A195820 Total number of smallest parts in all partitions of n that do not contain 1 as a part.
0, 1, 1, 3, 2, 7, 5, 12, 13, 22, 22, 43, 43, 67, 81, 117, 133, 195, 223, 312, 373, 492, 584, 782, 925, 1190, 1433, 1820, 2170, 2748, 3268, 4075, 4872, 5997, 7150, 8781, 10420, 12669, 15055, 18198, 21535, 25925, 30602, 36624, 43201, 51428, 60478, 71802, 84215
Offset: 1
Keywords
Examples
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.
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.
- K. Ono, Congruences for the Andrews spt-function
- Wikipedia, Spt function
Programs
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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 *) -
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
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
Extensions
More terms from D. S. McNeil, Oct 19 2011
Comments