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

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.

Sequence in context: A232533 A061263 A156209 * A296558 A309051 A056764

Adjacent sequences: A076273 A076274 A076275 * A076277 A076278 A076279

Keyword

nonn

Author

Floor van Lamoen, Oct 04 2002

Extensions

More terms from Vladeta Jovovic, Robert G. Wilson v, Dean Hickerson and Don Reble, Oct 06 2002

Status

approved