A117454 Triangle read by rows: T(n,k) is the number of partitions of n into distinct parts such that the difference between the largest and smallest parts is k (n>=1; 0<=k<=n-2 for n>=2).

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 0, 1, 2, 2, 1, 2, 0, 1, 1, 1, 0, 2, 2, 2, 1, 2, 0, 1, 1, 0, 2, 0, 3, 3, 2, 1, 2, 0, 1, 1, 1, 0, 2, 2, 3, 3, 2, 1, 2, 0, 1, 1, 0, 1, 2, 2, 3, 4, 3, 2, 1, 2, 0, 1, 1, 1, 1, 1, 3, 4, 3, 4, 3, 2, 1, 2, 0

Offset

1,14

Comments

Also number of partitions of n in which all integers smaller than the largest part occur and have k parts smaller than the largest part (n>=1, k>=0). Row 1 has one term; rows j (j>=2) have j-1 terms.

Links

John Tyler Rascoe, Rows n = 1..120, flattened

Formula

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

Sum_{k=0..n-2} k*T(n,k) = A117455(n).

Example

T(12,5)=3 because we have [7,3,2],[6,5,1] and [6,3,2,1].
Triangle starts:
  1;
  1;
  1,1;
  1,0,1;
  1,1,0,1;
  1,0,2,0,1;
  ...

Maple

g:=sum(t^(i-1)*x^(i*(i+1)/2)/(1-x^i)/product(1-t*x^j, j=1..i-1), i=1..20): gser:=simplify(series(g, x=0, 20)): for n from 1 to 16 do P[n]:=coeff(gser, x^n) od: 1; for n from 2 to 16 do seq(coeff(P[n], t, j), j=0..n-2) od; # yields sequence in triangular form

Mathematica

z = 20; d[n_] := d[n] = Select[IntegerPartitions[n], DeleteDuplicates[#] == # &]; p[n_, k_] := p[n, k] = d[n][[k]]; t = Table[Max[p[n, k]] - Min[p[n, k]], {n, 1, z}, {k, 1, PartitionsQ[n]}]; u = Table[Count[t[[n]], k], {n, 1, z}, {k, 0, n - 2}];
TableForm[u] (* A117454 as an array *)
Flatten[u]   (* A117454 as a sequence *)
(* Clark Kimberling, Mar 14 2014 *)

Prog

(PARI)
A_xy(N) = {my(x='x+O('x^(N+1)), A = sum(i=1, (sqrtint(8*N+1)-1)\2, x^(i*(i+1)/2)/(1-x^i) * prod(j=1, i-1, y/(1-y*x^j)))); vector(N, n, Vecrev(polcoeff(A, n)))} \\ John Tyler Rascoe, Jun 24 2026

Crossrefs

Cf. A000009 (row sums), A117455.

Sequence in context: A129691 A280750 A280748 * A115357 A171182 A333806

Adjacent sequences: A117451 A117452 A117453 * A117455 A117456 A117457

Keyword

nonn,tabf,easy

Author

Emeric Deutsch, Mar 18 2006

Status

approved