A091614 -id:A091614 - cp's OEIS frontend

A091623 Column 1 of triangle A091614.

1, -1, -3, -1, 5, 13, 27, 41, 43, 25, -23, -157, -447, -1011, -2087, -4081, -7685, -14141, -25707, -46251, -82729, -147295, -261317, -462403, -816653, -1439457, -2532779, -4448035, -7794637, -13624815, -23744693, -41230561, -71277787

Offset: 1

Christian G. Bower, Jan 23 2004

sign

G. C. Greubel, Table of n, a(n) for n = 1..120

Cf. A091614.

b[n_, l_, k_]:= b[n, l, k]= If[n==0, 1, Sum[If[i==l, 0, Sum[b[n-i*j, i, k], {j, 1, Min[k, n/i]}]], {i, n}]];
t[n_, k_]:= t[n, k]= If[k>n, 0, b[n, 0, k] - b[n, 0, k-1]]; (* A091613 *)
M:= With[{p=53}, Table[t[n, k], {n, p}, {k, p}]];
T := Inverse[M]; (* A091614 *)
Table[T[[n, 1]], {n, 50}] (* G. C. Greubel, Nov 27 2021 *)

A091613 Triangle: T(n,k) = number of compositions (ordered partitions) of n such that some part is repeated consecutively k times and no part is repeated consecutively more than k times.

Original entry on oeis.org

1, 1, 1, 3, 0, 1, 4, 3, 0, 1, 7, 6, 2, 0, 1, 14, 10, 5, 2, 0, 1, 23, 23, 11, 4, 2, 0, 1, 39, 50, 22, 10, 4, 2, 0, 1, 71, 99, 48, 22, 9, 4, 2, 0, 1, 124, 200, 105, 46, 21, 9, 4, 2, 0, 1, 214, 404, 223, 101, 46, 20, 9, 4, 2, 0, 1, 378, 805, 468, 218, 98, 45, 20, 9, 4, 2, 0, 1, 661, 1599, 979, 466, 213, 98, 44, 20, 9, 4, 2, 0, 1

Offset: 1

Christian G. Bower, Jan 23 2004

Cf. A232294 - A128695 = column 3. - Geoffrey Critzer, Mar 24 2014

			Triangle starts:
    1;
    1,   1;
    3,   0,   1;
    4,   3,   0,  1;
    7,   6,   2,  0,  1;
   14,  10,   5,  2,  0, 1;
   23,  23,  11,  4,  2, 0, 1;
   39,  50,  22, 10,  4, 2, 0, 1;
   71,  99,  48, 22,  9, 4, 2, 0, 1;
  124, 200, 105, 46, 21, 9, 4, 2, 0, 1;
  ...
In the partition 3+3+2+2+2+1+3+3+1, 2 is repeated consecutively 3 times, no part is repeated consecutively more than 3 times. (3 appears 4 times nonconsecutively.)

Alois P. Heinz, Rows n = 1..100, flattened

Row sums: A000079(n-1) (2^(n-1)).
Inverse: A091614.
Square: A091615.
Columns 1-6: A003242, A091616, A091617, A091618, A091619, A091620.
Convergent of columns: A034007.

b:= proc(n, l, k) option remember; `if`(n=0, 1, add(`if`(
      i=l, 0, add(b(n-i*j, i, k), j=1..min(k, n/i))), i=1..n))
    end:
T:= (n, k)-> b(n, 0, k)-b(n, 0, k-1):
seq(seq(T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Feb 08 2017

nn=15;Table[Take[Drop[Transpose[Map[PadRight[#,nn+1]&,Table[ CoefficientList[Series[1/(1-Sum[Sum[x^(j i),{i,1,k}]/Sum[x^(j i),{i,0,k}],{j,1,nn}])-1/(1-Sum[Sum[x^(j i),{i,1,k-1}]/Sum[x^(j i),{i,0,k-1}],{j,1,nn}]),{x,0,nn}],x],{k,1,nn}]]],1][[n]],n],{n,1,nn}]//Grid
(* or *)
Needs["Combinatorica`"];Table[Distribution[Map[Max,Map[Length,Map[Split, Level[Map[Permutations,IntegerPartitions[n,n]],{2}]],{2}]],Range[1,n]],{n,1,15}]//Grid (* Geoffrey Critzer, Mar 24 2014 *)
b[n_, l_, k_] := b[n, l, k] = If[n == 0, 1, Sum[If[i == l, 0,
     Sum[b[n - i*j, i, k], {j, 1, Min[k, n/i]}]], {i, 1, n}]];
T[n_, k_] := b[n, 0, k] - b[n, 0, k - 1];
Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 04 2021, after Alois P. Heinz *)

G.f. for column k: 1/(1 - Sum_{i>=1} (x^i + x^(2*i) + ... + x^(k*i))/( 1 + x^i + x^(2*i) + ... + x^(k*i)) ) - 1/(1 - Sum_{i>=1} (x^i + x^(2*i) + ... + x^((k-1)*i))/( 1 + x^i + x^(2*i) + ... + x^((k-1)*i))). - Geoffrey Critzer, Mar 24 2014

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A091623 Column 1 of triangle A091614.

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A091613 Triangle: T(n,k) = number of compositions (ordered partitions) of n such that some part is repeated consecutively k times and no part is repeated consecutively more than k times.

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