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 15 results. Next

A209902 E.g.f.: Product_{n>=1} 1/(1 - x^n)^(1/n!).

Original entry on oeis.org

1, 1, 3, 10, 50, 261, 1877, 13511, 122663, 1150988, 12656562, 142842855, 1882666887, 24961232401, 375233443223, 5784328028680, 98433762560780, 1704971188321787, 32593405802749763, 629093184347294419, 13243913786996162915, 283647771230983625422
Offset: 0

Views

Author

Paul D. Hanna, Mar 14 2012

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 10*x^3/3! + 50*x^4/4! + 261*x^5/5! +...
such that
A(x) = 1/((1-x) * (1-x^2)^(1/2) * (1-x^3)^(1/3!) * (1-x^4)^(1/4!) *...).

Links

Crossrefs

Cf. A087906.

Programs

  • PARI
    {a(n)=n!*polcoeff(prod(m=1,n,1/(1-x^m+x*O(x^n))^(1/m!)),n)}
for(n=0,21,print1(a(n),", "))

Formula

E.g.f.: exp( Sum_{n>=1} (exp(x^n) - 1)/n ).
E.g.f.: exp( Sum_{n>=1} A087906(n)*x^n/n! ) where A087906(n) = Sum_{d|n} (n-1)!/(d-1)!.
E.g.f.: Product_{n>=1} B(x^n)^(1/n) where B(x) = exp(exp(x)-1) = e.g.f. of Bell numbers.

A356009 a(n) = n! * Sum_{k=1..n} Sum_{d|k} 1/(d * (k/d)!).

Original entry on oeis.org

1, 4, 15, 73, 390, 2641, 19208, 164585, 1541746, 16158341, 181370552, 2283224065, 30160914446, 434715492485, 6655132252876, 109315669969217, 1879289179364690, 34719396682318021, 666070910669770400, 13590051478686198401, 289043813095242038422
Offset: 1

Views

Author

Seiichi Manyama, Jul 23 2022

Keywords

Crossrefs

Cf. A087906, A356004.

Programs

  • Mathematica
    Table[n! * Sum[Sum[1/(d*(k/d)!), {d,Divisors[k]}], {k,1,n}], {n,1,25}] (* Vaclav Kotesovec, Aug 11 2025 *)
  • PARI
    a(n) = n!*sum(k=1, n, sumdiv(k, d, 1/(d*(k/d)!)));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, (exp(x^k)-1)/k)/(1-x)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, log(1-x^k)/k!)/(1-x)))

Formula

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

A352012 a(n) = Sum_{p|n, p prime} (n-1)!/(p-1)!.

Original entry on oeis.org

0, 1, 1, 6, 1, 180, 1, 5040, 20160, 378000, 1, 59875200, 1, 6235669440, 47221574400, 1307674368000, 1, 533531142144000, 1, 126713646259200000, 1219830034655232000, 51090956251003468800, 1, 38778025108327464960000, 25852016738884976640000
Offset: 1

Views

Author

Seiichi Manyama, Feb 28 2022

Keywords

Links

Crossrefs

Cf. A087906, A352004, A352014.

Programs

  • Maple
    f:= proc(n) local p;
  add( (n-1)!/(p-1)!, p = numtheory:-factorset(n))
end proc:
map(f, [$1..30]): # Robert Israel, Nov 14 2024
  • Mathematica
    a[1] = 0; a[n_] := (n - 1)! * Plus @@ (1/(FactorInteger[n][[;; , 1]] - 1)!); Array[a, 25] (* Amiram Eldar, Mar 01 2022 *)
  • PARI
    a(n) = sumdiv(n, d, isprime(d)*(n-1)!/(d-1)!);
  • PARI
    my(N=40, x='x+O('x^N)); concat(0, Vec(serlaplace(-sum(k=1, N, isprime(k)*log(1-x^k)/k!))))
  • PARI
    a(n) = my(f=factor(n)); sum(k=1, #f~, (n-1)!/(f[k,1]-1)!); \\ Michel Marcus, Mar 01 2022

Formula

E.g.f.: -Sum_{p prime} log(1-x^p)/p!.
a(n) = 1 if and only if n is prime.

A354845 a(n) = n! * Sum_{d|n} (n/d)^(d-1) / d!.

Original entry on oeis.org

1, 3, 7, 49, 121, 2281, 5041, 134401, 907201, 13184641, 39916801, 3753509761, 6227020801, 393409336321, 7638997766401, 160474477363201, 355687428096001, 75792615407308801, 121645100408832001, 32459310892353945601, 475723576423839744001, 7306033564948620902401
Offset: 1

Views

Author

Seiichi Manyama, Jun 08 2022

Keywords

Links

Crossrefs

Cf. A087906, A327578, A354843.

Programs

  • Mathematica
    a[n_] := n! * DivisorSum[n, (n/#)^(#-1)/#! &]; Array[a, 20] (* Amiram Eldar, Jun 08 2022 *)
  • PARI
    a(n) = n!*sumdiv(n, d, (n/d)^(d-1)/d!);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, (exp(k*x^k)-1)/k)))

Formula

E.g.f.: Sum_{k>0} (exp(k * x^k) - 1)/k.
If p is prime, a(p) = 1 + p!.

A352013 a(n) = Sum_{d|n} (-1)^(n/d+1) * (n-1)!/(d-1)!.

