A116684 -id:A116684 - cp's OEIS frontend

A116682 Sum of the odd parts in all partitions of n into distinct parts.

0, 1, 0, 4, 4, 9, 10, 17, 26, 38, 50, 66, 92, 116, 154, 203, 264, 326, 416, 514, 644, 802, 986, 1198, 1474, 1784, 2156, 2608, 3124, 3728, 4454, 5286, 6266, 7420, 8736, 10279, 12062, 14106, 16472, 19214, 22330, 25914, 30032, 34714, 40058, 46208, 53136

Offset: 0

Emeric Deutsch, Feb 22 2006

nonn

			a(9)=38 because in the partitions of 9 into distinct parts, namely, [9],[8,1],[7,2],[6,3],[6,2,1],[5,4],[5,3,1] and [4,3,2], the sum of the odd parts is 9+1+7+3+1+5+5+3+1+3=38.

David A. Corneth, Table of n, a(n) for n = 0..10000

Cf. A000009, A000700, A066189, A116681, A116683, A116684.

f:=product(1+x^j,j=1..70)*sum((2*j-1)*x^(2*j-1)/(1+x^(2*j-1)),j=1..40): fser:=series(f,x=0,60): seq(coeff(fser,x,n),n=0..50);

d[n_] := d[n] = Select[IntegerPartitions[n], DeleteDuplicates[#] == # &]
Map[Total[Select[Flatten[d[#]], OddQ]] &, -1 + Range[30]]  (* Peter J. C. Moses, Mar 14 2014 *)
(* or *)
CoefficientList[Series[QPochhammer[-1, x]*(1 + EllipticTheta[2, 0, x]^4 - EllipticTheta[4, 0, x]^4)/48, {x, 0, 100}], x] (* Vaclav Kotesovec, Jun 24 2025 *)

a(n) = Sum_{k=0..n} k*A116681(n,k).
G.f.: (Product_{j>=1} 1+x^j)*(Sum_{j>=1} (2*j-1)*x^(2*j-1)/(1+x^(2*j-1))).
a(n) + A116684(n) = A066189(n) = n*A000009(n). - Vaclav Kotesovec, Jun 24 2025
a(n) = Sum_{k=0..floor(n/2)} A000700(n-2*k) * A000009(2*k) * (n - 2*k). - David A. Corneth, Jun 24 2025

A116683 Triangle read by rows: T(n,k) is the number of partitions of n into distinct parts, in which the sum of the even parts is k (n>=0, 0<=k<=n).

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 2, 1, 0, 1, 0, 1, 0, 2, 2, 0, 1, 0, 1, 0, 0, 0, 2, 2, 0, 1, 0, 1, 0, 2, 0, 2, 2, 0, 2, 0, 1, 0, 2, 0, 0, 0, 3, 2, 0, 2, 0, 1, 0, 2, 0, 2, 0, 3, 3, 0, 2, 0, 2, 0, 2, 0, 2, 0, 0, 0, 4, 3, 0, 2, 0, 2, 0, 2, 0, 2, 0, 3, 0, 4, 3, 0, 3, 0, 2, 0, 4

Offset: 0

Emeric Deutsch, Feb 22 2006

Row 2n-1 has 2n-1 terms; row 2n has 2n+1 terms. Row sums yield A000009. T(n,0)=A000700(n). Columns 2n-1 contain only 0's. Sum(k*T(n,k), k=0..n)=A116684(n).

			T(9,6)=2 because we have [6,3] and [4,3,2].
Triangle starts:
1;
1;
0,0,1;
1,0,1;
1,0,0,0,1;
1,0,1,0,1;
1,0,1,0,0,0,2

Cf. A000009, A000700, A116684.

g:=product((1+x^(2*j-1))*(1+(t*x)^(2*j)),j=1..30): gser:=simplify(series(g,x=0,20)): P[0]:=1: for n from 1 to 14 do P[n]:=sort(coeff(gser,x^n)) od: for n from 0 to 14 do seq(coeff(P[n],t,j),j=0..2*floor(n/2)) od; # yields sequence in triangular form

G.f.=product((1+x^(2j-1))(1+(tx)^(2j)), j=1..infinity).

A239292 (sum of all odd parts of all strict partitions of n) - (sum of all even parts of all strict partitions of n); for "strict", see Comments.

Original entry on oeis.org

0, 1, -2, 2, 0, 3, -4, -1, 4, 4, 0, 0, 4, -2, 0, 1, 16, 6, 4, 2, 8, 8, 14, 4, 20, 18, 22, 32, 32, 32, 28, 32, 52, 56, 64, 83, 76, 92, 112, 130, 140, 168, 172, 198, 212, 256, 288, 318, 368, 416, 456, 527, 564, 640, 712, 806, 884, 985, 1116, 1224, 1344, 1496

Offset: 0

Clark Kimberling, Mar 14 2014

A strict partition is one having distinct parts. a(n) < 0 if and only if n is one of these: 2,6,7,13.

			The strict partitions of 6 are 6, 51, 42, 321.  The sum of all the odd parts is 10 and the sum of all the even parts is 14, so that a(6) = -4.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000009, A116682, A116684.

b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0,
      `if`(n=0, [1, 0], b(n, i-1) +`if`(i>n, 0, (p->p+
      [0, p[1]*`if`(irem(i, 2)=1, i, -i)])(b(n-i, i-1)))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..80);  # Alois P. Heinz, Mar 15 2014

d[n_] := d[n] = Select[IntegerPartitions[n], DeleteDuplicates[#] == # &]; Map[Total[Select[#, OddQ]] - Total[Select[#, EvenQ]]&[Flatten[d[#]]] &, -1 + Range[55]]  (* Peter J. C. Moses, Mar 14 2014 *)
b[n_, i_] := b[n, i] = If[n > i (i + 1)/2, 0,
     If[n == 0, {1, 0}, b[n, i - 1] + If[i > n, 0, Function[p, p +
     {0, p[[1]]*If[Mod[i, 2] == 1, i, -i]}][b[n - i, i - 1]]]]];
a[n_] := b[n, n][[2]];
a /@ Range[0, 80] (* Jean-François Alcover, May 31 2021, after Alois P. Heinz *)

a(n) = A116682(n) - A116684(n) for n >= 0.

cp's OEIS Frontend

A116682 Sum of the odd parts in all partitions of n into distinct parts.

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A116683 Triangle read by rows: T(n,k) is the number of partitions of n into distinct parts, in which the sum of the even parts is k (n>=0, 0<=k<=n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

A239292 (sum of all odd parts of all strict partitions of n) - (sum of all even parts of all strict partitions of n); for "strict", see Comments.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula