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

0,3

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.

Formula

G.f.: Sum_{n>=1} (n*x^n/(1-x^n))*Product_{k=1..n} 1/(1-x^k).

Example

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.

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 *)

Crossrefs

Cf. A006128, A092314, A092322, A092269, A092309, A092313, A092310, A092311, A092268.

Sequence in context: A345432 A298950 A213494 * A366157 A366069 A026353

Adjacent sequences: A092318 A092319 A092320 * A092322 A092323 A092324

Keyword

easy,nonn

Author

Vladeta Jovovic, Feb 16 2004

Extensions

More terms from Robert G. Wilson v, Feb 18 2004

Status

approved