A076276 - OEIS

A076276

Number of + signs needed to write the partitions of n (A000041) as sums.

0, 0, 1, 3, 7, 13, 24, 39, 64, 98, 150, 219, 322, 455, 645, 892, 1232, 1668, 2259, 3008, 4003, 5260, 6897, 8951, 11599, 14893, 19086, 24284, 30827, 38888, 48959, 61293, 76578, 95223, 118152, 145993, 180037, 221175, 271186, 331402, 404208, 491521

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OFFSET

0,4

COMMENTS

Also, total number of parts in all partitions of n-1 plus the number of emergent parts of n, if n >= 1. Also, sum of largest parts of all partitions of n-1 plus the number of emergent parts of n, if n >= 1. - Omar E. Pol, Oct 30 2011

Also total number of parts that are not the largest part in all partitions of n. - Omar E. Pol, Apr 30 2012

Empirical: For n > 1, a(n) is the sum of the entries in the second column of the lower-triangular matrix of coefficients giving the expansion of degree-n complete homogeneous symmetric functions in the Schur basis of the algebra of symmetric functions. - John M. Campbell, Mar 18 2018

LINKS

Table of n, a(n) for n=0..41.

FORMULA

a(n) = (Sum_{k=1..n} tau(k)*numbpart(n-k))-numbpart(n) = A006128(n)-A000041(n), n>0. - Vladeta Jovovic, Oct 06 2002

G.f.: sum(n>=1, (n-1) * x^n / prod(k=1,n, 1-x^k ) ). - Joerg Arndt, Apr 17 2011

a(n) = A006128(n-1) + A182699(n), n >= 1. - Omar E. Pol, Oct 30 2011

EXAMPLE

4=1+3=2+2=1+1+2=1+1+1+1, 7 + signs are needed, so a(4)=7.

MATHEMATICA

a[0]=0; a[n_] := Sum[DivisorSigma[0, k]PartitionsP[n-k], {k, 1, n}]-PartitionsP[n]

CROSSREFS

Cf. A001475, A248475.