A240009 -id:A240009 - cp's OEIS frontend

A209423 Difference between the number of odd parts and the number of even parts in all the partitions of n.

1, 1, 4, 4, 10, 13, 24, 30, 52, 68, 105, 137, 202, 264, 376, 485, 669, 864, 1162, 1486, 1968, 2501, 3256, 4110, 5285, 6630, 8434, 10511, 13241, 16417, 20505, 25273, 31344, 38438, 47346, 57782, 70746, 85947, 104663, 126594, 153386, 184793, 222865, 267452

Offset: 1

Clark Kimberling, Mar 08 2012

nonn

a(n) = number of parts of odd multiplicity (each counted only once) in all partitions of n. Example: a(5) = 10 because we have [5'],[4',1'],[3',2'], [3',1,1],[2,2,1'],[2',1',1,1], and [1',1,1,1,1] (the 10 counted parts are marked). - Emeric Deutsch, Feb 08 2016

			The partitions of 5 are [5], [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1], and [1,1,1,1,1], a total of 15 odd parts and 5 even parts, so that a(5)=10.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)

Cf. A066897, A066898, A000041, A240009, A264398.

b:= proc(n, i) option remember; local m, f, g;
      m:= irem(i, 2);
      if n=0 then [1, 0, 0]
    elif i<1 then [0, 0, 0]
    else f:= b(n, i-1); g:= `if`(i>n, [0$3], b(n-i, i));
         [f[1]+g[1], f[2]+g[2]+m*g[1], f[3]+g[3]+(1-m)*g[1]]
      fi
    end:
a:= n-> b(n, n)[2] -b(n, n)[3]:
seq(a(n), n=1..50);  # Alois P. Heinz, Jul 09 2012
g := add(x^j/(1+x^j), j = 1 .. 80)/mul(1-x^j, j = 1 .. 80): gser := series(g, x = 0, 50): seq(coeff(gser, x, n), n = 0 .. 45); # Emeric Deutsch, Feb 08 2016

f[n_, i_] := Count[Flatten[IntegerPartitions[n]], i]
o[n_] := Sum[f[n, i], {i, 1, n, 2}]
e[n_] := Sum[f[n, i], {i, 2, n, 2}]
Table[o[n], {n, 1, 45}]  (* A066897 *)
Table[e[n], {n, 1, 45}]  (* A066898 *)
%% - %                   (* A209423 *)
b[n_, i_] := b[n, i] = Module[{m, f, g}, m = Mod[i, 2]; If[n==0, {1, 0, 0}, If[i<1, {0, 0, 0}, f = b[n, i-1]; g = If[i>n, {0, 0, 0}, b[n-i, i]]; {f[[1]] + g[[1]], f[[2]] + g[[2]] + m*g[[1]], f[[3]] + g[[3]] + (1-m)* g[[1]]}]]]; a[n_] := b[n, n][[2]] - b[n, n][[3]]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Nov 16 2015, after Alois P. Heinz *)

a(n) = A066897(n) - A066898(n) = A206563(n,1) - A206563(n,2). - Omar E. Pol, Mar 08 2012
G.f.: Sum_{j>0} x^j/(1+x^j)/Product_{k>0}(1 - x^k). - Emeric Deutsch, Feb 08 2016
a(n) = Sum_{i=1..n} (-1)^(i + 1)*A181187(n, i). - John M. Campbell, Mar 18 2018
a(n) ~ log(2) * exp(Pi*sqrt(2*n/3)) / (2^(3/2) * Pi * sqrt(n)). - Vaclav Kotesovec, May 25 2018
For n > 0, a(n) = A305121(n) + A305123(n). - Vaclav Kotesovec, May 26 2018
a(n) = Sum_{k=-floor(n/2)+(n mod 2)..n} k * A240009(n,k). - Alois P. Heinz, Oct 23 2018
a(n) = Sum_{k>0} k * A264398(n,k). - Alois P. Heinz, Aug 05 2020

A171967 Number of partitions of n with distinct numbers of odd and even parts.

Original entry on oeis.org

0, 1, 2, 2, 5, 5, 10, 12, 20, 25, 37, 49, 68, 90, 119, 158, 206, 269, 344, 446, 565, 722, 908, 1148, 1435, 1795, 2229, 2765, 3416, 4204, 5164, 6315, 7717, 9380, 11406, 13793, 16692, 20093, 24203, 29012, 34799, 41552, 49636, 59059, 70279, 83341, 98822

Offset: 0

Reinhard Zumkeller, Jan 21 2010

nonn

a(n) = A000041(n) - A045931(n) = A108949(n) + A108950(n).

a(n) = Sum_{k<>0} A240009(n,k). - Alois P. Heinz, Mar 30 2014

Alois P. Heinz, Table of n, a(n) for n = 0..3500

Cf. A130780, A171966.

