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%I #27 Feb 17 2021 07:45:08
%S 0,1,4,10,22,42,77,131,217,345,537,812,1211,1767,2547,3615,5078,7043,
%T 9687,13185,17815,23867,31766,41972,55146,71997,93519,120813,155358,
%U 198811,253374,321509,406436,511802,642264,803140,1001154,1243966,1541167,1903754,2345300,2881404,3531195,4316632,5264444,6405389
%N Expansion of Sum_{i>=1} i*x^i/(1 - x) * Product_{j=1..i} 1/(1 - x^j).
%C Total number of parts in all partitions of all positive integers <= n.
%C Sum of largest parts of all partitions of all positive integers <= n.
%C From _Omar E. Pol_, Feb 16 2021: (Start)
%C Apart from initial zero this is as follows:
%C Convolution of A341062 and A014153.
%C Convolution of A000005 and A000070.
%C Convolution of nonzero terms of A006218 and A000041.
%C a(n) is also the total number of divisors of all terms in the n-th row of triangle A340581. These divisors are also all parts of all partitions of all positive integers <= n. (End)
%H Alois P. Heinz, <a href="/A284870/b284870.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%F G.f.: Sum_{i>=1} i*x^i/(1 - x) * Product_{j=1..i} 1/(1 - x^j).
%F G.f.: (1/(1 - x)) * Sum_{i>=1} x^i/(1 - x^i) * Product_{j>=1} 1/(1 - x^j).
%F a(n) = Sum_{k=0..n} A006128(k).
%F a(n) = A124920(n+1) - 1.
%F a(n) = Sum_{k=1..n} k * A299779(n,k). - _Alois P. Heinz_, May 14 2018
%e a(4) = 22 because we have 1 = 1, 2 = 2, 1 + 1 = 2, 3 = 3, 2 + 1 = 3, 1 + 1 + 1 = 3, 4 = 4, 3 + 1 = 4, 2 + 2 = 4, 2 + 1 + 1 = 4 and 1 + 1 + 1 + 1 = 4 therefore 1 + 1 + 2 + 1 + 2 + 3 + 1 + 2 + 2 + 3 + 4 = 22 (total number of parts) or 1 + 2 + 1 + 3 + 2 + 1 + 4 + 3 + 2 + 2 + 1 = 22 (sum of largest parts).
%p b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n],
%p b(n, i-1) +(p-> p+[0, p[1]])(b(n-i, min(n-i, i))))
%p end:
%p a:= proc(n) a(n):= `if`(n<1, 0, a(n-1)+b(n$2)[2]) end:
%p seq(a(n), n=0..45); # _Alois P. Heinz_, Feb 16 2021
%t nmax = 45; CoefficientList[Series[Sum[i x^i /(1 - x) Product[1/(1 - x^j), {j, 1, i}], {i, 1, nmax}], {x, 0, nmax}], x]
%t nmax = 45; CoefficientList[Series[1/(1 - x) Sum[x^i /(1 - x^i), {i, 1, nmax}] Product[1/(1 - x^j), {j, 1, nmax}], {x, 0, nmax}], x]
%t Accumulate[Table[Sum[DivisorSigma[0, k] PartitionsP[n - k], {k, 1, n}], {n, 0, 45}]]
%Y Cf. A000070, A006128, A015724, A026905, A124920, A138137, A182738, A299779.
%Y Cf. A000041, A006218, A014153, A340581, A341062.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Apr 04 2017
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