 1 X 4 matrix:
[1234]
12 2 X 2 matrices:
[12] [12] [13] [13] [14] [14] [23] [23] [24] [24] [34] [34]
[34] [43] [24] [42] [23] [32] [14] [41] [13] [31] [12] [21]
and 24 4 X 1 matrices:
[1][1][1][1][1][1][2][2][2][2][2][2][3][3][3][3][3][3][4][4][4][4][4][4]
[2][2][3][3][4][4][1][1][3][3][4][4][1][1][2][2][4][4][1][1][2][2][3][3]
[3][4][2][4][2][3][3][4][1][4][1][3][2][4][1][4][1][2][2][3][1][3][1][2]
[4][3][4][2][3][2][4][3][4][1][3][1][4][2][4][1][2][1][3][2][3][1][2][1]
in total 1+12+24 = 37.
(End)

Examples

			a(4) = 4! (1 + 1/2! + 1/4!) = 24 (1 + 1/2 + 1/24) = 37.

Links

Crossrefs

Cf. A000005, A005225, A038041, A038048, A061095, A121860, A209903 (exp), A236696, A320444.
Row sums of A066387.

Programs

  • Mathematica
    a[n_] := n! DivisorSum[n, 1/#! &]; Array[a, 22] (* Jean-François Alcover, Dec 23 2015 *)
  • PARI
    a(n)=n! * sumdiv(n, d, 1/d! );  /* Joerg Arndt, Oct 07 2012 */

Formula

E.g.f.: Sum_{n>0} (exp(x^n)-1). - Vladeta Jovovic, Dec 30 2001
E.g.f.: Sum_{k>0} x^k/k!/(1-x^k). - Vladeta Jovovic, Oct 14 2003
Equals the logarithmic derivative of A209903. - Paul D. Hanna, Jul 26 2012

A330199 Expansion of e.g.f. Product_{k>=1} exp(1 - exp(x^k)).

Original entry on oeis.org

1, -1, -2, 1, 1, 98, -39, 3225, 1226, 6459, 12473, 821830, -214739887, -201448561, -8997850614, -514986723363, -1310942141971, -26465356716946, -931753364233567, -1858534483400559, 167210272584038942, -7112146717031426801, 312288595642509829797
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 05 2019

Keywords

Links

Crossrefs

Cf. A000587, A057625, A209903.

Programs

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

Formula

E.g.f.: A(x) = Product_{k>=1} B(x^k), where B(x) = e.g.f. of complementary Bell numbers (A000587).
E.g.f.: exp(-Sum_{j>=1} Sum_{i>=1} x^(i*j) / i!).
a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n-1,k-1) * A057625(k) * a(n-k).

A346055 Expansion of e.g.f. Product_{k>=1} B(x^k/k) where B(x) = exp(exp(x) - 1) = e.g.f. of Bell numbers.

Original entry on oeis.org

1, 1, 3, 10, 47, 246, 1617, 11586, 97463, 891610, 9189623, 102024396, 1250714445, 16351489116, 232261545869, 3499469551402, 56582677946675, 964734301550142, 17509882651329087, 333381717125596692, 6710286637806825557, 141167551783524139468
Offset: 0

Views

Author

Seiichi Manyama, Jul 02 2021

Keywords

Links

Crossrefs

Cf. A000110, A005225, A081066, A209903, A346056, A346057.

Programs

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

Formula

E.g.f.: exp( Sum_{k>=1} (exp(x^k/k) - 1) ).
E.g.f.: exp( Sum_{k>=1} A005225(k)*x^k/k! ).
a(n) = (n-1)! * Sum_{k=1..n} k * (Sum_{d|k} 1/(d! * (k/d)^d)) * a(n-k)/(n-k)! for n > 0.

A346056 Expansion of e.g.f. Product_{k>=1} B(x^k/k!) where B(x) = exp(exp(x) - 1) = e.g.f. of Bell numbers.

Original entry on oeis.org

1, 1, 3, 9, 38, 168, 915, 5225, 34228, 236622, 1805297, 14498751, 125907798, 1146476984, 11129874215, 112934907867, 1209762361679, 13499714095281, 157931096158594, 1918777335806274, 24309294470496502, 318987321135326838, 4346474397776153974
Offset: 0

Views

Author

Seiichi Manyama, Jul 02 2021

Keywords

Links

Crossrefs

Cf. A000110, A038041, A081066, A209903, A346055, A346058.

Programs

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

Formula

E.g.f.: exp( Sum_{k>=1} (exp(x^k/k!) - 1) ).
E.g.f.: exp( Sum_{k>=1} A038041(k)*x^k/k! ).
a(n) = (n-1)! * Sum_{k=1..n} k * (Sum_{d|k} 1/(d! * (k/d)!^d)) * a(n-k)/(n-k)! for n > 0.

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.

