A214258 -id:A214258 - cp's OEIS frontend

A214269 Number T(n,k) of compositions of n where the difference between largest and smallest parts equals k and adjacent parts are unequal; triangle T(n,k), n>=1, 0<=k

1, 1, 0, 1, 2, 0, 1, 1, 2, 0, 1, 3, 1, 2, 0, 1, 2, 8, 1, 2, 0, 1, 4, 7, 8, 1, 2, 0, 1, 2, 13, 12, 8, 1, 2, 0, 1, 4, 25, 18, 12, 8, 1, 2, 0, 1, 4, 27, 46, 23, 12, 8, 1, 2, 0, 1, 4, 43, 69, 51, 23, 12, 8, 1, 2, 0, 1, 3, 71, 111, 90, 56, 23, 12, 8, 1, 2, 0

Offset: 1

Alois P. Heinz, Jul 09 2012

			T(7,0) = 1: [7].
T(7,1) = 4: [4,3], [3,4], [2,3,2], [1,2,1,2,1].
T(7,2) = 7: [3,1,3], [3,1,2,1], [2,1,3,1], [1,3,2,1], [1,3,1,2], [1,2,3,1], [1,2,1,3].
T(7,3) = 8: [5,2], [4,2,1], [4,1,2], [2,5], [2,4,1], [2,1,4], [1,4,2], [1,2,4].
T(7,4) = 1: [1,5,1].
T(7,5) = 2: [6,1], [1,6].
Triangle T(n,k) begins:
  1;
  1,  0;
  1,  2,  0;
  1,  1,  2,  0;
  1,  3,  1,  2,  0;
  1,  2,  8,  1,  2,  0;
  1,  4,  7,  8,  1,  2,  0;
  1,  2, 13, 12,  8,  1,  2,  0;

Alois P. Heinz, Rows n = 1..150, flattened

Columns k=0-10 give: A000012, A214270, A214271, A214272, A214273, A214274, A214275, A214276, A214277, A214278, A214279.
Row sums give: A003242.
Cf. A214246, A214247, A214248, A214249, A214257, A214258, A214268.

b:= proc(n, k, s, t, l) option remember;
      `if`(n<0, 0, `if`(n=0, 1, add(`if`(j=l, 0, b(n-j, k,
       min(s, j), max(t, j), j)), j=max(1, t-k+1)..s+k-1)))
    end:
A:= proc(n, k) option remember;
      `if`(n=0, 1, add(b(n-j, k+1, j, j, j), j=1..n))
    end:
T:= (n, k)-> A(n, k) -`if`(k=0, 0, A(n, k-1)):
seq(seq(T(n, k), k=0..n-1), n=1..14);

b[n_, k_, s_, t_, l_] := b[n, k, s, t, l] = If[n < 0, 0, If[n == 0, 1, Sum [If[j == l, 0, b[n-j, k, Min[s, j], Max[t, j], j]], {j, Max[1, t-k+1], s+k-1}] ] ]; a[n_, k_] := a[n, k] = If[n == 0, 1, Sum[b[n - j, k+1, j, j, j], {j, 1, n}]]; t[n_, k_] := a[n, k] - If[k == 0, 0, a[n, k-1]]; Table[Table[t[n, k], {k, 0, n-1}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Dec 11 2013, translated from Maple *)

T(n,0) = 1, T(n,k) = A214268(n,k) - A214268(n,k-1) for k>0.

A214247 Number A(n,k) of compositions of n where differences between neighboring parts are in {-k,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 4, 4, 1, 1, 1, 1, 1, 2, 5, 2, 1, 1, 1, 1, 1, 3, 3, 5, 4, 1, 1, 1, 1, 1, 1, 2, 2, 7, 3, 1, 1, 1, 1, 1, 1, 3, 3, 6, 10, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 4, 9, 2

Offset: 0

Alois P. Heinz, Jul 08 2012

			A(5,0) = 2: [5], [1,1,1,1,1].
A(5,1) = 4: [5], [3,2], [2,3], [2,1,2].
A(5,2) = 2: [5], [1,3,1].
A(5,3) = 3: [5], [4,1], [1,4].
Square array A(n,k) begins:
  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
  2,  1,  1,  1,  1,  1,  1,  1,  1, ...
  2,  3,  1,  1,  1,  1,  1,  1,  1, ...
  3,  2,  3,  1,  1,  1,  1,  1,  1, ...
  2,  4,  2,  3,  1,  1,  1,  1,  1, ...
  4,  5,  3,  2,  3,  1,  1,  1,  1, ...
  2,  5,  2,  3,  2,  3,  1,  1,  1, ...
  4,  7,  6,  1,  3,  2,  3,  1,  1, ...

