OFFSET
1,4
COMMENTS
Total number of smallest parts in all partitions of the head of the last section of the set of partitions of n.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1000
G. E. Andrews, The number of smallest parts in the partitions of n
A. Folsom and K. Ono, The spt-function of Andrews
F. G. Garvan, Congruences for Andrews' spt-function modulo 32760 and extension of Atkin's Hecke-type partition congruences, arXiv:1011.1957 [math.NT], 2020.
F. G. Garvan, Congruences for Andrews' spt-function modulo powers of 5, 7 and 13, arXiv:1011.1955 [math.NT], 2010.
Wikipedia, Spt function
FORMULA
G.f.: Sum_{i>=2} x^i/(1 - x^i) * Product_{j>=i} 1/(1 - x^j). - Ilya Gutkovskiy, Apr 03 2017
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)*n) * (1 - (72 + 5*Pi^2)*sqrt(6) / (144*Pi*sqrt(n))). - Vaclav Kotesovec, Jul 31 2017
EXAMPLE
For n = 8 the seven partitions of 8 that do not contain 1 as a part are:
. (8)
. (4) + (4)
. 5 + (3)
. 6 + (2)
. 3 + 3 + (2)
. 4 + (2) + (2)
. (2) + (2) + (2) + (2)
Note that in every partition the smallest parts are shown between parentheses. The total number of smallest parts is 1+2+1+1+1+2+4 = 12, so a(8) = 12.
MAPLE
b:= proc(n, i) option remember;
`if`(n=0 or i<2, 0, b(n, i-1)+
add(`if`(n=i*j, j, b(n-i*j, i-1)), j=1..n/i))
end:
a:= n-> b(n, n):
seq(a(n), n=1..60); # Alois P. Heinz, Apr 09 2012
MATHEMATICA
Table[s = Select[IntegerPartitions[n], ! MemberQ[#, 1] &]; Plus @@ Table[Count[x, Min[x]], {x, s}], {n, 50}] (* T. D. Noe, Oct 19 2011 *)
b[n_, i_] := b[n, i] = If[n==0 || i<2, 0, b[n, i-1] + Sum[If[n== i*j, j, b[n-i*j, i-1]], {j, 1, n/i}]]; a[n_] := b[n, n]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Oct 12 2015, after Alois P. Heinz *)
PROG
(Sage)
def A195820(n):
return sum(list(p).count(min(p)) for p in Partitions(n, min_part=2))
# D. S. McNeil, Oct 19 2011
CROSSREFS
KEYWORD
nonn
AUTHOR
Omar E. Pol, Oct 19 2011
EXTENSIONS
More terms from D. S. McNeil, Oct 19 2011
STATUS
approved