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.

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A308345 Expansion of e.g.f. Sum_{k>=1} log(1/(1 - x^k/k)).

Original entry on oeis.org

1, 2, 4, 15, 48, 310, 1440, 11970, 85120, 821016, 7257600, 91707000, 958003200, 13440913200, 178919989248, 2809456650000, 41845579776000, 763629026160000, 12804747411456000, 257140635922025856, 4918792391884800000, 106876408948152480000
Offset: 1

Views

Author

Ilya Gutkovskiy, May 21 2019

Keywords

Links

Crossrefs

Cf. A007841, A038048, A182926, A182927, A308337.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Sum[Log[1/(1 - x^k/k)], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[n! Sum[1/(d (n/d)^d), {d, Divisors[n]}], {n, 1, 22}]

Formula

a(n) = n! * Sum_{d|n} 1/(d*(n/d)^d).
a(n) = A007841(n) - (1/n) * Sum_{k=1..n-1} k*binomial(n,k)*A007841(n-k)*a(k).
a(n) ~ 2 * (n-1)!. - Vaclav Kotesovec, Feb 16 2020

A318250 a(n) = (n - 1)! * sigma_2(n), where sigma_2(n) = sum of squares of divisors of n (A001157).

Original entry on oeis.org

1, 5, 20, 126, 624, 6000, 36000, 428400, 3669120, 47174400, 442713600, 8382528000, 81430272000, 1556755200000, 22666355712000, 445916959488000, 6067609067520000, 161837779783680000, 2317659281473536000, 66418224823222272000, 1216451004088320000000, 31165474724742758400000
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 22 2018

Keywords

Crossrefs

Cf. A000219, A001157, A038048, A318249.

Programs

  • Mathematica
    Table[(n - 1)! DivisorSigma[2, n], {n, 1, 22}]
nmax = 22; Rest[CoefficientList[Series[Sum[x^k/(k (1 - x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!]
nmax = 22; Rest[CoefficientList[Series[-Log[Product[(1 - x^k)^k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!]
  • PARI
    a(n) = (n-1)!*sigma(n,2); \\ Michel Marcus, Aug 22 2018

Formula

E.g.f.: Sum_{k>=1} x^k/(k*(1 - x^k)^2).
E.g.f.: -log(Product_{k>=1} (1 - x^k)^k).
E.g.f.: A(x) = log(B(x)), where B(x) = o.g.f. of A000219.
a(p^k) = (p^(2*k+2) - 1)*(p^k - 1)!/(p^2 - 1), where p is a prime.

A330505 Expansion of e.g.f. Sum_{k>=1} arctanh(x^k).

Original entry on oeis.org

1, 2, 8, 24, 144, 960, 5760, 40320, 524160, 4354560, 43545600, 638668800, 6706022400, 99632332800, 2092278988800, 20922789888000, 376610217984000, 9247873130496000, 128047474114560000, 2919482409811968000, 77852864261652480000
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 16 2019

Keywords

Crossrefs

Cf. A002131, A002448, A010050, A015128, A015680, A038048, A054785, A228274, A265024, A330504.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[Sum[ArcTanh[x^k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
nmax = 21; CoefficientList[Series[-Log[EllipticTheta[4, 0, x]]/2, {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[(n - 1)! DivisorSum[n, # &, OddQ[n/#] &], {n, 1, 21}]

Formula

E.g.f.: -log(theta_4(x)) / 2.
E.g.f.: (1/2) * Sum_{k>=1} log((1 + x^k) / (1 - x^k)).
E.g.f.: log(Product_{k>=1} ((1 + x^k) / (1 - x^k))^(1/2)).
E.g.f.: Sum_{k>=1} x^(2*k - 1) / ((2*k - 1) * (1 - x^(2*k - 1))).
exp(2 * Sum_{n>=1} a(n) * x^n / n!) = g.f. of A015128.
a(n) = (n - 1)! * Sum_{d|n, n/d odd} d.

A087905 a(n) = n! * Sum_{d|n} (d/n)^d.

Original entry on oeis.org

1, 3, 8, 36, 144, 1010, 5760, 50400, 416640, 4250232, 43545600, 553106400, 6706022400, 95865541200, 1410695430144, 22720842144000, 376610217984000, 6888030445296000, 128047474114560000, 2587520533615041024
Offset: 1

Views

Author

Vladeta Jovovic, Oct 14 2003

Keywords

Links

Crossrefs

Cf. A061095, A038041, A057625, A038048, A005225.