b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t<>0, 1, 0), `if`(i<1, 0, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i, t+(2*irem(i, 2)-1)))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Mar 30 2014

$RecursionLimit = 1000; b[n_, i_, t_] := b[n, i, t] = If[n==0, If[t != 0, 1, 0], If[i < 1, 0, b[n, i-1, t] + If[i>n, 0, b[n-i, i, t+(2*Mod[i, 2]-1)]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jun 30 2015, after Alois P. Heinz *)

A239832 Number of partitions of n having 1 more even part than odd, so that there is an ordering of parts for which the even and odd parts alternate and the first and last terms are even.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 2, 2, 4, 3, 7, 6, 11, 11, 17, 19, 27, 31, 41, 51, 62, 79, 95, 121, 142, 182, 212, 269, 314, 393, 459, 570, 665, 816, 958, 1160, 1364, 1639, 1928, 2297, 2706, 3200, 3768, 4434, 5212, 6105, 7170, 8361, 9799, 11396, 13322, 15450, 18022

Offset: 0

Clark Kimberling, Mar 29 2014

Let c(n) be the number of partitions of n having 1 more odd part than even, so that there is an ordering of parts for which the even and odd parts alternate and the first and last terms are odd. Then c(n) = a(n+1) for n >= 0.

			The three partitions counted by a(10) are [10], [4,1,2,1,2], and [2,3,2,1,2].

Cf. A239833, A239835, A045931, A239871.

Column k=-1 of A240009.

p[n_] := p[n] = Select[IntegerPartitions[n], Count[#, ?OddQ] == -1 + Count[#, ?EvenQ] &]; t = Table[p[n], {n, 0, 10}]
TableForm[t] (* shows the partitions *)
Table[Length[p[n]], {n, 0, 30}]  (* A239832 *)
(* Peter J. C. Moses, Mar 10 2014 *)

A239833 Number of partitions of n having an ordering of parts in which no parts of equal parity are adjacent and the first and last terms have the same parity.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 3, 4, 6, 7, 10, 13, 17, 22, 28, 36, 46, 58, 72, 92, 113, 141, 174, 216, 263, 324, 394, 481, 583, 707, 852, 1029, 1235, 1481, 1774, 2118, 2524, 3003, 3567, 4225, 5003, 5906, 6968, 8202, 9646, 11317, 13275, 15531, 18160, 21195, 24718, 28772

Offset: 0

Clark Kimberling, Mar 29 2014

			a(10) counts these 10 partitions:  [10], [1,8,1], [7,2,1], [3,6,1], [5,4,1], [5,3,2], [3,4,3], [4,1,2,1,2], [2,3,2,1,2], [1,2,1,2,1,2,1].

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

Cf. A239832, A239835, A045931, A239871.

b:= proc(n, i, t) option remember; `if`(abs(t)>n, 0,
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i, t+(2*irem(i, 2)-1))))))
    end:
a:= n-> b(n$2, -1) +b(n$2, 1):
seq(a(n), n=0..80);  # Alois P. Heinz, Apr 02 2014

p[n_] := p[n] = Select[IntegerPartitions[n], Abs[Count[#, ?OddQ] - Count[#, ?EvenQ]] == 1 &]; t = Table[p[n], {n, 0, 10}]
TableForm[t] (* shows the partitions*)
t = Table[Length[p[n]], {n, 0, 60}] (* A239833 *)
(* Peter J. C. Moses, Mar 10 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[Abs[t]>n, 0, If[n==0, 1, If[i<1, 0, b[n, i-1, t] + If[i>n, 0, b[n-i, i, t+(2*Mod[i, 2]-1)]]]]]; a[n_] := b[n, n, -1] + b[n, n, 1]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Oct 12 2015, after Alois P. Heinz *)

a(n) = A239832(n) + A239832(n+1) for n >= 0.

a(n) = A240009(n,-1) + A240009(n,1). - Alois P. Heinz, Apr 02 2014

A239835 Number of partitions of n such that the absolute value of the difference between the number of odd parts and the number of even parts is <=1.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 4, 7, 8, 12, 15, 20, 26, 33, 44, 54, 71, 86, 113, 136, 175, 211, 268, 323, 403, 487, 601, 726, 885, 1068, 1292, 1556, 1867, 2244, 2678, 3208, 3809, 4547, 5379, 6398, 7542, 8937, 10506, 12404, 14542, 17110, 20011, 23465, 27381, 32006, 37267

Offset: 0

Clark Kimberling, Mar 29 2014

Number of partitions of n having an ordering of parts in which no parts of equal parity are adjacent, as in Example.

			a(8) counts these 8 partitions:  8, 161, 521, 341, 4121, 323, 3212, 21212.

