A025162 -id:A025162 - cp's OEIS frontend

A194543 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) is the number of partitions of n into parts p_i such that |p_i - p_j| >= k for i != j.

1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 5, 2, 2, 1, 1, 7, 3, 2, 2, 1, 1, 11, 4, 3, 2, 2, 1, 1, 15, 5, 3, 3, 2, 2, 1, 1, 22, 6, 4, 3, 3, 2, 2, 1, 1, 30, 8, 5, 4, 3, 3, 2, 2, 1, 1, 42, 10, 6, 4, 4, 3, 3, 2, 2, 1, 1, 56, 12, 7, 5, 4, 4, 3, 3, 2, 2, 1, 1, 77, 15, 9, 6, 5, 4, 4, 3, 3, 2, 2, 1, 1

Offset: 0

Alois P. Heinz, Aug 29 2011

T(n,k) = 1 for n >= 0 and k >= n.

In general, column k > 0 is asymptotic to c^(1/4) * r * exp(2*sqrt(c*n)) / (2*sqrt(Pi*(1-r)*(r + k*(1-r))) * n^(3/4)), where r is the smallest real root of the equation r^k + r = 1 and c = k*log(r)^2/2 + polylog(2, 1-r). - Vaclav Kotesovec, Jan 02 2016

			T(7,3) = 3: [7], [6,1], [5,2].
T(23,6) = 11: [23], [22,1], [21,2], [20,3], [19,4], [18,5], [17,6], [16,7], [15,8], [15,7,1], [14,8,1].
Triangle begins:
   1;
   1, 1;
   2, 1, 1;
   3, 2, 1, 1;
   5, 2, 2, 1, 1;
   7, 3, 2, 2, 1, 1;
  11, 4, 3, 2, 2, 1, 1;
  15, 5, 3, 3, 2, 2, 1, 1;

Alois P. Heinz, Rows n = 0..140

Columns 0-8 give: A000041, A000009, A003114, A025157, A025158, A025159, A025160, A025161, A025162. T(n,0)-T(n,1) = A047967(n).

b:= proc(n, i, k) option remember;
      if n<0 then 0
    elif n=0 then 1
    else add(b(n-i-j, i+j, k), j=k..n-i)
      fi
    end:
T:= (n, k)-> `if`(n=0, 1, 0) +add(b(n-i, i, k), i=1..n):
seq(seq(T(n, k), k=0..n), n=0..20);

b[n_, i_, k_] := b[n, i, k] = If[n<0, 0, If[n == 0, 1, Sum[b[n-i-j, i+j, k], {j, k, n-i}]]]; T[n_, k_] := If[n == 0, 1, 0] + Sum[b[n-i, i, k], {i, 1, n}]; Table[ Table[T[n, k], {k, 0, n}], {n, 0, 20}] // Flatten (* Jean-François Alcover, Jan 19 2015, after Alois P. Heinz *)

G.f. of column k: Sum_{j>=0} x^(j*((j-1)*k/2+1))/Product_{i=1..j} (1-x^i).

A025158 Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 4.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 7, 8, 9, 11, 12, 14, 16, 18, 20, 23, 25, 28, 31, 35, 38, 43, 47, 53, 58, 65, 71, 80, 87, 97, 106, 118, 128, 142, 154, 170, 185, 203, 220, 242, 262, 287, 311, 340, 368, 402, 435, 474, 513, 558, 603, 656, 708, 768, 829, 898, 968, 1048

Offset: 1

Clark Kimberling

nonn

Also number of partitions of n such that if k is the largest part, then each 1,2,...,k-1 occur at least 4 times. Example: a(8)=3 because we have [2,2,1,1,1,1], [2,1,1,1,1,1,1] and [1,1,1,1,1,1,1,1]. - Emeric Deutsch, Apr 17 2006

			a(8) = 3 because we have [8], [7,1] and [6,2].

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

Cf. A003114, A025157-A025162.

Column k=4 of A194543.

g:=sum(x^(2*k^2-k)/product(1-x^j,j=1..k),k=1..7): gser:=series(g,x=0,70): seq(coeff(gser,x,n),n=1..66); # Emeric Deutsch, Apr 17 2006

nmax = 100; Rest[CoefficientList[1 + Series[Sum[x^(j*(2*j - 1))/Product[1 - x^i, {i, 1, j}], {j, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jan 02 2016 *)

G.f.: Sum(x^(2*k^2-k)/Product(1-x^i, i=1..k), k=1..infinity). - Vladeta Jovovic, Aug 12 2004

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*n^(3/4)*sqrt(Pi*r^3*(1+4*r^3))), where r = 0.72449195900051561158837228218703656578649448135001101727... is the root of the equation r^4 + r = 1 and c = 2*log(r)^2 + polylog(2, 1-r) = 0.50498141294472195442598916817438524920370382784609501495065... . - Vaclav Kotesovec, Jan 02 2016

More terms from Vladeta Jovovic, Aug 12 2004

A025159 Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 5.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 8, 9, 10, 12, 13, 15, 17, 19, 21, 24, 26, 29, 32, 35, 38, 42, 46, 50, 55, 60, 66, 72, 79, 86, 95, 103, 113, 123, 135, 146, 160, 173, 189, 204, 222, 239, 260, 280, 303, 326, 353, 379, 410, 440, 475, 510, 550, 590, 636, 682

Offset: 1

Clark Kimberling

nonn

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

Cf. A003114, A025157-A025162.

