A239239 -id:A239239 - cp's OEIS frontend

A239241 Number of partitions of n into distinct parts for which (number of odd parts) = (number of even parts).

1, 0, 0, 1, 0, 2, 0, 3, 0, 4, 1, 5, 2, 6, 5, 7, 8, 8, 14, 9, 20, 11, 30, 13, 40, 17, 55, 23, 70, 32, 91, 45, 112, 65, 140, 91, 169, 128, 206, 177, 245, 241, 295, 323, 350, 429, 419, 559, 499, 722, 600, 921, 721, 1162, 874, 1452, 1062, 1800, 1299, 2210

Offset: 0

Clark Kimberling, Mar 13 2014

a(n) = A240021(n,0). - Alois P. Heinz, Apr 02 2014

			a(9) = 4 counts these partitions:  81, 72, 63, 54.

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

Cf. A239239, A239240, A239242, A239243, A000009.

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
     `if`(n=0, `if`(t=0, 1, 0 ), b(n, i-1, t)+`if`(i>n, 0,
      b(n-i, i-1, t+`if`(irem(i, 2)=1, 1, -1)))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 15 2014

z = 55; p[n_] := p[n] = IntegerPartitions[n]; d[u_] := d[u] = DeleteDuplicates[u]; g[u_] := g[u] = Length[u];
Table[g[Select[Select[p[n], d[#] == # &], Count[#, ?OddQ] < Count[#, ?EvenQ] &]], {n, 0, z}] (* A239239 *)
Table[g[Select[Select[p[n], d[#] == # &], Count[#, ?OddQ] <= Count[#, ?EvenQ] &]], {n, 0, z}] (* A239240 *)
Table[g[Select[Select[p[n], d[#] == # &], Count[#, ?OddQ] == Count[#, ?EvenQ] &]], {n, 0, z}] (* A239241 *)
Table[g[Select[Select[p[n], d[#] == # &], Count[#, ?OddQ] > Count[#, ?EvenQ] &]], {n, 0, z}] (* A239242 *)
Table[g[Select[Select[p[n], d[#] == # &], Count[#, ?OddQ] >= Count[#, ?EvenQ] &]], {n, 0, z}] (* A239243 *)
(* Peter J. C. Moses, Mar 10 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[n > i*(i+1)/2, 0, If[n==0, If[t==0, 1, 0], b[n, i-1, t] + If[i>n, 0, b[n-i, i-1, t + If[Mod[i, 2]==1, 1, -1]]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Dec 27 2015, after Alois P. Heinz *)

a(n) + A239239(n) + A239242(n) = A000009(n) for n >=1.

a(n) = [x^n y^0] Product_{i>=1} 1+x^i*y^(2*(i mod 2)-1). - Alois P. Heinz, Apr 03 2014

A240021 Number T(n,k) of partitions of n into distinct parts, where k is the difference between the number of odd parts and the number of even parts; triangle T(n,k), n>=0, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 0, 0, 1, 2, 1, 1, 1, 0, 1, 1, 1, 3, 1, 1, 1, 0, 2, 2, 2, 4, 1, 0, 1, 2, 1, 1, 4, 2, 4, 5, 1, 1, 1, 1, 2, 1, 2, 6, 3, 1, 6, 6, 1, 2, 2, 1, 3, 1, 5, 9, 3, 2, 9, 7, 2, 4, 3, 2, 3, 2, 8, 12, 4, 0, 1, 4, 12, 8, 3, 7, 4, 3, 4, 3, 14, 16, 4, 1, 1, 7, 16, 9, 6, 11, 5, 1, 4, 4, 6, 20, 20, 5, 2, 2

