|
%I #16 Jan 29 2021 20:48:17
%S 1,1,3,9,36,158,802,4434,26978,176637,1243528,9316519,74065506,
%T 621187700,5480130494,50662481722,489552042241,4931215686119,
%U 51668848043427,561981734692781,6333882472789914,73850048237680936,889461218944314524,11051067390893340510
%N Expansion of e.g.f. exp(Sum_{k>=1} d(k)*x^k/k!), where d(k) is the number of divisors of k (A000005).
%C Exponential transform of A000005.
%H Seiichi Manyama, <a href="/A295739/b295739.txt">Table of n, a(n) for n = 0..553</a>
%H M. Bernstein and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0205301">Some canonical sequences of integers</a>, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
%H 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 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ExponentialTransform.html">Exponential Transform</a>
%F E.g.f.: exp(Sum_{k>=1} A000005(k)*x^k/k!).
%F a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1)*A000005(k)*a(n-k).
%p a:=series(exp(add(tau(k)*x^k/k!,k=1..100)),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # _Paolo P. Lava_, Mar 27 2019
%t nmax = 23; CoefficientList[Series[Exp[Sum[DivisorSigma[0, k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
%t a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] DivisorSigma[0, k] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]
%Y Cf. A000005, A006171, A028342, A038200, A129921, A160399, A274804, A294363.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Nov 26 2017
|