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

Columns k=0-2 give: A000005, A173258, A214254.
Rows n=0, 1 and main diagonal give: A000012.
Cf. A000108, A214246, A214248, A214249, A214257, A214258, A214268, A214269, A227543.

b:= proc(n, i, k) option remember;
      `if`(n<1 or i<1, 0, `if`(n=i, 1, add(b(n-i, i+j, k), j={-k, k})))
    end:
A:= (n, k)-> `if`(n=0, 1, add(b(n, j, k), j=1..n)):
seq(seq(A(n, d-n), n=0..d), d=0..15);

b[n_, i_, k_] := b[n, i, k] = If[n < 1 || i < 1, 0, If[n == i, 1, Sum[b[n - i, i + j, k], { j, Union[{-k, k}]}]]]; a[n_, k_] := If[n == 0, 1, Sum[b[n, j, k], {j, 1, n}]]; Table[Table[a[n, d - n], {n, 0, d}], {d, 0, 15}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from Maple *)

tri(k) = {(k*(k+1)/2)}
ra(n) = {(sqrt(1+8*n)-1)/2}
C(q,n) = {c = if(n<=1, return(1)); return(sum(i=0, n-1, C(q, i) * C(q, n-1-i) * q^((i+1)*(n-1 -i))));}
Cw_q(i,k) = {return(q^(((k+2)*i)+1) * polrecip(C(q^(2*k),i)))}
A_qt(k,N) = {1 + sum(i=0,N/(k+1), t^((2*i)+1) * Cw_q(i,k) * (sum(j=0,ra(N+2)+1, prod(u=1,j, sum(v=0,(N-(tri(u)*k))/(k+2), t^((2*v)+1) * q^(((2*v)+1)*u*k) * Cw_q(v,k)))))^2)}
A_q(k,N) = {my(q='q+O('q^N), Aqt = A_qt(k,N), Aq = 1); for(i=1,poldegree(Aqt,t), Aq += polcoef(Aqt,i,t)/(1-q^i)); Aq}
A214247_array(maxrow,maxcolumn) = {my(m=concat([1],vector(maxrow,n,numdiv(n)))~); for(k=1,maxcolumn, m = matconcat([m,Vec(A_q(k,maxrow))~])); m}
A214247_array(10,10) \\ John Tyler Rascoe, Oct 15 2024

G.f. for column k > 0: A(k,q) is A(k,q,t) = Sum_{n,m>=0} (q^n)*(t^m) under the transform (q^n)*(t^m) -> (q^n)/(1-q^m) for all m > 0 where A(k,q,t) = 1 + Sum_{i>=0} ( t^((2*i)+1) * Cw(i,k,q) * (Sum_{j>=0} (Product_{u=1..j} (Sum_{v>=0} t^((2*v)+1) * q^(((2*v)+1)*k*u) * Cw(v,k,q))))^2 ), Cw(i,k,q) = q^(((k+2)*i)+1) * Ca(i,q^(2*k)), and Ca(i,q) is the i-th Carlitz-Riordan q-Catalan number (i-th row polynomial of A227543). - John Tyler Rascoe, Sep 13 2024

A214248 Number A(n,k) of compositions of n where differences between neighboring parts are in {-k,...,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 4, 3, 1, 1, 2, 4, 6, 2, 1, 1, 2, 4, 8, 11, 4, 1, 1, 2, 4, 8, 14, 17, 2, 1, 1, 2, 4, 8, 16, 27, 29, 4, 1, 1, 2, 4, 8, 16, 30, 49, 47, 3, 1, 1, 2, 4, 8, 16, 32, 59, 92, 78, 4, 1, 1, 2, 4, 8, 16, 32, 62, 113, 170, 130, 2

