A195820 - OEIS

%I #53 Feb 20 2023 07:51:30

%S 0,1,1,3,2,7,5,12,13,22,22,43,43,67,81,117,133,195,223,312,373,492,

%T 584,782,925,1190,1433,1820,2170,2748,3268,4075,4872,5997,7150,8781,

%U 10420,12669,15055,18198,21535,25925,30602,36624,43201,51428,60478,71802,84215

%N Total number of smallest parts in all partitions of n that do not contain 1 as a part.

%C Total number of smallest parts in all partitions of the head of the last section of the set of partitions of n.

%H Alois P. Heinz, <a href="/A195820/b195820.txt">Table of n, a(n) for n = 1..1000</a>

%H G. E. Andrews, <a href="https://web.archive.org/web/20171111124217/http://www.math.psu.edu/vstein/alg/antheory/preprint/andrews/17.pdf">The number of smallest parts in the partitions of n</a>

%H A. Folsom and K. Ono, <a href="https://web.archive.org/web/20130602180858/http://www.math.wisc.edu/~ono/reprints/111.pdf">The spt-function of Andrews</a>

%H F. G. Garvan, <a href="http://arxiv.org/abs/1011.1957">Congruences for Andrews' spt-function modulo 32760 and extension of Atkin's Hecke-type partition congruences</a>, arXiv:1011.1957 [math.NT], 2020.

%H F. G. Garvan, <a href="https://arxiv.org/abs/1011.1955">Congruences for Andrews' spt-function modulo powers of 5, 7 and 13</a>, arXiv:1011.1955 [math.NT], 2010.

%H K. Ono, <a href="https://web.archive.org/web/20190304031914/http://www.mathcs.emory.edu/~ono/publications-cv/pdfs/131.pdf">Congruences for the Andrews spt-function</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Spt_function">Spt function</a>

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

%F G.f.: Sum_{i>=2} x^i/(1 - x^i) * Product_{j>=i} 1/(1 - x^j). - _Ilya Gutkovskiy_, Apr 03 2017

%F 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

%e For n = 8 the seven partitions of 8 that do not contain 1 as a part are:

%e .  (8)

%e .  (4) + (4)

%e .   5  + (3)

%e .   6  + (2)

%e .   3  +  3  + (2)

%e .   4  + (2) + (2)

%e .  (2) + (2) + (2) + (2)

%e 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.

%p b:= proc(n, i) option remember;

%p       `if`(n=0 or i<2, 0, b(n, i-1)+

%p        add(`if`(n=i*j, j, b(n-i*j, i-1)), j=1..n/i))

%p     end:

%p a:= n-> b(n, n):

%p seq(a(n), n=1..60); # _Alois P. Heinz_, Apr 09 2012

%t Table[s = Select[IntegerPartitions[n], ! MemberQ[#, 1] &]; Plus @@ Table[Count[x, Min[x]], {x, s}], {n, 50}] (* _T. D. Noe_, Oct 19 2011 *)

%t 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_ *)

%o (Sage)

%o def A195820(n):

%o     return sum(list(p).count(min(p)) for p in Partitions(n,min_part=2))

%o # _D. S. McNeil_, Oct 19 2011

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

%K nonn

%O 1,4

%A _Omar E. Pol_, Oct 19 2011

%E More terms from _D. S. McNeil_, Oct 19 2011