A356458 Expansion of e.g.f. ( Product_{k>0} B(x^k) )^(1/(1-x)) where B(x) = exp(exp(x)-1) = e.g.f. of Bell numbers.

Original entry on oeis.org

1, 1, 6, 38, 319, 3117, 36359, 476121, 7025708, 114118746, 2029450055, 39078892305, 810834093733, 17998186069489, 425672049713174, 10676653292086790, 283014906314277059, 7901659174554937925, 231719030698518379003, 7118469816302381503209
Offset: 0

Views

Author

Seiichi Manyama, Aug 08 2022

Keywords

Crossrefs

Cf. A356025, A356461.
Cf. A000110, A209903, A355886.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, exp(exp(x^k)-1))^(1/(1-x))))
  • PARI
    a355886(n) = n!*sum(k=1, n, n\k/k!);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a355886(j)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A355886(k) * binomial(n-1,k-1) * a(n-k).

A356460 Expansion of e.g.f. Product_{k>0} B(x^k)^k where B(x) = exp(exp(x)-1) = e.g.f. of Bell numbers.

Original entry on oeis.org

1, 1, 6, 35, 303, 2772, 32903, 410335, 6051692, 95183187, 1675869175, 31437027030, 644157830077, 13976891765137, 325719071472590, 8007861177420275, 208953947981129027, 5725964099963426924, 165258064179632753563, 4987477844227598529047
Offset: 0

Views

Author

Seiichi Manyama, Aug 08 2022

Keywords

Crossrefs

Cf. A209902, A209903.
Cf. A000110, A354863, A356461.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, exp(exp(x^k)-1)^k)))
  • PARI
    a354863(n) = n!*sumdiv(n, d, n/d/d!);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a354863(j)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

E.g.f.: Product_{k>0} exp(k * (exp(x^k)-1)).
a(0) = 1; a(n) = Sum_{k=1..n} A354863(k) * binomial(n-1,k-1) * a(n-k).

A356494 Expansion of e.g.f. Product_{k>0} B(k * x^k) where B(x) = exp(exp(x)-1) = e.g.f. of Bell numbers.

Original entry on oeis.org

1, 1, 6, 35, 327, 2892, 37943, 459895, 7330172, 116054835, 2168292295, 41072348550, 898738816957, 19782331776937, 487091519709590, 12305361661242275, 337777113607935587, 9528258228302443724, 289373132780801591323, 9016757353084706862647
Offset: 0

Views

Author

Seiichi Manyama, Aug 09 2022

Keywords

Crossrefs

Cf. A209903, A346055, A356495.
Cf. A000110, A354843, A356460.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, exp(exp(k*x^k)-1))))
  • PARI
    a354843(n) = n!*sumdiv(n, d, (n/d)^d/d!);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a354843(j)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A354843(k) * binomial(n-1,k-1) * a(n-k).

A356495 Expansion of e.g.f. Product_{k>0} B((k * x)^k) where B(x) = exp(exp(x)-1) = e.g.f. of Bell numbers.

Original entry on oeis.org

1, 1, 10, 191, 7287, 424292, 37434683, 4512452023, 726390985036, 149098938941283, 38187088904721655, 11903871288193251930, 4442392007373264794677, 1953788894138983864638457, 1000334575509506861927067378, 589712001176601700420819946615
Offset: 0

Views

Author

Seiichi Manyama, Aug 09 2022

Keywords

Crossrefs

Cf. A209903, A346055, A356494.
Cf. A000110, A354892.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, exp(exp((k*x)^k)-1))))
  • PARI
    a354892(n) = n!*sumdiv(n, d, d^n/(n/d)!);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a354892(j)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A354892(k) * binomial(n-1,k-1) * a(n-k).

A330198 Expansion of e.g.f. Product_{k>=1} 1 / (2 - exp(x^k)).

Original entry on oeis.org

1, 1, 5, 25, 195, 1521, 16713, 179425, 2432139, 33902149, 546239793, 9158893173, 173742256251, 3402217292137, 73413011744985, 1653326843775193, 40118677865954475, 1014971456865241197, 27429061245764539521, 770776923753566642365, 22928146838491708702395
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 05 2019

Keywords

Crossrefs

Cf. A000262, A000670, A209903.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Product[1/(2 - Exp[x^k]), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[Exp[Sum[Sum[(Exp[x^(k/d)] - 1)^d/d, {d, Divisors[k]}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: A(x) = Product_{k>=1} B(x^k), where B(x) = e.g.f. of Fubini numbers (A000670).
E.g.f.: Product_{j>=1} 1 / (1 - Sum_{i>=1} x^(i*j) / i!).
E.g.f.: exp(Sum_{k>=1} Sum_{d|k} (exp(x^(k/d)) - 1)^d / d).
a(n) ~ n! * c / (2 * (log(2))^(n+1)), where c = Product_{k>=2} 1/(2 - exp(log(2)^k)) = 10.38787833857244631... - Vaclav Kotesovec, Dec 11 2019
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