A213359 Sum of all parts that are not the smallest part (counted with multiplicity) of all partitions of n.
0, 0, 2, 5, 16, 27, 59, 96, 164, 260, 415, 606, 923, 1336, 1911, 2698, 3787, 5203, 7142, 9646, 12962, 17295, 22902, 30063, 39315, 51104, 66013, 84898, 108658, 138397, 175593, 221872, 279207, 350248, 437607, 545093, 676764, 837873, 1033961, 1272730, 1562137
Offset: 1
Keywords
Examples
a(4) = 5 because the partitions of 4 are [1,1,1,1], [1,1,2], [1,3], [2,2], and [4], having sum of parts that are not the smallest 0, 2, 3, 0, and 0, respectively, and 0 + 2 + 3 + 0 + 0 = 5. - _Emeric Deutsch_, Feb 02 2016
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..1000
Programs
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Maple
g := add(x^i*add(j*x^j/(1-x^j), j = i+1 .. 80)/((1-x^i)*mul(1-x^j, j = i+1 .. 80)), i = 1 .. 80): gser := series(g, x = 0, 55): seq(coeff(gser, x, n), n = 1 .. 40); # Emeric Deutsch, Feb 02 2016
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Mathematica
max = 42; gser = Sum[x^i*Sum[j*x^j/(1-x^j), {j, i+1, max}]/((1-x^i)* Product[1-x^j, {j, i+1, max}]), {i, 1, max}]+O[x]^max; CoefficientList[ gser, x] // Rest (* Jean-François Alcover, Feb 21 2017, after Emeric Deutsch *)
Formula
G.f.: Sum_{i>0}(x^i/(1-x^i))(Sum_{j>i}(j*x^j/(1-x^j))/Product_{j>i}(1-x^j)) (obtained by logarithmic differentiation of the bivariate g.f. given in A268189). - Emeric Deutsch, Feb 02 2016