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 1-3 of 3 results.

A202139 Expansion of e.g.f. log(1/(1-artanh(x))).

Original entry on oeis.org

0, 1, 1, 4, 14, 88, 544, 4688, 41712, 459520, 5333376, 71876352, 1027670016, 16428530688, 278818065408, 5167215464448, 101437811718144, 2140879726411776, 47698275298050048, 1130276555155243008, 28167446673847812096, 740796870212763254784
Offset: 0

Views

Author

Vladimir Kruchinin, Dec 12 2011

Keywords

Links

Crossrefs

Cf. A003704, A226968.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[Log[1/(1-ArcTanh[x])],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Sep 10 2022 *)
  • Maxima
    a(n):=n!*sum(((m-1)!*sum((stirling1(k+m,m)*2^k*binomial(n-1,k+m-1))/(k+m)!,k,0,n-m)),m,1,n);
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=0; for(i=1, n, v[i+1]=(i%2)*(i-1)!+sum(j=1, i\2, (2*j-2)!*binomial(i-1, 2*j-1)*v[i-2*j+2])); v; \\ Seiichi Manyama, Apr 30 2022

Formula

a(n) = n! * Sum_{m=1..n} (m-1)! * Sum_{k=0..n-m} Stirling1(k+m,m) * 2^k * binomial(n-1,k+m-1)/(k+m)!.
E.g.f.: log(2) - log(2 + log((1-x)/(1+x))). - Arkadiusz Wesolowski, Feb 19 2013
a(n) ~ n! * ((exp(2)+1)/(exp(2)-1))^n/n. - Vaclav Kotesovec, Jun 13 2013
a(0) = 0; a(n) = (n mod 2) * (n-1)! + Sum_{k=1..floor(n/2)} (2*k-2)! * binomial(n-1,2*k-1) * a(n-2*k+1). - Seiichi Manyama, Apr 30 2022

Extensions

Zero prepended by Harvey P. Dale, Sep 10 2022

A274805 The logarithmic transform of sigma(n).

Original entry on oeis.org

1, 2, -3, -6, 45, 11, -1372, 4298, 59244, -573463, -2432023, 75984243, -136498141, -10881169822, 100704750342, 1514280063802, -36086469752977, -102642110690866, 11883894518252419, -77863424962770751, -3705485804176583500, 71306510264347489177
Offset: 1

Views

Author

Johannes W. Meijer, Jul 27 2016

Keywords

Comments

The logarithmic transform [LOG] transforms an input sequence b(n) into the output sequence a(n). The LOG transform is the inverse of the exponential transform [EXP], see the Weisstein link and the Sloane and Plouffe reference. This relation goes by the name of Riddell’s formula. For information about the EXP transform see A274804. The logarithmic transform is related to the inverse multinomial transform, see A274844 and the first formula.
The definition of the LOG transform, see the second formula, shows that n >= 1. To preserve the identity EXP[LOG[b(n)]] = b(n) for n >= 0 for a sequence b(n) with offset 0 the shifted sequence b(n-1) with offset 1 has to be used as input for the LOG transform, otherwise information about b(0) will be lost in transformation.
In the a(n) formulas, see the examples, the cumulant expansion numbers A127671 appear.
We observe that the logarithmic transform leaves the value of a(0) undefined.
The Maple programs can be used to generate the logarithmic transform of a sequence. The first program uses a formula found by Alois P. Heinz, see A001187 and the first formula. The second program uses the definition of the logarithmic transform, see the Weisstein link and the second formula. The third program uses information about the inverse of the logarithmic transform, see A274804.

Examples

			Some a(n) formulas, see A127671:
a(0) = undefined
a(1) = 1*x(1)
a(2) = 1*x(2) - x(1)^2
a(3) = 1*x(3) - 3*x(1)*x(2) + 2*x(1)^3
a(4) = 1*x(4) - 4*x(1)*x(3) - 3*x(2)^2 + 12*x(1)^2*x(2) - 6*x(1)^4
a(5) = 1*x(5) - 5*x(1)*x(4) - 10*x(2)*x(3) + 20*x(1)^2*x(3) + 30*x(1)*x(2)^2 - 60*x(1)^3*x(2) + 24*x(1)^5

References

  • Frank Harary and Edgar M. Palmer, Graphical Enumeration, 1973.
  • Robert James Riddell, Contributions to the theory of condensation, Dissertation, University of Michigan, Ann Arbor, 1951.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 1995, pp. 18-23.

