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%I #30 Feb 27 2020 03:01:06
%S 0,1,4,8,17,26,49,69,115,164,249,343,513,686,974,1314,1806,2382,3232,
%T 4208,5597,7244,9456,12118,15687,19899,25422,32079,40589,50796,63805,
%U 79303,98817,122179,151145,185820,228598,279476,341807,416051,506205,613244,742720
%N Sum of largest parts (counted with multiplicity) of all partitions of n.
%H Vaclav Kotesovec, <a href="/A092321/b092321.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)
%H Margaret Archibald, A. Blecher, C. Brennan, A. Knopfmacher and T. Mansour, <a href="https://ajc.maths.uq.edu.au/pdf/66/ajc_v66_p104.pdf">Partitions according to multiplicities and part sizes</a>, Australasian Journal of Combinatorics, Volume 66(1) (2016), Pages 104-119.
%H Ljuben Mutafchiev, <a href="https://arxiv.org/abs/1712.03233">On the Largest Part Size and Its Multiplicity of a Random Integer Partition</a>, arXiv:1712.03233 [math.PR], 2017.
%F G.f.: Sum_{n>=1} (n*x^n/(1-x^n))*Product_{k=1..n} 1/(1-x^k).
%e Partitions of 4 are [1,1,1,1], [1,1,2], [2,2], [1,3], [4]; thus a(4) = 4*1 + 1*2 + 2*2 + 1*3 + 1*4 = 17.
%p b:= proc(n, i, t) option remember; `if`(n=0, [1, 0],
%p `if`(i<1, [0$2], b(n, i-1, t) +add((l->`if`(t, l,
%p l+[0, l[1]*i*j]))(b(n-i*j, i-1, true)), j=1..n/i)))
%p end:
%p a:= n-> b(n$2, false)[2]:
%p seq(a(n), n=0..50); # _Alois P. Heinz_, Jan 29 2014
%t f[n_] := Block[{c = 2n, k = 2, p = IntegerPartitions[n]}, m = Max @@@ p; l = Length[p]; While[k < l, c = c + m[[k]]*Count[p[[k]], m[[k]]]; k++ ]; If[n == 1, 1, c]]; Table[ f[n], {n, 41}] (* _Robert G. Wilson v_, Feb 18 2004, updated by _Jean-François Alcover_, Jan 29 2014 *)
%t nmax = 50; CoefficientList[Series[Sum[n*x^n/(1-x^n) * Product[1/(1 - x^k), {k, 1, n}], {n, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jul 06 2019 *)
%t Join[{0},Table[Total[Flatten[First[Split[#]]&/@IntegerPartitions[n]]],{n,50}]] (* _Harvey P. Dale_, Oct 29 2019 *)
%Y Cf. A006128, A092314, A092322, A092269, A092309, A092313, A092310, A092311, A092268.
%K easy,nonn
%O 0,3
%A _Vladeta Jovovic_, Feb 16 2004
%E More terms from _Robert G. Wilson v_, Feb 18 2004
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