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A103628
Total sum of parts of multiplicity 1 in all partitions of n.
8
0, 1, 2, 6, 10, 21, 33, 59, 89, 145, 212, 325, 463, 680, 948, 1348, 1845, 2558, 3446, 4681, 6219, 8306, 10901, 14352, 18632, 24230, 31151, 40077, 51074, 65088, 82290, 103986, 130517, 163679, 204078, 254174, 314975, 389839, 480369, 591133, 724600, 886965
OFFSET
0,3
COMMENTS
Total number of parts of multiplicity 1 in all partitions of n is A024786(n+1).
Equals A000041 convolved with A026741. - Gary W. Adamson, Jun 11 2009
LINKS
FORMULA
G.f.: x*(1+x+x^2)/(1-x^2)^2 /Product_{k>0}(1-x^k).
a(n) = A066186(n) - A194544(n). - Omar E. Pol, Nov 20 2011
a(n) = 3*A014153(n)/4 - 3*A000070(n)/4 - A270143(n+1)/4 + A087787(n)/4. - Vaclav Kotesovec, Nov 05 2016
a(n) ~ 3^(3/2) * exp(Pi*sqrt(2*n/3)) / (8*Pi^2) * (1 - Pi/(24*sqrt(6*n))). - Vaclav Kotesovec, Nov 05 2016
EXAMPLE
Partitions of 4 are [1,1,1,1], [1,1,2], [2,2], [1,3], [4] and a(4) = 0 + 2 + 0 + (1+3) + 4 = 10.
MAPLE
gf:=x*(1+x+x^2)/(1-x^2)^2/product((1-x^k), k=1..500): s:=series(gf, x, 100): for n from 0 to 60 do printf(`%d, `, coeff(s, x, n)) od: # James A. Sellers, Apr 22 2005
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
`if`(i<1, [0, 0], add((l->`if`(j=1, [l[1],
l[2]+l[1]*i], l))(b(n-i*j, i-1)), j=0..n/i)))
end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..50); # Alois P. Heinz, Feb 03 2013
MATHEMATICA
b[n_, p_] := b[n, p] = If[n == 0 && p == 0, {1, 0}, If[p == 0, Array[0&, n+2], Sum[Function[l, ReplacePart[l, m+2 -> p*l[[1]] + l[[m+2]]]][Join[b[n-p*m, p-1], Array[0&, p*m]]], {m, 0, n/p}]]]; a[n_] := b[n, n][[3]]; a[0] = 0; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jan 24 2014, after Alois P. Heinz *)
CROSSREFS
Cf. A026741. - Gary W. Adamson, Jun 11 2009
Column k=1 of A222730. - Alois P. Heinz, Mar 03 2013
Sequence in context: A067716 A125518 A083176 * A207382 A334344 A272952
KEYWORD
easy,nonn
AUTHOR
Vladeta Jovovic, Mar 25 2005
EXTENSIONS
More terms from James A. Sellers, Apr 22 2005
STATUS
approved