Offset: 0

Alois P. Heinz, Jul 08 2012

			A(3,0) = 2: [3], [1,1,1].
A(4,1) = 6: [4], [2,2], [2,1,1], [1,2,1], [1,1,2], [1,1,1,1].
A(5,2) = 14: [5], [3,2], [3,1,1], [2,3], [2,2,1], [2,1,2], [2,1,1,1], [1,3,1], [1,2,2], [1,2,1,1], [1,1,3], [1,1,2,1], [1,1,1,2], [1,1,1,1,1].
Square array A(n,k) begins:
  1,  1,  1,  1,  1,  1,  1,  1, ...
  1,  1,  1,  1,  1,  1,  1,  1, ...
  2,  2,  2,  2,  2,  2,  2,  2, ...
  2,  4,  4,  4,  4,  4,  4,  4, ...
  3,  6,  8,  8,  8,  8,  8,  8, ...
  2, 11, 14, 16, 16, 16, 16, 16, ...
  4, 17, 27, 30, 32, 32, 32, 32, ...
  2, 29, 49, 59, 62, 64, 64, 64, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-2 give: A000005, A034297, A214255.
Main diagonal gives: A011782.
Cf. A214246, A214247, A214249, A214257, A214258, A214268, A214269.

b:= proc(n, i, k) option remember;
      `if`(n<1 or i<1, 0, `if`(n=i, 1, add(b(n-i, i+j, k), j={$-k..k})))
    end:
A:= (n, k)-> `if`(n=0, 1, add(b(n, j, k), j=1..n)):
seq(seq(A(n, d-n), n=0..d), d=0..15);

b[n_, i_, k_] := b[n, i, k] = If[n < 1 || i < 1, 0, If[n == i, 1, Sum[b[n - i, i + j, k], {j, -k, k}]]]; A[n_, k_] := If[n == 0, 1, Sum[b[n, j, k], {j, 1, n}]]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 15}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

A214249 Number A(n,k) of compositions of n where differences between neighboring parts are in {-k,...,k} \ {0}; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 3, 4, 4, 1, 1, 1, 1, 3, 4, 5, 5, 1, 1, 1, 1, 3, 4, 7, 11, 5, 1, 1, 1, 1, 3, 4, 7, 12, 14, 7, 1, 1, 1, 1, 3, 4, 7, 14, 20, 18, 10, 1, 1, 1, 1, 3, 4, 7, 14, 21, 30, 36, 9, 1, 1, 1, 1, 3, 4, 7, 14, 23, 36, 50, 49, 14, 1

Offset: 0

Alois P. Heinz, Jul 08 2012

			A(3,0) = 1: [3].
A(4,1) = 2: [4], [1,2,1].
A(5,2) = 5: [5], [3,2], [2,3], [2,1,2], [1,3,1].
A(6,3) = 12: [6], [4,2], [3,2,1], [3,1,2], [2,4], [2,3,1], [2,1,3], [2,1,2,1], [1,4,1], [1,3,2], [1,2,3], [1,2,1,2].
Square array A(n,k) begins:
  1,  1,  1,  1,  1,  1,  1,  1, ...
  1,  1,  1,  1,  1,  1,  1,  1, ...
  1,  1,  1,  1,  1,  1,  1,  1, ...
  1,  3,  3,  3,  3,  3,  3,  3, ...
  1,  2,  4,  4,  4,  4,  4,  4, ...
  1,  4,  5,  7,  7,  7,  7,  7, ...
  1,  5, 11, 12, 14, 14, 14, 14, ...
  1,  5, 14, 20, 21, 23, 23, 23, ...

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

Columns k=0-2 give: A000012, A173258, A214256.
Main diagonal gives: A003242.
Cf. A214246, A214247, A214248, A214257, A214258, A214268, A214269.

b:= proc(n, i, k) option remember; `if`(n<1 or i<1, 0,
      `if`(n=i, 1, add(b(n-i, i+j, k), j={$-k..k} minus{0})))
    end:
A:= (n, k)-> `if`(n=0, 1, add(b(n, j, min(n, k)), j=1..n)):
seq(seq(A(n, d-n), n=0..d), d=0..15);

b[n_, i_, k_] := b[n, i, k] = If[n<1 || i<1, 0, If[n == i, 1, Sum[b[n-i, i+j, k], {j, Range[-k, -1] ~Join~ Range[k]}]]]; A[n_, k_] := If[n == 0, 1, Sum[b[n, j, Min[n, k]], {j, 1, n}]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 15}] // Flatten (* Jean-François Alcover, Jan 15 2014, translated from Maple *)