Programs

  • Mathematica
    a[n_]:= n!*DivisorSum[n, (#/n)^# &]; Array[a, 50] (* G. C. Greubel, May 16 2018 *)
  • PARI
    {a(n)= n!*sumdiv(n, d, (d/n)^d)};
for(n=1, 30, print1(a(n), ", ")) \\ G. C. Greubel, May 16 2018

Formula

E.g.f.: Sum_{k>0} x^k/(k-x^k).

A300512 Expansion of e.g.f. log(Sum_{k>=0} p(k)*x^k/k!), where p(k) = number of partitions of k (A000041).

Original entry on oeis.org

0, 1, 1, -1, -1, 6, -1, -77, 203, 1344, -10692, -15862, 579611, -1518768, -32884753, 283168220, 1550435633, -38615194078, 44538307279, 4920513118440, -39485852954288, -546206846420721, 11322395643617278, 23746787652752639, -2713550731461618505, 17064642256532964421
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 07 2018

Keywords

Comments

Logarithmic transform of A000041.

Examples

			E.g.f.: A(x) = x/1! + x^2/2! - x^3/3! - x^4/4! + 6*x^5/5! - x^6/6! - 77*x^7/7! + 203*x^8/8! + ...

Links

Crossrefs

Cf. A000041, A038048, A300511.

Programs

  • Maple
    a:= proc(n) option remember; (t-> `if`(n=0, 0, t(n)-add(j*a(j)*
      binomial(n, j)*t(n-j), j=1..n-1)/n))(combinat[numbpart])
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 07 2018
  • Mathematica
    nmax = 25; CoefficientList[Series[Log[Sum[PartitionsP[k] x^k/k!, {k, 0, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = PartitionsP[n] - Sum[k Binomial[n, k] PartitionsP[n - k] a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 0, 25}]

Formula

E.g.f.: log(Sum_{k>=0} A000041(k)*x^k/k!).

A354848 a(n) = (n-1)! * Sum_{d|n} d^(n/d + 1).

Original entry on oeis.org

1, 5, 20, 150, 624, 9600, 36000, 811440, 6572160, 105235200, 442713600, 39437798400, 81430272000, 4956708556800, 137741700096000, 3014189418240000, 6067609067520000, 1977977787641856000, 2317659281473536000, 1297953221362237440000
Offset: 1

Views

Author

Seiichi Manyama, Jun 08 2022

Keywords

Crossrefs

Cf. A038048, A078308, A354849, A354851.

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+1));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, log(1-k*x^k))))

Formula

a(n) = (n-1)! * A078308(n).
E.g.f.: -Sum_{k>0} log(1 - k * x^k).
If p is prime, a(p) = (p-1)! + p * p!.

A078521 Signed triangle of D'Arcais numbers (A008298) : coefficients of r in the polynomials generated by the series coefficients of z^n in Product[(1-z^k)^r, {k,1,Inf}]*(n!).

Original entry on oeis.org

1, 0, -1, 0, -3, 1, 0, -8, 9, -1, 0, -42, 59, -18, 1, 0, -144, 450, -215, 30, -1, 0, -1440, 3394, -2475, 565, -45, 1, 0, -5760, 30912, -28294, 9345, -1225, 63, -1, 0, -75600, 293292, -340116, 147889, -27720, 2338, -84, 1, 0, -524160, 3032208, -4335596, 2341332, -579369, 69552, -4074, 108, -1, 0, -6531840
Offset: 1

Views

Author

Wouter Meeussen, Jan 07 2003

Keywords

Comments

Also the Bell transform of -A038048(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

Examples

			The z-expansion of Product[(1-z^k)^r, {k,1,3}] is 1 - r*z + ((-3+r)*r*z^2)/2 -(r*(8-9*r +r^2)*z^3)/6, so the third row of the triangle is 0,-8,9,-1.
Triangle begins
1,
0, -1,
0, -3, 1,
0, -8, 9, -1,
0, -42, 59, -18, 1,
0, -144, 450, -215, 30, -1,
0, -1440, 3394, -2475, 565, -45, 1,
0, -5760, 30912, -28294, 9345, -1225, 63, -1,
0, -75600, 293292, -340116, 147889, -27720, 2338, -84, 1
...

Crossrefs

Cf. A008298, A010815, A038048.