Column k=5 of A194543.

G.f.: Sum(x^(5/2*k^2-3/2*k)/Product(1-x^i, i=1..k), k=1..infinity). - Vladeta Jovovic, Aug 12 2004

a(n) ~ c^(1/4) * r * exp(2*sqrt(c*n)) / (2*sqrt(Pi*(1-r)*(5-4*r)) * n^(3/4)), where r = 0.754877666246692760049508896358528691894606617772793143989... is the root of the equation r^5 + r = 1 and c = 5*log(r)^2/2 + polylog(2, 1-r) = 0.45973143655369174108251201834952526825516678... . - Vaclav Kotesovec, Jan 02 2016

More terms from Vladeta Jovovic, Aug 12 2004

A025160 Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 6.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 9, 10, 11, 13, 14, 16, 18, 20, 22, 25, 27, 30, 33, 36, 39, 43, 46, 50, 54, 59, 63, 69, 74, 81, 87, 95, 102, 112, 120, 131, 141, 154, 165, 180, 193, 210, 225, 244, 261, 283, 302, 326, 348, 375, 400, 430, 458, 492

Offset: 1

Clark Kimberling

nonn

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

Cf. A003114, A025157-A025162.

Column k=6 of A194543.

G.f.: Sum(x^(3*k^2-2*k)/Product(1-x^i, i=1..k), k=1..infinity). - Vladeta Jovovic, Aug 12 2004

More terms from Vladeta Jovovic, Aug 12 2004

A025161 Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 7.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 10, 11, 12, 14, 15, 17, 19, 21, 23, 26, 28, 31, 34, 37, 40, 44, 47, 51, 55, 59, 63, 68, 73, 78, 84, 90, 97, 104, 112, 120, 130, 139, 150, 161, 174, 186, 201, 215, 232, 248, 267, 285, 307, 327, 351, 374, 401

Offset: 1

Clark Kimberling

nonn

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

Cf. A003114, A025157-A025162.

Column k=7 of A194543.

G.f.: Sum(x^(7/2*k^2-5/2*k)/Product(1-x^i, i=1..k), k=1..infinity). - Vladeta Jovovic, Aug 12 2004

More terms from Naohiro Nomoto, Feb 27 2002

A179046 Partitions into distinct parts with minimal difference 3 and minimal part >= 3.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 7, 8, 9, 11, 12, 14, 16, 18, 20, 23, 25, 28, 32, 35, 39, 44, 49, 54, 61, 67, 75, 83, 92, 101, 113, 123, 136, 150, 165, 180, 199, 217, 239, 261, 286, 312, 343, 373, 408, 445, 486, 528, 577, 626, 682, 741, 805, 873, 949, 1027, 1114

Offset: 0

Joerg Arndt, Jan 04 2011

nonn

			a(13)=4 because there are 4 such partitions of 13: 3+10=4+9=5+8=13.
a(0)=1 because the condition is void for the empty list.

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

Cf. A003106 (min diff=2, min part=2), A000009 (min diff=1, min part=1).

Cf. A003114 (min diff=2), A025157 (min diff=3), A025158 (min diff=4), A025159 (min diff=5), A025160 (min diff=6), A025161 (min diff=7), A025162 (min diff=8).

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(n>i*(i+1)/2-3, 0, b(n, i-1)+
      `if`(i>n, 0, b(n-i, i-3))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..80);  # Alois P. Heinz, Apr 02 2014

b[n_, i_] := b[n, i] = If[n == 0, 1,
  If[n > i(i+1)/2 - 3, 0, b[n, i - 1] +
  If[i > n, 0, b[n - i, i - 3]]]];
a[n_] := b[n, n];
a /@ Range[0, 80] (* Jean-François Alcover, Nov 20 2020, after Alois P. Heinz *)

N=66; x='x+O('x^N);
gf = sum(n=0,N, x^(3*n*(n+1)/2)/prod(k=1,n,1-x^k));
v = Vec(gf)
/* Joerg Arndt, Apr 07 2011 */

A179046 = lambda n: Partitions(n,max_slope=-3).filter(lambda x: not x or min(x) >= 3).cardinality() # D. S. McNeil, Jan 04 2011

G.f.: sum(n>=0, x^(3*n*(n+1)/2) / prod(k=1,n,1-x^k) ), this is a special case of the g.f. sum(n>=0, x^(D*n*(n+1)/2) / prod(k=1,n,1-x^k) ) for partitions into distinct parts with minimal difference D and minimal part >= D. - Joerg Arndt, Apr 07 2011

The g.f. for partitions into distinct parts with minimal difference D and no restriction on the minimal part is sum(n>=0, x^(D*n*(n+1)/2 - (D-1)*n) / prod(k=1..n, 1-x^k) ). - Joerg Arndt, Mar 31 2014

cp's OEIS Frontend

A194543 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) is the number of partitions of n into parts p_i such that |p_i - p_j| >= k for i != j.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A025158 Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A025159 Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 5.

Views

Author

Keywords

Links

Crossrefs

Formula

Extensions

A025160 Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 6.

Views

Author

Keywords

Links

Crossrefs

Formula

Extensions

A025161 Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 7.

Views

Author

Keywords

Links

Crossrefs

Formula

Extensions

A179046 Partitions into distinct parts with minimal difference 3 and minimal part >= 3.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Sage

Formula