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 A195820 Total number of smallest parts in all partitions of n that do not contain 1 as a part. 13
 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 (list; graph; refs; listen; history; text; internal format)
 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 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: A130922 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

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Last modified May 11 21:35 EDT 2021. Contains 343808 sequences. (Running on oeis4.)