Programs

  • Maple
    # The function BellMatrix is defined in A264428.
BellMatrix(n -> -n!*numtheory:-sigma(n+1), 9); # Peter Luschny, Jan 26 2016
# Alternative:
P := proc(n, x) option remember; if n = 0 then 1 else
-(1/n)*x*add(numtheory:-sigma(n-k)*P(k,x), k=0..n-1) fi end:
Trow := n -> seq(n!*coeff(P(n, x), x, k), k=0..n):
seq(Trow(n), n=0..9); # Peter Luschny, Oct 03 2018
  • Mathematica
    w=16;(CoefficientList[ #, r]&/@ CoefficientList[Series[Product[(1-z^k)^r, {k, 1, w}], {z, 0, w}], z])Range[0, w]!
(* Second program: *)
BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[Function[n, -n!*DivisorSigma[1, n + 1]], rows = 12];
Table[B[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

Formula

See Mathematica line.
Row sums give A010815 * n!.

A110376 a(n) = Sum_{r < n, gcd(r,n)=1} n!/r.

Original entry on oeis.org

1, 2, 9, 32, 250, 864, 12348, 67584, 804816, 5760000, 116915040, 686776320, 19323757440, 157991178240, 2951575200000, 42301705420800, 1202482800691200, 10048607738265600, 425162773111910400, 4541227794432000000
Offset: 1

Views

Author

Amarnath Murthy, Jul 24 2005

Keywords

Examples

			a(6) = 6!(1/1 + 1/5) = 864.

Crossrefs

Cf. A038048, A110373, A110374.

Programs

  • Maple
    a:=proc(n) local s,r: s:=0: for r from 1 to n do if gcd(r,n)=1 then s:=s+1/r else s:=s: fi: od: n!*s end: seq(a(n),n=1..23); # Emeric Deutsch, Jul 25 2005
  • Mathematica
    Do[Print[n! * Plus @@ Map[(1/#)&, Select[Range[n], GCD[ #, n] == 1 &]]], {n, 1, 30}] (* Ryan Propper, Jul 25 2005 *)

Extensions

More terms from Emeric Deutsch, Ryan Propper and Reinhard Zumkeller, Jul 25 2005

A110377 a(n) = Sum_{r < n, gcd(r,n)=1} n!/r!.

Original entry on oeis.org

1, 2, 9, 28, 205, 726, 8659, 47384, 562545, 4234330, 68588311, 483088332, 10699776685, 102434734598, 2016289908585, 24588487650736, 611171244308689, 6456997293606738, 209020565553571999, 2838875160624256460
Offset: 1

Views

Author

Amarnath Murthy, Jul 24 2005

Keywords

Examples

			a(6) = 6!(1/1! + 1/5!) = 726.

Crossrefs

Cf. A038048, A110276, A110373, A110374.

Programs

  • Maple
    a:=proc(n) local s,r: s:=0: for r from 1 to n do if gcd(r,n)=1 then s:=s+1/r! else s:=s: fi: od: n!*s end: seq(a(n),n=1..23); # Emeric Deutsch, Jul 25 2005

Extensions

More terms from Emeric Deutsch, Jul 25 2005

A305305 Expansion of e.g.f. 1/(1 - Sum_{k>=1} x^k/(k*(1 - x^k))).

Original entry on oeis.org

1, 1, 5, 32, 292, 3174, 42758, 659028, 11725656, 233646240, 5183599152, 126353158656, 3362529785712, 96896454983184, 3007687250735568, 100017757744279584, 3547903924884082176, 133715849506895518848, 5336112511923188151168, 224772952826373341478912, 9966476790792153522756864
Offset: 0

Views

Author

Ilya Gutkovskiy, May 29 2018

Keywords

Comments

a(n)/n! is the invert transform of [1, 3/2, 4/3, 7/4, 6/5, ... = sums of reciprocals of divisors of 1, 2, 3, 4, 5, ...].

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 32*x^3/3! + 292*x^4/4! + 3174*x^5/5! + 42758*x^6/6! + ...

Links

Crossrefs

Cf. A000041, A000203, A017665, A017666, A038048, A180305.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n=0, 1, add(add(
      1/d, d=numtheory[divisors](j))*b(n-j), j=1..n))
    end:
a:= n-> b(n)*n!:
seq(a(n), n=0..20);  # Alois P. Heinz, May 29 2018
  • Mathematica
    nmax = 20; CoefficientList[Series[1/(1 - Sum[x^k/(k (1 - x^k)), {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[1/(1 - Sum[DivisorSigma[-1, k] x^k, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[DivisorSigma[-1, k] a[n - k], {k, 1, n}]; Table[n! a[n], {n, 0, 20}]

Formula

E.g.f.: 1/(1 - Sum_{k>=1} (sigma(k)/k)*x^k), where sigma() = A000203.
E.g.f.: 1/(1 - Sum_{k>=1} (A017665(k)/A017666(k))*x^k).
E.g.f.: 1/(1 - log(f(x))), where f(x) = o.g.f. for A000041, Product_{k>=1} 1/(1 - x^k).
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