A214246 Number A(n,k) of compositions of n where differences between neighboring parts are in {-k,0,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 4, 3, 1, 1, 2, 2, 6, 2, 1, 1, 2, 2, 5, 11, 4, 1, 1, 2, 2, 3, 5, 17, 2, 1, 1, 2, 2, 3, 4, 10, 29, 4, 1, 1, 2, 2, 3, 2, 7, 10, 47, 3, 1, 1, 2, 2, 3, 2, 6, 8, 21, 78, 4, 1, 1, 2, 2, 3, 2, 4, 5, 9, 22, 130, 2

Offset: 0

Alois P. Heinz, Jul 08 2012

			A(3,0) = 2: [3], [1,1,1].
A(4,1) = 6: [4], [2,2], [2,1,1], [1,2,1], [1,1,2], [1,1,1,1].
A(5,2) = 5: [5], [3,1,1], [1,3,1], [1,1,3], [1,1,1,1,1].
A(6,3) = 7: [6], [4,1,1], [3,3], [2,2,2], [1,4,1], [1,1,4], [1,1,1,1,1,1].
Square array A(n,k) begins:
  1,  1,  1,  1,  1,  1,  1,  1, ...
  1,  1,  1,  1,  1,  1,  1,  1, ...
  2,  2,  2,  2,  2,  2,  2,  2, ...
  2,  4,  2,  2,  2,  2,  2,  2, ...
  3,  6,  5,  3,  3,  3,  3,  3, ...
  2, 11,  5,  4,  2,  2,  2,  2, ...
  4, 17, 10,  7,  6,  4,  4,  4, ...
  2, 29, 10,  8,  5,  4,  2,  2, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Column k=0 and main diagonal give: A000005.
Columns k=1, 2 give: A034297, A214253.
Cf. A214247, A214248, A214249, A214257, A214258, A214268, A214269.

b:= proc(n, i, k) option remember;
      `if`(n<1 or i<1, 0, `if`(n=i, 1, add(b(n-i, i+j, k), j={-k, 0, k})))
    end:
A:= (n, k)-> `if`(n=0, 1, add(b(n, j, k), j=1..n)):
seq(seq(A(n, d-n), n=0..d), d=0..15);

b[n_, i_, k_] := b[n, i, k] = If[n < 1 || i < 1, 0, If[n == i, 1, Sum[b[n - i, i + j, k], {j, Union[{-k, 0, k}]}]]]; A[n_, k_] := If[n == 0, 1, Sum[b[n, j, k], {j, 1, n}]]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 15}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

A214257 Number A(n,k) of compositions of n where the difference between largest and smallest parts is <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 4, 3, 1, 1, 2, 4, 6, 2, 1, 1, 2, 4, 8, 11, 4, 1, 1, 2, 4, 8, 14, 15, 2, 1, 1, 2, 4, 8, 16, 27, 27, 4, 1, 1, 2, 4, 8, 16, 30, 47, 39, 3, 1, 1, 2, 4, 8, 16, 32, 59, 88, 63, 4, 1, 1, 2, 4, 8, 16, 32, 62, 111, 158, 100, 2

Offset: 0

Alois P. Heinz, Jul 08 2012

			A(3,0) =  2: [3], [1,1,1].
A(4,1) =  6: [4], [2,2], [2,1,1], [1,2,1], [1,1,2], [1,1,1,1].
A(5,1) =  8: [5], [3,2], [2,3], [2,2,1], [2,1,2], [2,1,1,1], [1,2,2], [1,2,1,1], [1,1,2,1], [1,1,1,2], [1,1,1,1,1],
A(5,2) = 14: [5], [3,2], [3,1,1], [2,3], [2,2,1], [2,1,2], [2,1,1,1], [1,3,1], [1,2,2], [1,2,1,1], [1,1,3], [1,1,2,1], [1,1,1,2], [1,1,1,1,1].
Square array A(n,k) begins:
  1,  1,  1,  1,  1,  1,  1,  1, ...
  1,  1,  1,  1,  1,  1,  1,  1, ...
  2,  2,  2,  2,  2,  2,  2,  2, ...
  2,  4,  4,  4,  4,  4,  4,  4, ...
  3,  6,  8,  8,  8,  8,  8,  8, ...
  2, 11, 14, 16, 16, 16, 16, 16, ...
  4, 15, 27, 30, 32, 32, 32, 32, ...
  2, 27, 47, 59, 62, 64, 64, 64, ...

