A092309 -id:A092309 - cp's OEIS frontend

A097940 Sum of smallest parts (counted with multiplicity) of all compositions of n.

1, 4, 8, 20, 37, 86, 173, 372, 788, 1680, 3550, 7554, 15994, 33820, 71374, 150376, 316151, 663474, 1389760, 2906116, 6066899, 12645608, 26318870, 54700044, 113536171, 235363832, 487342781, 1007969620, 2082597193, 4298660754, 8864505305, 18263797648, 37597869188

Offset: 1

Vladeta Jovovic, Sep 05 2004

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Knopfmacher, Arnold; Munagi, Augustine O. Smallest parts in compositions, Kotsireas, Ilias S. (ed.) et al., Advances in combinatorics. 3rd Waterloo workshop on computer algebra (WWCA, W80) 2011, Waterloo, Canada, May 26-29, 2011. Berlin: Springer. 197-207 (2013).

Cf. A046746, A092309.

Drop[ CoefficientList[ Series[(1 - x)^2*Sum[k*x^k/(1 - x - x^k)^2, {k, 50}], {x, 0, 30}], x], 1] (* Robert G. Wilson v, Sep 08 2004 *)

G.f.: (1-x)^2 * Sum_{k>=1} k*x^k/(1-x-x^k)^2.
a(n) ~ n * 2^(n-3). - Vaclav Kotesovec, Sep 05 2014
a(n) = Sum_{k=1..n} A308630(n,k). - R. J. Mathar, Jun 12 2019

More terms from Robert G. Wilson v, Sep 08 2004

A213359 Sum of all parts that are not the smallest part (counted with multiplicity) of all partitions of n.

Original entry on oeis.org

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

Omar E. Pol, Jan 08 2013

nonn

			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

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A066186, A092269, A092309, A268189.

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

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 *)

a(n) = A066186(n) - A092309(n).

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

cp's OEIS Frontend

A097940 Sum of smallest parts (counted with multiplicity) of all compositions of n.

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