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

Cf. A239832, A239833, A045931, A239871.

b:= proc(n, i, t) option remember; `if`(abs(t)-n>1, 0,
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i, t+(2*irem(i, 2)-1))))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Apr 01 2014

p[n_] := p[n] = Select[IntegerPartitions[n], Abs[Count[#, ?OddQ] - Count[#, ?EvenQ]] <= 1 &]; t = Table[p[n], {n, 0, 10}]
TableForm[t] (* shows the partitions *)
Table[Length[p[n]], {n, 0, 60}] (* A239835 *)
(* Peter J. C. Moses, Mar 10 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[Abs[t]-n>1, 0, If[n==0, 1, If[i<1, 0, b[n, i-1, t] + If[i>n, 0, b[n-i, i, t+(2*Mod[i, 2]-1)]]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Nov 16 2015, after Alois P. Heinz *)

a(n) = A045931(n) + A239833(n) for n >= 0.

a(n) = Sum_{k=-1..1} A240009(n,k). - Alois P. Heinz, Apr 01 2014

A240010 Number of partitions of n, where the difference between the number of odd parts and the number of even parts is 1.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 4, 3, 7, 6, 11, 11, 17, 19, 27, 31, 41, 51, 62, 79, 95, 121, 142, 182, 212, 269, 314, 393, 459, 570, 665, 816, 958, 1160, 1364, 1639, 1928, 2297, 2706, 3200, 3768, 4434, 5212, 6105, 7170, 8361, 9799, 11396, 13322, 15450, 18022, 20850

Offset: 1

Alois P. Heinz, Mar 30 2014

nonn

With offset 2 number of partitions of n, where the difference between the number of odd parts and the number of even parts is -1.

			a(9) = 3: [9], [4,2,1,1,1], [3,2,2,1,1].
a(10) = 7: [8,1,1], [7,2,1], [6,3,1], [5,4,1], [5,3,2], [4,3,3], [2,2,2,1,1,1,1].

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

Column k=1 of A240009.

b:= proc(n, i, t) option remember; `if`(abs(t)>n, 0,
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i, t+(2*irem(i, 2)-1))))))
    end:
a:= n-> b(n$2, -1):
seq(a(n), n=1..80);

b[n_, i_, t_] := b[n, i, t] = If[Abs[t] > n, 0, If[n == 0, 1, If[i < 1, 0, b[n, i - 1, t] + If[i > n, 0, b[n - i, i, t + 2 Mod[i, 2] - 1]]]]];
a[n_] := b[n, n, -1];
Array[a, 80] (* Jean-François Alcover, Dec 10 2020, after Alois P. Heinz *)

A240011 Number of partitions of n, where the difference between the number of odd parts and the number of even parts is 2.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 3, 4, 5, 8, 8, 13, 14, 21, 23, 34, 37, 52, 60, 79, 93, 120, 143, 178, 216, 263, 321, 386, 470, 560, 684, 806, 980, 1154, 1395, 1636, 1969, 2304, 2758, 3225, 3835, 4480, 5305, 6186, 7288, 8495, 9961, 11594, 13545, 15742, 18325, 21269, 24675

Offset: 2

Alois P. Heinz, Mar 30 2014

nonn

With offset 4 number of partitions of n, where the difference between the number of odd parts and the number of even parts is -2.

			a(10) = 5: [9,1], [7,3], [5,5], [4,2,1,1,1,1], [3,2,2,1,1,1].
a(11) = 8: [8,1,1,1], [7,2,1,1], [6,3,1,1], [5,4,1,1], [5,3,2,1], [4,3,3,1], [3,3,3,2], [2,2,2,1,1,1,1,1].

Alois P. Heinz, Table of n, a(n) for n = 2..1000

Column k=2 of A240009.

b:= proc(n, i, t) option remember; `if`(abs(t)>n, 0,
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i, t+(2*irem(i, 2)-1))))))
    end:
a:= n-> b(n$2, -2):
seq(a(n), n=2..80);

b[n_, i_, t_] := b[n, i, t] = If[Abs[t] > n, 0, If[n == 0, 1, If[i < 1, 0, b[n, i - 1, t] + If[i > n, 0, b[n - i, i, t + 2 Mod[i, 2] - 1]]]]];
a[n_] := b[n, n, -2];
a /@ Range[2, 80] (* Jean-François Alcover, Dec 10 2020, after Alois P. Heinz *)

A240012 Number of partitions of n, where the difference between the number of odd parts and the number of even parts is 3.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 4, 6, 8, 10, 14, 17, 23, 27, 38, 43, 59, 69, 91, 106, 139, 162, 207, 245, 306, 364, 449, 534, 650, 778, 934, 1117, 1334, 1592, 1887, 2251, 2652, 3155, 3705, 4391, 5139, 6075, 7086, 8347, 9720, 11406, 13252, 15505, 17978, 20965, 24272

Offset: 3

Alois P. Heinz, Mar 30 2014

nonn

With offset 6 number of partitions of n, where the difference between the number of odd parts and the number of even parts is -3.