Alois P. Heinz, Antidiagonals n = 0..150, flattened

Columns k=0-1 give: A000005, A072951.
Main diagonal gives: A011782.
Cf. A214246, A214247, A214248, A214249, A214258, A214268, A214269.

b:= proc(n, k, s, t) option remember;
      `if`(n<0, 0, `if`(n=0, 1, add(b(n-j, k,
       min(s,j), max(t,j)), j=max(1, t-k+1)..s+k-1)))
    end:
A:= (n, k)-> `if`(n=0, 1, add(b(n-j, k+1, j, j), j=1..n)):
seq(seq(A(n, d-n), n=0..d), d=0..11);
# second Maple program:
b:= proc(n, s, t) option remember; `if`(n=0, x^(t-s),
      add(b(n-j, min(s, j), max(t, j)), j=1..n))
    end:
T:= (n, k)-> coeff(b(n$2, 0), x, k):
A:= proc(n, k) option remember; `if`(k<0, 0,
      `if`(k>n, A(n$2), A(n, k-1)+T(n, k)))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..11);  # Alois P. Heinz, Jan 05 2019

b[n_, k_, s_, t_] := b[n, k, s, t] = If[n < 0, 0, If[n == 0, 1, Sum [b[n - j, k, Min[s, j], Max[t, j]], {j, Max[1, t - k + 1], s + k - 1}]]]; A[n_, k_] := If[n == 0, 1, Sum[b[n - j, k + 1, j, j], {j, 1, n}]]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 11}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

T(n,k) = Sum_{i=0..k} A214258(n,i).

A214268 Number A(n,k) of compositions of n where the difference between largest and smallest parts is <= k and adjacent parts are unequal; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 3, 4, 4, 1, 1, 1, 1, 3, 4, 5, 3, 1, 1, 1, 1, 3, 4, 7, 11, 5, 1, 1, 1, 1, 3, 4, 7, 12, 12, 3, 1, 1, 1, 1, 3, 4, 7, 14, 20, 16, 5, 1, 1, 1, 1, 3, 4, 7, 14, 21, 28, 30, 5, 1

Offset: 0

Alois P. Heinz, Jul 09 2012

			A(3,0) = 1: [3].
A(4,1) = 2: [4], [1,2,1].
A(5,2) = 5: [5], [3,2], [2,3], [2,1,2], [1,3,1].
A(6,3) = 12: [6], [4,2], [3,2,1], [3,1,2], [2,4], [2,3,1], [2,1,3], [2,1,2,1], [1,4,1], [1,3,2], [1,2,3], [1,2,1,2].
A(7,4) = 21: [7], [5,2], [4,3], [4,2,1], [4,1,2], [3,4], [3,1,3], [3,1,2,1], [2,5], [2,4,1], [2,3,2], [2,1,4], [2,1,3,1], [1,5,1], [1,4,2], [1,3,2,1], [1,3,1,2], [1,2,4], [1,2,3,1], [1,2,1,3], [1,2,1,2,1].
Square array A(n,k) begins:
  1,  1,  1,  1,  1,  1,  1,  1, ...
  1,  1,  1,  1,  1,  1,  1,  1, ...
  1,  1,  1,  1,  1,  1,  1,  1, ...
  1,  3,  3,  3,  3,  3,  3,  3, ...
  1,  2,  4,  4,  4,  4,  4,  4, ...
  1,  4,  5,  7,  7,  7,  7,  7, ...
  1,  3, 11, 12, 14, 14, 14, 14, ...
  1,  5, 12, 20, 21, 23, 23, 23, ...

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

Columns k=0, 1 give: A000012, 1+A214270(n).
Main diagonal gives: A003242.
Cf. A214246, A214247, A214248, A214249, A214257, A214258, A214269.

b:= proc(n, k, s, t, l) option remember;
      `if`(n<0, 0, `if`(n=0, 1, add(`if`(j=l, 0, b(n-j, k,
       min(s, j), max(t, j), j)), j=max(1, t-k+1)..s+k-1)))
    end:
A:= (n, k)-> `if`(n=0, 1, add(b(n-j, k+1, j, j, j), j=1..n)):
seq(seq(A(n,d-n), n=0..d), d=0..14);

b[n_, k_, s_, t_, l_] := b[n, k, s, t, l] = If[n < 0, 0, If[n == 0, 1, Sum[If[j == l, 0, b[n - j, k, Min[s, j], Max[t, j], j]], {j, Max[1, t - k + 1], s + k - 1}]]]; A[n_, k_] := If[n == 0, 1, Sum[b[n - j, k + 1, j, j, j], {j, 1, n}]]; Table[Table[A [n, d - n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

A323111 Number of compositions of 2n where the difference between largest and smallest parts equals n.