Links

Crossrefs

Some LOG transform pairs are, n >= 1: A006125(n-1) and A033678(n); A006125(n) and A001187(n); A006125(n+1) and A062740(n); A000045(n) and A112005(n); A000045(n+1) and A007553(n); A000040(n) and A007447(n); A000051(n) and (-1)*A263968(n-1); A002416(n) and A062738(n); A000290(n) and A033464(n-1); A029725(n-1) and A116652(n-1); A052332(n) and A002031(n+1); A027641(n)/A027642(n) and (-1)*A060054(n+1)/(A075180(n-1).
Cf. A274804, A274844, A127671, A000203.
Cf. A112005, A007553, A062740, A007447, A062738, A033464, A116652, A002031, A003704, A003707, A155585, A000142, A226968.

Programs

  • Maple
    nmax:=22: with(numtheory): b := proc(n): sigma(n) end: a:= proc(n) option remember; b(n) - add(k*binomial(n, k)*b(n-k)*a(k), k=1..n-1)/n: end: seq(a(n), n=1..nmax); # End first LOG program.
nmax:=22: with(numtheory): b := proc(n): sigma(n) end: t1 := log(1 + add(b(n)*x^n/n!, n=1..nmax+1)): t2 := series(t1, x, nmax+1): a := proc(n): n!*coeff(t2, x, n) end: seq(a(n), n=1..nmax); # End second LOG program.
nmax:=22: with(numtheory): b := proc(n): sigma(n) end: f := series(exp(add(r(n)*x^n/n!, n=1..nmax+1)), x, nmax+1): d := proc(n): n!*coeff(f, x, n) end: a(1):=b(1): r(1):= b(1): for n from 2 to nmax+1 do r(n) := solve(d(n)-b(n), r(n)): a(n):=r(n): od: seq(a(n), n=1..nmax); # End third LOG program.
  • Mathematica
    a[1] = 1; a[n_] := a[n] = DivisorSigma[1, n] - Sum[k*Binomial[n, k] * DivisorSigma[1, n-k]*a[k], {k, 1, n-1}]/n; Table[a[n], {n, 1, 22}] (* Jean-François Alcover, Feb 27 2017 *)
  • PARI
    N=33; x='x+O('x^N); Vec(serlaplace(log(1+sum(n=1,N,sigma(n)*x^n/n!)))) \\ Joerg Arndt, Feb 27 2017

Formula

a(n) = b(n) - Sum_{k = 1..n-1}((k*binomial(n, k)*b(n-k)*a(k))/n), n >= 1, with b(n) = A000203(n) = sigma(n).
E.g.f. log(1+ Sum_{n >= 1}(b(n)*x^n/n!)), n >= 1, with b(n) = A000203(n) = sigma(n).

A328055 Expansion of e.g.f. -log(1 - x / (1 - x)^2).

Original entry on oeis.org

0, 1, 5, 32, 270, 2904, 38400, 605520, 11113200, 232888320, 5488560000, 143704108800, 4138573824000, 130020673305600, 4425201196416000, 162194862064435200, 6369479157000960000, 266808274486161408000, 11874724379464826880000, 559591797303082672128000
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 03 2019

Keywords

Comments

a(n) is the number of ways to choose one element from each branch of labeled octupi with n nodes (cf. A029767 and example below). - Enrique Navarrete, Oct 29 2023

Examples

			For n=2, the 3 labeled octupi are the following, and there are 2+2+1 ways to choose one element from each branch:
O-1-2;
O-2-1;
1-O-2. - _Enrique Navarrete_, Oct 29 2023

Crossrefs

Cf. A001906, A004146, A005248, A005443, A029767, A052567 (exponential transform), A100404, A226968, A328054.

Programs

  • Magma
    [0] cat [Factorial(n - 1)*(Lucas(2*n)-2):n in [1..20]]; // Marius A. Burtea, Oct 03 2019
  • Mathematica
    nmax = 19; CoefficientList[Series[-Log[1 - x/(1 - x)^2], {x, 0, nmax}], x] Range[0, nmax]!
Join[{0}, Table[(n - 1)! (LucasL[2 n] - 2), {n, 1, 19}]]
  • PARI
    my(x='x+O('x^20)); concat(0, Vec(serlaplace(-log(1 - x / (1 - x)^2)))) \\ Michel Marcus, Oct 03 2019

Formula

E.g.f.: log(1 + Sum_{k>=1} Fibonacci(2*k) * x^k).
a(n) = (n - 1)! * (Lucas(2*n) - 2) for n > 0.
Showing 1-3 of 3 results.

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