A116856 -id:A116856 - cp's OEIS frontend

A092314 Sum of smallest parts of all partitions of n into odd parts.

1, 1, 4, 2, 7, 6, 11, 8, 18, 16, 24, 23, 34, 36, 51, 48, 66, 74, 90, 98, 126, 137, 164, 182, 220, 247, 294, 324, 380, 434, 496, 556, 650, 728, 835, 938, 1068, 1204, 1372, 1531, 1736, 1956, 2198, 2462, 2784, 3104, 3482, 3890, 4358, 4864, 5441, 6048, 6748, 7516

Offset: 1

Vladeta Jovovic, Feb 15 2004

a(n) = Sum_{k>=1} k*A116856(n,k). - Emeric Deutsch, Feb 24 2006

			a(5)=7 because the partitions of 5 into odd parts are [5],[3,1,1] and [1,1,1,1,1] and the smallest parts add up to 5+1+1=7.

Vaclav Kotesovec, Table of n, a(n) for n = 1..2500

Cf. A092322, A092269, A092309, A092321, A092313, A092310, A092311, A092268.

Cf. A116856, A092322.

g:=sum((2*n-1)*x^(2*n-1)/Product(1-x^(2*k-1),k=n..30),n=1..30): gser:=series(g,x=0,57): seq(coeff(gser,x^n),n=1..54); # Emeric Deutsch, Feb 24 2006

nmax = 50; Rest[CoefficientList[Series[Sum[(2*n - 1)*x^(2*n - 1) / Product[(1 - x^(2*k - 1)), {k, n, nmax}], {n, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 06 2019 *)

G.f.: Sum((2*n-1)*x^(2*n-1)/Product(1-x^(2*k-1), k = n .. infinity), n = 1 .. infinity).

a(n) ~ exp(Pi*sqrt(n/3)) / (4 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jul 07 2019

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 25 2004

A116799 Triangle read by rows: T(n,k) is the number of partitions of n into odd parts such that the largest part is k (n>=1, k>=1).

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 1, 0, 2, 0, 1, 0, 1, 1, 0, 2, 0, 2, 0, 1, 1, 0, 3, 0, 2, 0, 1, 0, 1, 1, 0, 3, 0, 3, 0, 2, 0, 1, 1, 0, 3, 0, 4, 0, 2, 0, 1, 0, 1, 1, 0, 4, 0, 4, 0, 3, 0, 2, 0, 1, 1, 0, 4, 0, 5, 0, 4, 0, 2, 0, 1, 0, 1, 1, 0, 4, 0, 6, 0, 5, 0, 3, 0, 2, 0, 1, 1, 0, 5, 0, 7, 0, 6

Offset: 1

Emeric Deutsch, Feb 24 2006

Both rows 2n-1 and 2n have 2n-1 terms. Row sums yield A000009. T(n,2k)=0. T(n,3)=A002264(n). Sum(k*T(n,k),k>=1)=A092322(n).

			T(10,5)=3 because we have [3,3,3,1], [3,3,1,1,1,1] and [3,1,1,1,1,1,1,1].
Triangle starts:
1;
1;
1,0,1;
1,0,1;
1,0,1,0,1;
1,0,2,0,1;

Cf. A000009, A002264, A092322, A116856.

g:=sum(t^(2*j-1)*x^(2*j-1)/product(1-x^(2*i-1),i=1..j),j=1..40): gser:=simplify(series(g,x=0,22)): for n from 1 to 16 do P[n]:=sort(coeff(gser,x^n)) od: for n from 1 to 16 do seq(coeff(P[n],t^j),j=1..2*ceil(n/2)-1) od; # yields sequence in triangular form

G.f.=sum(t^(2j-1)*x^(2j-1)/product(1-x^(2i-1), i=1..j), j=1..infinity).

cp's OEIS Frontend

A092314 Sum of smallest parts of all partitions of n into odd parts.

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