A092321 Sum of largest parts (counted with multiplicity) of all partitions of n.
0, 1, 4, 8, 17, 26, 49, 69, 115, 164, 249, 343, 513, 686, 974, 1314, 1806, 2382, 3232, 4208, 5597, 7244, 9456, 12118, 15687, 19899, 25422, 32079, 40589, 50796, 63805, 79303, 98817, 122179, 151145, 185820, 228598, 279476, 341807, 416051, 506205, 613244, 742720
Offset: 0
Examples
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.
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
- Margaret Archibald, A. Blecher, C. Brennan, A. Knopfmacher and T. Mansour, Partitions according to multiplicities and part sizes, Australasian Journal of Combinatorics, Volume 66(1) (2016), Pages 104-119.
- Ljuben Mutafchiev, On the Largest Part Size and Its Multiplicity of a Random Integer Partition, arXiv:1712.03233 [math.PR], 2017.
Programs
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Maple
b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i<1, [0$2], b(n, i-1, t) +add((l->`if`(t, l, l+[0, l[1]*i*j]))(b(n-i*j, i-1, true)), j=1..n/i))) end: a:= n-> b(n$2, false)[2]: seq(a(n), n=0..50); # Alois P. Heinz, Jan 29 2014 -
Mathematica
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 *) 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 *) Join[{0},Table[Total[Flatten[First[Split[#]]&/@IntegerPartitions[n]]],{n,50}]] (* Harvey P. Dale, Oct 29 2019 *)
Formula
G.f.: Sum_{n>=1} (n*x^n/(1-x^n))*Product_{k=1..n} 1/(1-x^k).
Extensions
More terms from Robert G. Wilson v, Feb 18 2004