A215513 spt(n) - p(n): total number of smallest parts in all partitions of n minus the number of partitions of n.

0, 1, 2, 5, 7, 15, 20, 35, 50, 77, 105, 161, 214, 305, 413, 570, 751, 1022, 1330, 1772, 2295, 2996, 3837, 4970, 6305, 8050, 10155, 12844, 16065, 20169, 25055, 31197, 38549, 47650, 58540, 71960, 87916, 107424, 130655, 158830, 192260, 232642, 280406

Offset

1,3

Comments

Also total number of smallest parts that are not on the right border in all partitions of n.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Formula

a(n) = A092269(n) - A000041(n).

a(n) = A000070(n-2) + A220479(n), n >= 2.

a(n) ~ exp(Pi*sqrt(2*n/3)) / (2*Pi*sqrt(2*n)) * (1 - 25*Pi/(24*sqrt(6*n)) + (25/48 + 49*Pi^2/6912)/n). - Vaclav Kotesovec, Jul 31 2017

Example

For n = 6 the partitions of 6 with the smallest parts that are not in the right border in brackets are
-----------------------------------------
.      Partitions of 6            Value
-----------------------------------------
.                       6           0
.                  [3]+ 3           1
.                   4 + 2           0
.              [2]+[2]+ 2           2
.                   5 + 1           0
.               3 + 2 + 1           0
.               4 +[1]+ 1           1
.           2 + 2 +[1]+ 1           1
.           3 +[1]+[1]+ 1           2
.       2 +[1]+[1]+[1]+ 1           3
.  [1]+[1]+[1]+[1]+[1]+ 1           5
--------------------------------------
.                           Total: 15
On the other hand the total number of smallest parts in all partitions of 6 is 26 and the number of partitions of 6 is 11, so a(6) = 26 - 11 = 15.

Mathematica

b[n_, i_] := b[n, i] = If[n==0 || i==1, n, {q, r} = QuotientRemainder[n, i]; If[r == 0, q, 0] + Sum[b[n - i*j, i - 1], {j, 0, n/i}]];
a[n_] := b[n, n] - PartitionsP[n];
Array[a, 50] (* Jean-François Alcover, Jun 05 2021, using Alois P. Heinz's code for A092269 *)

Crossrefs

Cf. A000041, A000070, A002865, A092269, A120452, A220479.

Sequence in context: A257025 A076720 A245921 * A358878 A306956 A337655

Adjacent sequences: A215510 A215511 A215512 * A215514 A215515 A215516

Keyword

nonn

Author

Omar E. Pol, Jan 13 2013

Status

approved