Original entry on oeis.org

1, 0, 3, -11, 25, -59, 721, -10919, 60481, -15119, 3628801, -93471839, 479001601, -8648639, 134399865601, -2833553923199, 20922789888001, -174888473759999, 6402373705728001, -228084898487846399, 3652732042831872001, -14079294028799
Offset: 1

Views

Author

Seiichi Manyama, Feb 28 2022

Keywords

Links

Crossrefs

Cf. A087906, A298906, A327243, A352014.

Programs

  • Maple
    restart;
f:= proc(n) local d;
  add((-1)^(n/d + 1) * (n-1)!/(d-1)!, d = numtheory:-divisors(n))
end proc:
map(f, [$1..30]); # Robert Israel, Nov 14 2024
  • Mathematica
    a[n_] := DivisorSum[n, (-1)^(n/#+1) * (n-1)!/(#-1)! &]; Array[a, 22] (* Amiram Eldar, Aug 30 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (-1)^(n/d+1)*(n-1)!/(d-1)!);
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, log(1+x^k)/k!)))

Formula

E.g.f.: Sum_{k>0} log(1+x^k)/k!.
E.g.f.: -Sum_{k>0} (-1)^k * (exp(x^k) - 1)/k. - Seiichi Manyama, Jun 18 2023

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

Views

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.

A370579 a(n) = n! * Sum_{d|n} 1/(d-1)!.

Original entry on oeis.org

1, 4, 9, 52, 125, 1806, 5047, 87368, 544329, 7408810, 39916811, 1281329292, 6227020813, 174477663374, 2015997984015, 45336862771216, 355687428096017, 16059446167564818, 121645100408832019, 5372665305815808020, 76707372899469312021, 2248001765299683993622
Offset: 1

Views

Author

Seiichi Manyama, Feb 23 2024

Keywords

Crossrefs

Cf. A087906, A370580.
Cf. A370602, A370603.

Programs

  • PARI
    a(n) = n!*sumdiv(n, d, 1/(d-1)!);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, x^k*exp(x^k))))

Formula

a(n) = n * A087906(n).
If p is prime, a(p) = p + p!.
E.g.f.: Sum_{k>0} x^k * exp(x^k).

A370580 a(n) = (n-1)! * Sum_{d|n} d/(d-1)!.

Original entry on oeis.org

1, 3, 5, 22, 29, 546, 727, 18488, 100809, 1164250, 3628811, 208232652, 479001613, 18741602894, 236107872015, 4796881689616, 20922789888017, 1618457192352018, 6402373705728019, 471378116297088020, 6105908234409984021, 153272981387362636822
Offset: 1

Views

Author

Seiichi Manyama, Feb 23 2024

Keywords

Crossrefs

Cf. A087906, A370579.

Programs

  • PARI
    a(n) = (n-1)!*sumdiv(n, d, d/(d-1)!);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, x^k/k*exp(x^k))))

Formula

If p is prime, a(p) = p + (p-1)!.
E.g.f.: Sum_{k>0} x^k/k * exp(x^k).

A352059 a(n) = Sum_{p|n, p prime} (n-1)!/(n/p-1)!.

Original entry on oeis.org

0, 1, 2, 6, 24, 180, 720, 840, 20160, 378000, 3628800, 6985440, 479001600, 6235669440, 47221574400, 259459200, 20922789888000, 2972883513600, 6402373705728000, 20274518622758400, 1219830034655232000, 51090956251003468800, 1124000727777607680000
Offset: 1

Views

Author

Seiichi Manyama, Mar 02 2022

Keywords

Crossrefs

Cf. A087906, A352012, A352058, A352060.

Programs

  • Mathematica
    a[1] = 0; a[n_] := Plus @@ ((n-1)!/(n/FactorInteger[n][[;;,1]] - 1)!); Array[a, 25] (* Amiram Eldar, Mar 02 2022 *)
  • PARI
    a(n) = my(f=factor(n)); sum(k=1, #f~, (n-1)!/(n/f[k, 1]-1)!);
  • PARI
    my(N=40, x='x+O('x^N)); concat(0, Vec(serlaplace(sum(k=1, N, isprime(k)*(exp(x^k)-1)/k))))

Formula

E.g.f.: Sum_{p prime} (exp(x^p) - 1)/p.

A354849 a(n) = (n-1)! * Sum_{d|n} d^(n/d) / (d-1)!.

Original entry on oeis.org

1, 3, 5, 34, 29, 1626, 727, 99128, 584649, 12353050, 3628811, 4648976652, 479001613, 803709466574, 11133394272015, 391883024332816, 20922789888017, 312756670075449618, 6402373705728019, 148614866400768768020, 2663970255433783296021
Offset: 1

Views

Author

Seiichi Manyama, Jun 08 2022

Keywords

Crossrefs

Cf. A087906.

Programs

  • Mathematica
    a[n_] := (n - 1)! * DivisorSum[n, #^(n/#)/(# - 1)! &]; Array[a, 20] (* Amiram Eldar, Jun 08 2022 *)
  • PARI
    a(n) = (n-1)!*sumdiv(n, d, d^(n/d)/(d-1)!);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, log(1-k*x^k)/k!)))

Formula

E.g.f.: -Sum_{k>0} log(1 - k * x^k)/k!.
If p is prime, a(p) = p + (p-1)!.
Showing 1-10 of 15 results. Next

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

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