Offset: 0

Alois P. Heinz, Mar 31 2014

T(n,k) is defined for all n >= 0, k in A001057.  Row n contains all terms from the leftmost to the rightmost nonzero term.  All other terms (not in the triangle) are equal to 0. First nonzero term of column k>=0 is at n = k^2, first nonzero term of column k<=0 is at n = k*(k+1).
T(n,k) = T(n+k,-k).
T(2n*(2n+1),2n) = A000041(n).
T(4n^2+14n+11,2n+2) = A000070(n).
T(n^2,n) = 1.
T(n^2,n-1) = 0.
T(n^2,n-2) = A209815(n+1).
T(n^2+1,n-1) = A000065(n).
T(n,0) = A239241(n).
Sum_{k<=-1} T(n,k) = A239239(n).
Sum_{k<=0} T(n,k) = A239240(n).
Sum_{k>=1} T(n,k) = A239242(n).
Sum_{k>=0} T(n,k) = A239243(n).
Sum_{k=-1..1} T(n,k) = A239881(n).
T(n,-1) + T(n,1) = A239880(n).
Sum_{k=-n..n} T(n,k) = A000009 (row sums).

			T(12,-3) = 1: [6,4,2].
T(12,-2) = 2: [10,2], [8,4].
T(12,-1) = 1: [12].
T(12,0) = 2: [6,3,2,1], [5,4,2,1].
T(12,1) = 6: [9,2,1], [8,3,1], [7,4,1], [7,3,2], [6,5,1], [5,4,3].
T(12,2) = 3: [11,1], [9,3], [7,5].
T(13,-1) = 6: [10,2,1], [8,4,1], [8,3,2], [7,4,2], [6,5,2], [6,4,3].
T(14,-2) = 3: [12,2], [10,4], [8,6].
Triangle T(n,k) begins:
: n\k : -3 -2 -1  0  1  2  3  ...
+-----+--------------------------
:  0  :           1
:  1  :              1
:  2  :        1
:  3  :           1, 1
:  4  :        1, 0, 0, 1
:  5  :           2, 1
:  6  :     1, 1, 0, 1, 1
:  7  :        1, 3, 1
:  8  :     1, 1, 0, 2, 2
:  9  :        2, 4, 1, 0, 1
: 10  :     2, 1, 1, 4, 2
: 11  :        4, 5, 1, 1, 1
: 12  :  1, 2, 1, 2, 6, 3
: 13  :     1, 6, 6, 1, 2, 2
: 14  :  1, 3, 1, 5, 9, 3

Alois P. Heinz, Rows n = 0..500, flattened

Columns k=0-10 give: A239241, A239871(n+1), A240138, A240139, A240140, A240141, A240142, A240143, A240144, A240145, A240146.

Cf. A240009.

b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0, `if`(n=0, 1,
      expand(b(n, i-1)+`if`(i>n, 0, b(n-i, i-1)*x^(2*irem(i, 2)-1)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=ldegree(p)..degree(p)))(b(n$2)):
seq(T(n), n=0..20);

b[n_, i_] := b[n, i] = If[n>i*(i+1)/2, 0, If[n == 0, 1, Expand[b[n, i-1] + If[i>n, 0, b[n-i, i-1]*x^(2*Mod[i, 2]-1)]]]]; T[n_] := Function[{p}, Table[ Coefficient[p, x, i], {i, Exponent[p, x, Min], Exponent[p, x]}]][b[n, n]]; Table[ T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Feb 11 2015, after Alois P. Heinz *)

N=20; q='q+O('q^N);
e(n) = if(n%2!=0, u, 1/u);
gf = prod(n=1,N, 1 + e(n)*q^n );
V = Vec( gf );
{ for (j=1, #V,  \\ print triangle, including leading zeros
    for (i=0, N-j, print1("   "));  \\ padding
    for (i=-j+1, j-1, print1(polcoeff(V[j], i, u),", "));
    print();
); }
/* Joerg Arndt, Apr 01 2014 */

G.f.: prod(n>=1, 1 + e(n)*q^n ) = 1 + sum(n>=1, e(n)*q^n * prod(k=1..n-1, 1+e(k)*q^k) ) where e(n) = u if n odd, otherwise 1/u; see Pari program. [Joerg Arndt, Apr 01 2014]

