A239871 -id:A239871 - cp's OEIS frontend

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.

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]

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

A239872 Number of strict partitions of 2n having 1 more even part than odd, so that there is at least one ordering of the parts in which the even and odd parts alternate, and the first and last terms are even.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 2, 3, 6, 10, 17, 26, 40, 57, 81, 110, 148, 193, 250, 316, 397, 491, 603, 732, 885, 1061, 1268, 1508, 1790, 2120, 2510, 2970, 3517, 4170, 4950, 5887, 7013, 8371, 10005, 11979, 14353, 17217, 20654, 24785, 29725, 35637, 42672, 51046, 60962

Offset: 0

Clark Kimberling, Mar 29 2014

Let c(n) be the number of strict partitions (that is, every part has multiplicity 1) of 2n 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. This sequence is nondecreasing, unlike A239871, of which it is a bisection; the other bisection is A239873.

			a(9) counts these 3 partitions of 18:  [18], [8,3,4,1,2], [6,5,4,1,2].

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

Cf. A239241, A239871, A239873, A239832.

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2 or
      abs(t)-n>0, 0, `if`(n=0, 1, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i-1, t+(2*irem(i, 2)-1)))))
    end:
a:= n-> b(2*n$2, 1):
seq(a(n), n=0..60);  # Alois P. Heinz, Apr 01 2014

d[n_] := Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &]; p[n_] := p[n] = Select[d[n], Count[#, ?OddQ] == -1 + Count[#, ?EvenQ] &]; t = Table[p[n], {n, 0, 20}]
TableForm[t] (* shows the partitions *)
u = Table[Length[p[2 n]], {n, 0, 40}]  (* A239872 *)
(* Peter J. C. Moses, Mar 10 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[n > i*(i+1)/2 || Abs[t]-n > 0, 0, If[n == 0, 1, b[n, i-1, t] + If[i>n, 0, b[n-i, i-1, t + (2*Mod[i, 2] - 1)]]]]; a[n_] := b[2*n, 2*n, 1]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Oct 28 2015, after Alois P. Heinz *)

A239873 Number of strict partitions of 2n + 1 having 1 more even part than odd, so that there is at least one ordering of the parts in which the even and odd parts alternate, and the first and last terms are even.

Original entry on oeis.org

0, 0, 0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 43, 51, 61, 74, 91, 113, 144, 184, 239, 311, 407, 530, 692, 895, 1155, 1478, 1882, 2375, 2983, 3715, 4602, 5660, 6925, 8418, 10187, 12257, 14686, 17514, 20809, 24624, 29049, 34154, 40051, 46842, 54668, 63667

Offset: 0

Clark Kimberling, Mar 29 2014

Let c(n) be the number of strict partitions (that is, every part has multiplicity 1) of 2n 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. This sequence is nondecreasing, unlike A239871, of which it is a bisection; the other bisection is A239872.

			a(7) counts these 9 partitions of 15:  [12,1,2], [10,1,4], [10,3,2], [4,9,2], [8,1,6], [8,5,2], [8,3,4], [6,7,2], [6,5,4].

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

Cf. A239241, A239871, A239872, A239832.

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

d[n_] := Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &]; p[n_] := p[n] = Select[d[n], Count[#, ?OddQ] == -1 + Count[#, ?EvenQ] &]; t = Table[p[n], {n, 0, 20}]
TableForm[t] (* shows the partitions *)
u = Table[Length[p[2 n + 1]], {n, 0, 38}]  (* A239873 *)
(* Peter J. C. Moses, Mar 10 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[n > i (i + 1)/2 || Abs[t] > n, 0, If[n == 0, 1, b[n, i-1, t] + If[i>n, 0, b[n-i, i-1, t + (2 Mod[i, 2] - 1)]]]]; a[n_] := b[2n+1, 2n+1, 1]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)

cp's OEIS Frontend

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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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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A239872 Number of strict partitions of 2n having 1 more even part than odd, so that there is at least one ordering of the parts in which the even and odd parts alternate, and the first and last terms are even.

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A239873 Number of strict partitions of 2n + 1 having 1 more even part than odd, so that there is at least one ordering of the parts in which the even and odd parts alternate, and the first and last terms are even.

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