|
%I #25 Apr 30 2022 12:19:03
%S 1,1,4,22,170,1588,18236,240840,3662424,62456136,1185150768,
%T 24714979584,562659843984,13870798275072,368324715871680,
%U 10478253239415552,317975367247809408,10252138622419702656,349999438215928660992,12612365665457524786944,478414908509124826439424
%N Expansion of e.g.f. 1/(1 - Sum_{k>=1} d(k) * x^k / k), where d(n) = number of divisors of n (A000005).
%F a(0) = 1; a(n) = Sum_{k=1..n} A318249(k) * binomial(n,k) * a(n-k).
%t d[k_] := d[k] = DivisorSigma[0, k]; a[0] = 1; a[n_] := a[n] = Sum[(k - 1)! * d[k] * Binomial[n, k] * a[n - k], {k, 1, n}]; Array[a, 21, 0] (* _Amiram Eldar_, Apr 30 2022 *)
%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=1, N, numdiv(k)*x^k/k))))
%o (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, (j-1)!*numdiv(j)*binomial(i, j)*v[i-j+1])); v;
%Y Cf. A000005, A028342, A129921, A305305, A318249, A340903.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Apr 29 2022
|
