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A215513 spt(n) - p(n): total number of smallest parts in all partitions of n minus the number of partitions of n. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
Also total number of smallest parts that are not on the right border in all partitions of n.
LINKS
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
Sequence in context: A257025 A076720 A245921 * A358878 A306956 A337655
KEYWORD
nonn
AUTHOR
Omar E. Pol, Jan 13 2013
STATUS
approved

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Last modified February 21 02:19 EST 2024. Contains 370219 sequences. (Running on oeis4.)