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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. 7
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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

LINKS

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

FORMULA

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

EXAMPLE

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;

MAPLE

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);

MATHEMATICA

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-Fran├žois Alcover, Jan 19 2015, after Alois P. Heinz *)

CROSSREFS

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

Sequence in context: A114282 A112739 A225640 * A287920 A027293 A104762

Adjacent sequences:  A194540 A194541 A194542 * A194544 A194545 A194546

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Aug 29 2011

STATUS

approved

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Last modified June 24 22:04 EDT 2019. Contains 324337 sequences. (Running on oeis4.)