A239240 Number of partitions of n into distinct parts for which (number of odd parts) <= (number of even parts).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 4, 2, 6, 4, 9, 6, 13, 10, 18, 15, 24, 24, 32, 35, 43, 51, 56, 72, 74, 100, 97, 136, 128, 183, 168, 241, 222, 315, 290, 408, 381, 522, 497, 664, 647, 839, 837, 1054, 1081, 1317, 1384, 1641, 1767, 2035, 2242, 2519, 2831, 3108, 3555, 3828

Offset: 0

Clark Kimberling, Mar 13 2014

a(n) = Sum_{k<=0} A240021(n,k). - Alois P. Heinz, Apr 02 2014

			a(7) = 4 counts these partitions:  61, 52, 43, 421.

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

Cf. A239239, A239241, A239242, A239243, A000009.

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
     `if`(n=0, `if`(t<=0, 1, 0 ), b(n, i-1, t)+`if`(i>n, 0,
      b(n-i, i-1, t+`if`(irem(i, 2)=1, 1, -1)))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 15 2014

z = 55; p[n_] := p[n] = IntegerPartitions[n]; d[u_] := d[u] = DeleteDuplicates[u]; g[u_] := g[u] = Length[u];
Table[g[Select[Select[p[n], d[#] == # &], Count[#, ?OddQ] < Count[#, ?EvenQ] &]], {n, 0, z}] (* A239239 *)
Table[g[Select[Select[p[n], d[#] == # &], Count[#, ?OddQ] <= Count[#, ?EvenQ] &]], {n, 0, z}] (* A239240 *)
Table[g[Select[Select[p[n], d[#] == # &], Count[#, ?OddQ] == Count[#, ?EvenQ] &]], {n, 0, z}] (* A239241 *)
Table[g[Select[Select[p[n], d[#] == # &], Count[#, ?OddQ] > Count[#, ?EvenQ] &]], {n, 0, z}] (* A239242 *)
Table[g[Select[Select[p[n], d[#] == # &], Count[#, ?OddQ] >= Count[#, ?EvenQ] &]], {n, 0, z}] (* A239243 *)
(* Peter J. C. Moses, Mar 10 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[n>i*(i+1)/2, 0, If[n==0, If[t <= 0, 1, 0], b[n, i-1, t] + If[i>n, 0, b[n-i, i-1, t+If[Mod[i, 2]==1, 1, -1]]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Aug 30 2016, after Alois P. Heinz *)

a(n) + A239242(n) = A000009(n) for n >=1.

A239243 Number of partitions of n into distinct parts for which (number of odd parts) >= (number of even parts).

Original entry on oeis.org

1, 1, 0, 2, 1, 3, 2, 4, 4, 6, 7, 8, 11, 11, 17, 16, 25, 22, 36, 31, 49, 44, 68, 61, 90, 85, 120, 118, 156, 160, 204, 217, 261, 291, 337, 386, 429, 507, 548, 662, 694, 854, 882, 1096, 1112, 1396, 1406, 1765, 1768, 2219, 2223, 2776, 2784, 3451, 3484, 4275

Offset: 0

Clark Kimberling, Mar 13 2014

a(n) = Sum_{k>=0} A240021(n,k). - Alois P. Heinz, Apr 02 2014

			a(8) = 4 counts these partitions:  71, 53, 521, 431.

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

Cf. A239239, A239240, A239241, A239242, A000009.

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
     `if`(n=0, `if`(t>=0, 1, 0 ), b(n, i-1, t)+`if`(i>n, 0,
      b(n-i, i-1, t+`if`(irem(i, 2)=1, 1, -1)))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 15 2014

