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A195820 Total number of smallest parts in all partitions of n that do not contain 1 as a part.

%I A195820 #53 Feb 20 2023 07:51:30
%S A195820 0,1,1,3,2,7,5,12,13,22,22,43,43,67,81,117,133,195,223,312,373,492,
%T A195820 584,782,925,1190,1433,1820,2170,2748,3268,4075,4872,5997,7150,8781,
%U A195820 10420,12669,15055,18198,21535,25925,30602,36624,43201,51428,60478,71802,84215
%N A195820 Total number of smallest parts in all partitions of n that do not contain 1 as a part.
%C A195820 Total number of smallest parts in all partitions of the head of the last section of the set of partitions of n.
%H A195820 Alois P. Heinz, <a href="/A195820/b195820.txt">Table of n, a(n) for n = 1..1000</a>
%H A195820 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 A195820 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 A195820 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 A195820 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 A195820 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 A195820 Wikipedia, <a href="http://en.wikipedia.org/wiki/Spt_function">Spt function</a>
%F A195820 a(n) = A092269(n) - A000070(n-1).
%F A195820 G.f.: Sum_{i>=2} x^i/(1 - x^i) * Product_{j>=i} 1/(1 - x^j). - _Ilya Gutkovskiy_, Apr 03 2017
%F A195820 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 A195820 For n = 8 the seven partitions of 8 that do not contain 1 as a part are:
%e A195820 .  (8)
%e A195820 .  (4) + (4)
%e A195820 .   5  + (3)
%e A195820 .   6  + (2)
%e A195820 .   3  +  3  + (2)
%e A195820 .   4  + (2) + (2)
%e A195820 .  (2) + (2) + (2) + (2)
%e A195820 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 A195820 b:= proc(n, i) option remember;
%p A195820       `if`(n=0 or i<2, 0, b(n, i-1)+
%p A195820        add(`if`(n=i*j, j, b(n-i*j, i-1)), j=1..n/i))
%p A195820     end:
%p A195820 a:= n-> b(n, n):
%p A195820 seq(a(n), n=1..60); # _Alois P. Heinz_, Apr 09 2012
%t A195820 Table[s = Select[IntegerPartitions[n], ! MemberQ[#, 1] &]; Plus @@ Table[Count[x, Min[x]], {x, s}], {n, 50}] (* _T. D. Noe_, Oct 19 2011 *)
%t A195820 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 A195820 (Sage)
%o A195820 def A195820(n):
%o A195820     return sum(list(p).count(min(p)) for p in Partitions(n,min_part=2))
%o A195820 # _D. S. McNeil_, Oct 19 2011
%Y A195820 Cf. A000041, A000070, A002865, A092269, A135010, A138121, A138135, A138137, A182984.
%K A195820 nonn
%O A195820 1,4
%A A195820 _Omar E. Pol_, Oct 19 2011
%E A195820 More terms from _D. S. McNeil_, Oct 19 2011