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A304964 Expansion of 1/(1 - Sum_{i>=1, j>=1, k>=1, l>=1} x^(i*j*k*l)).

%I A304964 #6 May 22 2018 20:33:49
%S A304964 1,1,5,13,47,133,443,1333,4263,13143,41419,128791,403815,1259639,
%T A304964 3941579,12310299,38492034,120271953,375964616,1174935195,3672413322,
%U A304964 11477465221,35872928244,112117013835,350417746650,1095202995267,3422999582632,10698350241417,33437065631262,104505382585023
%N A304964 Expansion of 1/(1 - Sum_{i>=1, j>=1, k>=1, l>=1} x^(i*j*k*l)).
%C A304964 Invert transform of A007426.
%H A304964 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A304964 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F A304964 G.f.: 1/(1 - Sum_{k>=1} A007426(k)*x^k).
%p A304964 A:= proc(n, k) option remember; `if`(k=1, 1,
%p A304964       add(A(d, k-1), d=numtheory[divisors](n)))
%p A304964     end:
%p A304964 a:= proc(n) option remember; `if`(n=0, 1,
%p A304964       add(A(j, 4)*a(n-j), j=1..n))
%p A304964     end:
%p A304964 seq(a(n), n=0..35);  # _Alois P. Heinz_, May 22 2018
%t A304964 nmax = 29; CoefficientList[Series[1/(1 - Sum[x^(i j k l), {i, 1, nmax}, {j, 1, nmax/i}, {k, 1, nmax/i/j}, {l, 1, nmax/i/j/k}]), {x, 0, nmax}], x]
%t A304964 nmax = 29; CoefficientList[Series[1/(1 - Sum[Sum[DivisorSigma[0, d] DivisorSigma[0, k/d], {d, Divisors[k]}] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
%t A304964 a[0] = 1; a[n_] := a[n] = Sum[Sum[DivisorSigma[0, d] DivisorSigma[0, k/d], {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 29}]
%Y A304964 Cf. A000005, A007426, A011782, A129921, A280486, A280487, A304963.
%K A304964 nonn
%O A304964 0,3
%A A304964 _Ilya Gutkovskiy_, May 22 2018