A116799 -id:A116799 - cp's OEIS frontend

A092322 Sum of largest parts of all partitions of n into odd parts.

1, 1, 4, 4, 9, 12, 19, 24, 36, 48, 64, 83, 108, 140, 179, 224, 280, 352, 432, 532, 652, 795, 960, 1160, 1392, 1669, 1992, 2368, 2804, 3320, 3908, 4592, 5388, 6300, 7349, 8560, 9940, 11524, 13340, 15401, 17752, 20436, 23472, 26920, 30840, 35256, 40252, 45900

Offset: 1

Vladeta Jovovic, Feb 15 2004

a(n) = Sum_{k>=1} k*A116799(n,k). - Emeric Deutsch, Feb 24 2006

			a(5)=9 because the partitions of 5 into odd parts are [5],[3,1,1] and [1,1,1,1,1] and the largest parts add up to 5+3+1=9.

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

Cf. A092314, A092269, A092309, A092321, A092313, A092310, A092311, A092268.

Cf. A116799.

g:=sum((2*n-1)*x^(2*n-1)/Product(1-x^(2*k-1),k=1..n),n=1..30): gser:=series(g,x=0,50): seq(coeff(gser,x^n),n=1..48); # 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, 1, n}], {n, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 06 2019 *)

G.f.: Sum_{n>=1} (2*n-1)*x^(2*n-1)/Product_{k=1..n} (1-x^(2*k-1)).

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

A116856 Triangle read by rows: T(n,k) is the number of partitions of n into odd parts such that the smallest part is k (n>=1, k>=1).

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 2, 0, 0, 0, 1, 3, 0, 1, 4, 0, 0, 0, 0, 0, 1, 5, 0, 1, 6, 0, 1, 0, 0, 0, 0, 0, 1, 8, 0, 1, 0, 1, 10, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 12, 0, 2, 0, 1, 15, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 18, 0, 2, 0, 1, 0, 1, 22, 0, 3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 27, 0, 3, 0, 1, 0, 1, 32, 0, 4, 0, 1, 0

Offset: 1

Emeric Deutsch, Feb 24 2006

Row 2n-1 has 2n-1 terms; row 4n+2 has 2n+1 terms; row 4n has 2n-1 terms. Row sums yield A000009.

			T(12,3)=2 because we have [9,3] and [3,3,3,3].
Triangle starts:
1;
1;
1,0,1;
2;
2,0,0,0,1;
3,0,1;
4,0,0,0,0,0,1

Cf. A000009, A092314, A116799.

g:=sum(t^(2*j-1)*x^(2*j-1)/product(1-x^(2*i-1),i=j..20),j=1..30): gser:=simplify(series(g,x=0,20)): for n from 1 to 17 do P[n]:=sort(coeff(gser,x^n)) od: d:=proc(n) if n mod 2 = 1 then n elif n mod 4 = 2 then n/2 else n/2-1 fi end: for n from 1 to 17 do seq(coeff(P[n],t^j),j=1..d(n)) od; # yields sequence in triangular form; d(n) is the degree of the polynomial P[n]

imax = 18;
Rest@CoefficientList[#, t]& /@ Rest@CoefficientList[Sum[t^(2j-1)*x^(2j-1)/ Product[1 - x^(2i-1), {i, j, imax}], {j, 1, imax}] + O[x]^imax, x] // Flatten (* Jean-François Alcover, Aug 26 2024 *)

G.f.: Sum_{j=1..oo} t^(2*j-1)*x^(2*j-1)/Product_{i=j..oo} 1-x^(2*i-1).
T(n,2k)=0.
Sum_{k>=1} k*T(n,k) = A092314(n).

cp's OEIS Frontend

A092322 Sum of largest parts of all partitions of n into odd parts.

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