A025158 -id:A025158 - 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).

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

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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.

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