%I #39 Oct 02 2021 19:38:55

%S 0,0,1,3,7,13,24,39,64,98,150,219,322,455,645,892,1232,1668,2259,3008,

%T 4003,5260,6897,8951,11599,14893,19086,24284,30827,38888,48959,61293,

%U 76578,95223,118152,145993,180037,221175,271186,331402,404208,491521

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

%C 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

%C Also total number of parts that are not the largest part in all partitions of n. - _Omar E. Pol_, Apr 30 2012

%C 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

%F a(n) = (Sum_{k=1..n} tau(k)*numbpart(n-k))-numbpart(n) = A006128(n)-A000041(n), n>0. - _Vladeta Jovovic_, Oct 06 2002

%F G.f.: sum(n>=1, (n-1) * x^n / prod(k=1,n, 1-x^k ) ). - _Joerg Arndt_, Apr 17 2011

%F a(n) = A006128(n-1) + A182699(n), n >= 1. - _Omar E. Pol_, Oct 30 2011

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

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

%Y Cf. A001475, A248475.

%K nonn

%O 0,4

%A _Floor van Lamoen_, Oct 04 2002

%E More terms from _Vladeta Jovovic_, _Robert G. Wilson v_, _Dean Hickerson_ and _Don Reble_, Oct 06 2002