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.
%I A274805 #26 Feb 16 2025 08:33:36
%S A274805 1,2,-3,-6,45,11,-1372,4298,59244,-573463,-2432023,75984243,
%T A274805 -136498141,-10881169822,100704750342,1514280063802,-36086469752977,
%U A274805 -102642110690866,11883894518252419,-77863424962770751,-3705485804176583500,71306510264347489177
%N A274805 The logarithmic transform of sigma(n).
%C A274805 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.
%C A274805 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.
%C A274805 In the a(n) formulas, see the examples, the cumulant expansion numbers A127671 appear.
%C A274805 We observe that the logarithmic transform leaves the value of a(0) undefined.
%C A274805 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.
%D A274805 Frank Harary and Edgar M. Palmer, Graphical Enumeration, 1973.
%D A274805 Robert James Riddell, Contributions to the theory of condensation, Dissertation, University of Michigan, Ann Arbor, 1951.
%D A274805 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 1995, pp. 18-23.
%H A274805 Alois P. Heinz, <a href="/A274805/b274805.txt">Table of n, a(n) for n = 1..451</a>
%H A274805 M. Bernstein and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0205301">Some Canonical Sequences of Integers</a>, Linear Algebra and its Applications, Vol. 226-228 (1995), pp. 57-72. Erratum 320 (2000), 210. [Link to arXiv version]
%H A274805 M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
%H A274805 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.
%H A274805 Eric W. Weisstein MathWorld, <a href="https://mathworld.wolfram.com/LogarithmicTransform.html">Logarithmic Transform</a>.
%F A274805 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).
%F A274805 E.g.f. log(1+ Sum_{n >= 1}(b(n)*x^n/n!)), n >= 1, with b(n) = A000203(n) = sigma(n).
%e A274805 Some a(n) formulas, see A127671:
%e A274805 a(0) = undefined
%e A274805 a(1) = 1*x(1)
%e A274805 a(2) = 1*x(2) - x(1)^2
%e A274805 a(3) = 1*x(3) - 3*x(1)*x(2) + 2*x(1)^3
%e A274805 a(4) = 1*x(4) - 4*x(1)*x(3) - 3*x(2)^2 + 12*x(1)^2*x(2) - 6*x(1)^4
%e A274805 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
%p A274805 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.
%p A274805 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.
%p A274805 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.
%t A274805 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 *)
%o A274805 (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
%Y A274805 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).
%Y A274805 Cf. A274804, A274844, A127671, A000203.
%Y A274805 Cf. A112005, A007553, A062740, A007447, A062738, A033464, A116652, A002031, A003704, A003707, A155585, A000142, A226968.
%K A274805 sign
%O A274805 1,2
%A A274805 _Johannes W. Meijer_, Jul 27 2016