Alois P. Heinz, Table of n, a(n) for n = 3..1000

Column k=3 of A240009.

b:= proc(n, i, t) option remember; `if`(abs(t)>n, 0,
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i, t+(2*irem(i, 2)-1))))))
    end:
a:= n-> b(n$2, -3):
seq(a(n), n=3..80);

b[n_, i_, t_] := b[n, i, t] = If[Abs[t] > n, 0, If[n == 0, 1, If[i < 1, 0, b[n, i - 1, t] + If[i > n, 0, b[n - i, i, t + 2 Mod[i, 2] - 1]]]]];
a[n_] := b[n, n, -3];
a /@ Range[3, 80] (* Jean-François Alcover, Dec 10 2020, after Alois P. Heinz *)

A240013 Number of partitions of n, where the difference between the number of odd parts and the number of even parts is 4.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 11, 14, 19, 24, 30, 40, 48, 63, 76, 98, 117, 151, 178, 227, 269, 337, 399, 496, 586, 720, 854, 1036, 1228, 1481, 1752, 2096, 2480, 2946, 3481, 4115, 4850, 5707, 6717, 7868, 9237, 10789, 12632, 14707, 17181, 19947, 23243, 26925

Offset: 4

Alois P. Heinz, Mar 30 2014

nonn

With offset 8 number of partitions of n, where the difference between the number of odd parts and the number of even parts is -4.

Alois P. Heinz, Table of n, a(n) for n = 4..1000

Column k=4 of A240009.

b:= proc(n, i, t) option remember; `if`(abs(t)>n, 0,
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i, t+(2*irem(i, 2)-1))))))
    end:
a:= n-> b(n$2, -4):
seq(a(n), n=4..80);

b[n_, i_, t_] := b[n, i, t] = If[Abs[t] > n, 0, If[n == 0, 1, If[i < 1, 0, b[n, i - 1, t] + If[i > n, 0, b[n - i, i, t + 2 Mod[i, 2] - 1]]]]];
a[n_] := b[n, n, -4];
a /@ Range[4, 80] (* Jean-François Alcover, Dec 10 2020, after Alois P. Heinz *)

A240014 Number of partitions of n, where the difference between the number of odd parts and the number of even parts is 5.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 20, 24, 32, 41, 51, 65, 81, 102, 125, 158, 190, 239, 287, 357, 426, 528, 626, 769, 913, 1110, 1314, 1590, 1877, 2255, 2660, 3174, 3738, 4439, 5215, 6162, 7230, 8502, 9954, 11666, 13626, 15911, 18551, 21590, 25118, 29154

Offset: 5

Alois P. Heinz, Mar 30 2014

nonn

With offset 10 number of partitions of n, where the difference between the number of odd parts and the number of even parts is -5.

Alois P. Heinz, Table of n, a(n) for n = 5..1000

Column k=5 of A240009.

b:= proc(n, i, t) option remember; `if`(abs(t)>n, 0,
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i, t+(2*irem(i, 2)-1))))))
    end:
a:= n-> b(n$2, -5):
seq(a(n), n=5..80);

b[n_, i_, t_] := b[n, i, t] = If[Abs[t] > n, 0, If[n == 0, 1, If[i < 1, 0, b[n, i - 1, t] + If[i > n, 0, b[n - i, i, t + 2 Mod[i, 2] - 1]]]]];
a[n_] := b[n, n, -5];
a /@ Range[5, 80] (* Jean-François Alcover, Dec 10 2020, after Alois P. Heinz *)

cp's OEIS Frontend

A209423 Difference between the number of odd parts and the number of even parts in all the partitions of n.

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A171967 Number of partitions of n with distinct numbers of odd and even parts.

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A239832 Number of partitions of n having 1 more even part than odd, so that there is an ordering of parts for which the even and odd parts alternate and the first and last terms are even.

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A239833 Number of partitions of n having an ordering of parts in which no parts of equal parity are adjacent and the first and last terms have the same parity.

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A239835 Number of partitions of n such that the absolute value of the difference between the number of odd parts and the number of even parts is <=1.

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A240010 Number of partitions of n, where the difference between the number of odd parts and the number of even parts is 1.

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A240011 Number of partitions of n, where the difference between the number of odd parts and the number of even parts is 2.

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A240012 Number of partitions of n, where the difference between the number of odd parts and the number of even parts is 3.

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