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

1,4

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.

K. Ono, Congruences for the Andrews spt-function

Wikipedia, Spt function

Formula

a(n) = A092269(n) - A000070(n-1).

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

Cf. A000041, A000070, A002865, A092269, A135010, A138121, A138135, A138137, A182984.

Sequence in context: A353170 A263018 A215622 * A006921 A292204 A292203

Adjacent sequences: A195817 A195818 A195819 * A195821 A195822 A195823

Keyword

nonn

Author

Omar E. Pol, Oct 19 2011

Extensions

More terms from D. S. McNeil, Oct 19 2011

Status

approved