cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Views

Author

Alois P. Heinz, Mar 31 2014

Keywords

Comments

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).

Examples

			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

Links

Crossrefs

Columns k=0-10 give: A239241, A239871(n+1), A240138, A240139, A240140, A240141, A240142, A240143, A240144, A240145, A240146.
Cf. A240009.

Programs

  • Maple
    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);
  • Mathematica
    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 *)
  • PARI
    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 */

Formula

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]

A239880 Number of strict partitions of n having an ordering in which no parts of equal parity are juxtaposed and the first and last terms have the same parity.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 5, 5, 7, 7, 10, 11, 14, 15, 19, 22, 26, 30, 35, 42, 47, 56, 62, 76, 83, 100, 108, 132, 142, 171, 184, 222, 239, 284, 306, 363, 394, 460, 500, 581, 636, 730, 802, 914, 1010, 1139, 1262, 1415, 1577, 1753, 1956, 2163, 2423, 2663
Offset: 0

Views

Author

Clark Kimberling, Mar 29 2014

Keywords

Comments

A strict partition is one in which every part has multiplicity 1.
a(n) = A240021(n,-1) + A240021(n,1). - Alois P. Heinz, Apr 02 2014

Examples

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

Links

Crossrefs

Cf. A239833, A239835, A239881.

Programs

  • Maple
    b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2 or
       abs(t)-n>1, 0, `if`(n=0, abs(t), b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i-1, t+(2*irem(i, 2)-1)))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Apr 02 2014
  • Mathematica
    d[n_] := Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
p[n_] := p[n] = Select[d[n], Abs[Count[#, ?OddQ] - Count[#, ?EvenQ]] == 1 &]; t =  Table[p[n], {n, 0, 12}]
TableForm[t] (* shows the partitions *)
u = Table[Length[p[n]], {n, 0, 60}]  (* A239880 *)
(* Peter J. C. Moses, Mar 10 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[n > i*(i+1)/2 || Abs[t]-n>1, 0, If[n==0, Abs[t], b[n, i-1, t] + If[i>n, 0, b[n-i, i-1, t + (2*Mod[i, 2]-1)]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)

A239882 Number of strict partitions of 2n having an ordering of the parts in which no two neighboring parts have the same parity.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 9, 15, 22, 33, 46, 65, 87, 117, 153, 199, 254, 324, 408, 512, 639, 795, 986, 1221, 1509, 1862, 2298, 2830, 3485, 4285, 5267, 6460, 7920, 9687, 11836, 14426, 17557, 21310, 25823, 31204, 37632, 45262, 54326, 65029, 77678, 92549, 110035, 130509
Offset: 0

Views

Author

Clark Kimberling, Mar 29 2014

Keywords

Comments

a(n) = number of strict partitions (that is, every part has multiplicity 1) of 2n having an ordering of the parts in which no two neighboring parts have the same parity. This sequence is nondecreasing, unlike A239881, of which it is a bisection; the other bisection is A239883.

Examples

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

Links

Crossrefs

Cf. A239881, A239883, A239872.

Programs

  • Mathematica
    d[n_] := Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &]; p[n_] := p[n] = Select[d[n], Abs[Count[#, ?OddQ] - Count[#, ?EvenQ]] <= 1 &]; t =  Table[p[n], {n, 0, 12}]
TableForm[t] (* shows the partitions *)
u = Table[Length[p[2 n]], {n, 0, 40}]  (* A239882 *)
(* Peter J. C. Moses, Mar 10 2014 *)

Extensions

More terms from Alois P. Heinz, Mar 31 2014

A239883 Number of strict partitions of 2n + 1 having an ordering of the parts in which no two neighboring parts have the same parity.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 13, 18, 23, 31, 41, 55, 73, 99, 132, 177, 236, 313, 412, 540, 701, 904, 1159, 1473, 1861, 2336, 2915, 3615, 4463, 5478, 6698, 8152, 9887, 11944, 14391, 17280, 20703, 24739, 29506, 35115, 41730, 49501, 58650, 69389, 82009, 96807, 114175
Offset: 0

Views

Author

Clark Kimberling, Mar 29 2014

Keywords

Comments

a(n) = number of strict partitions (that is, every part has multiplicity 1) of 2n + 1 having an ordering of the parts in which no two neighboring parts have the same parity. This sequence is nondecreasing, unlike A239881, of which it is a bisection; the other bisection is A239882.

Examples

			a(5) counts these 10 partitions of 11:  [11], [10,1], [9,2], [8,3], [8,1,2], [7,4], [6,5], [6,1,4], [6,3,2], [4,5,2].

Links

Crossrefs

Cf. A239881, A239882, A239872.

Programs

  • Mathematica
    d[n_] := Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &]; p[n_] := p[n] = Select[d[n], Abs[Count[#, ?OddQ] - Count[#, ?EvenQ]] <= 1 &]; t =  Table[p[n], {n, 0, 12}]
TableForm[t] (* shows the partitions *)
u = Table[Length[p[2 n + 1]], {n, 0, 20}]  (* A239883 *)
(* Peter J. C. Moses, Mar 10 2014 *)

Extensions

More terms from Alois P. Heinz, Mar 31 2014
Showing 1-4 of 4 results.

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