z = 55; p[n_] := p[n] = IntegerPartitions[n]; d[u_] := d[u] = DeleteDuplicates[u]; g[u_] := g[u] = Length[u];
Table[g[Select[Select[p[n], d[#] == # &], Count[#, ?OddQ] < Count[#, ?EvenQ] &]], {n, 0, z}] (* A239239 *)
Table[g[Select[Select[p[n], d[#] == # &], Count[#, ?OddQ] <= Count[#, ?EvenQ] &]], {n, 0, z}] (* A239240 *)
Table[g[Select[Select[p[n], d[#] == # &], Count[#, ?OddQ] == Count[#, ?EvenQ] &]], {n, 0, z}] (* A239241 *)
Table[g[Select[Select[p[n], d[#] == # &], Count[#, ?OddQ] > Count[#, ?EvenQ] &]], {n, 0, z}] (* A239242 *)
Table[g[Select[Select[p[n], d[#] == # &], Count[#, ?OddQ] >= Count[#, ?EvenQ] &]], {n, 0, z}] (* A239243 *)
(* Peter J. C. Moses, Mar 10 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[n>i*(i+1)/2, 0, If[n==0, If[t>=0, 1, 0], b[n, i-1, t]+If[i>n, 0, b[n-i, i-1, t+If[Mod[i, 2]==1, 1, -1]]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Aug 30 2016, after Alois P. Heinz *)

a(n) + A239239(n) = A000009(n) for n >=1.

A239242 Number of partitions of n into distinct parts for which (number of odd parts) > (number of even parts).

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 2, 1, 4, 2, 6, 3, 9, 5, 12, 9, 17, 14, 22, 22, 29, 33, 38, 48, 50, 68, 65, 95, 86, 128, 113, 172, 149, 226, 197, 295, 260, 379, 342, 485, 449, 613, 587, 773, 762, 967, 987, 1206, 1269, 1497, 1623, 1855, 2063, 2289, 2610, 2823, 3280, 3471

Offset: 0

Clark Kimberling, Mar 13 2014

a(n) = Sum_{k>=1} A240021(n,k). - Alois P. Heinz, Apr 02 2014

			a(8) = 4 counts these partitions:  71, 53, 521, 431.

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

Cf. A239239, A239240, A239241, A239243, A000009.

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
     `if`(n=0, `if`(t>0, 1, 0 ), b(n, i-1, t)+`if`(i>n, 0,
      b(n-i, i-1, t+`if`(irem(i, 2)=1, 1, -1)))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 15 2014

z = 55; p[n_] := p[n] = IntegerPartitions[n]; d[u_] := d[u] = DeleteDuplicates[u]; g[u_] := g[u] = Length[u];
Table[g[Select[Select[p[n], d[#] == # &], Count[#, ?OddQ] < Count[#, ?EvenQ] &]], {n, 0, z}] (* A239239 *)
Table[g[Select[Select[p[n], d[#] == # &], Count[#, ?OddQ] <= Count[#, ?EvenQ] &]], {n, 0, z}] (* A239240 *)
Table[g[Select[Select[p[n], d[#] == # &], Count[#, ?OddQ] == Count[#, ?EvenQ] &]], {n, 0, z}] (* A239241 *)
Table[g[Select[Select[p[n], d[#] == # &], Count[#, ?OddQ] > Count[#, ?EvenQ] &]], {n, 0, z}] (* A239242 *)
Table[g[Select[Select[p[n], d[#] == # &], Count[#, ?OddQ] >= Count[#, ?EvenQ] &]], {n, 0, z}] (* A239243 *)
(* Peter J. C. Moses, Mar 10 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[n>i*(i+1)/2, 0, If[n==0, If[t>0, 1, 0], b[n, i-1, t]+If[i>n, 0, b[n-i, i-1, t+If[Mod[i, 2]==1, 1, -1]]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Aug 30 2016, after Alois P. Heinz *)

a(n) + A239240(n) = A000009(n) for n >=1.

cp's OEIS Frontend

A239241 Number of partitions of n into distinct parts for which (number of odd parts) = (number of even parts).

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A240021 Number T(n,k) of partitions of n into distinct parts, where k is the difference between the number of odd parts and the number of even parts; triangle T(n,k), n>=0, read by rows.

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A239240 Number of partitions of n into distinct parts for which (number of odd parts) <= (number of even parts).

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A239243 Number of partitions of n into distinct parts for which (number of odd parts) >= (number of even parts).

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A239242 Number of partitions of n into distinct parts for which (number of odd parts) > (number of even parts).

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