Original entry on oeis.org

0, 2, 3, 12, 23, 61, 135, 306, 679, 1499, 3257, 7053, 15167, 32443, 69080, 146533, 309743, 652762, 1371897, 2876284, 6017100, 12562511, 26180328, 54469099, 113151406, 234723068, 486276265, 1006195365, 2079647131, 4293758081, 8856361762, 18250277543

Offset: 1

Alois P. Heinz, Jan 04 2019

nonn

			a(2) = 2: 13, 31.
a(3) = 3: 114, 141, 411.
a(4) = 12: 1115, 1151, 1511, 5111, 125, 152, 215, 251, 512, 521, 26, 62.
a(5) = 23: 11116, 11161, 11611, 16111, 61111, 1126, 1162, 1216, 1261, 1612, 1621, 2116, 2161, 2611, 6112, 6121, 6211, 136, 163, 316, 361, 613, 631.

Alois P. Heinz, Table of n, a(n) for n = 1..200

Cf. A117989 (the same for partitions), A214258.

a(n) = A214258(2n,n).

Conjecture: a(n) ~ n * 2^(n-3). - Vaclav Kotesovec, Jan 07 2019

A214259 Number of compositions of n where the difference between largest and smallest parts equals one.

Original entry on oeis.org

0, 0, 2, 3, 9, 11, 25, 35, 60, 96, 157, 241, 401, 637, 1019, 1639, 2651, 4258, 6870, 11075, 17891, 28895, 46678, 75412, 121915, 197109, 318724, 515414, 833590, 1348301, 2181020, 3528138, 5707564, 9233625, 14938477, 24168522, 39102322, 63264680, 102358836

Offset: 1

Alois P. Heinz, Jul 08 2012

nonn

			a(3) = 2: [2,1], [1,2].
a(4) = 3: [2,1,1], [1,2,1], [1,1,2].
a(5) = 9: [3,2], [2,3], [2,2,1], [2,1,2], [1,2,2], [2,1,1,1], [1,2,1,1], [1,1,2,1], [1,1,1,2].
a(6) = 11: [2,2,1,1], [2,1,2,1], [2,1,1,2], [1,2,2,1], [1,2,1,2], [1,1,2,2], [2,1,1,1,1], [1,2,1,1,1], [1,1,2,1,1], [1,1,1,2,1], [1,1,1,1,2].

Alois P. Heinz, Table of n, a(n) for n = 1..4785

Column k=1 of A214258.

Cf. A000005, A001622, A072951.

with(numtheory):
a:= n-> add(binomial(t, n mod t), t=1..n) -tau(n):
seq(a(n), n=1..50);

a(n) = A072951(n) - A000005(n).

a(n) ~ phi^(n+1) / sqrt(5), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jan 07 2019

A323119 Number of compositions of n where the difference between largest and smallest parts equals two.

Original entry on oeis.org

0, 0, 2, 3, 12, 20, 49, 95, 188, 366, 714, 1347, 2565, 4851, 9121, 17082, 31960, 59637, 111065, 206485, 383411, 711109, 1317528, 2438838, 4510987, 8338099, 15403007, 28439107, 52483921, 96819153, 178542957, 329146111, 606618210, 1117730709, 2059048379

Offset: 2

Alois P. Heinz, Jan 05 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..3779

Column k=2 of A214258.

a(n) ~ c * d^n, where d = A058265 = 1.8392867552141611325518525646532866... is the real root of the equation d^3 - d^2 - d - 1 = 0 and c = 0.618419922319392550945330438071061626105588310617942936855358363357952137... is the real root of the equation 44*c^3 - 44*c^2 + 12*c - 1 = 0. - Vaclav Kotesovec, Jan 07 2019

cp's OEIS Frontend

A214269 Number T(n,k) of compositions of n where the difference between largest and smallest parts equals k and adjacent parts are unequal; triangle T(n,k), n>=1, 0<=k

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A214247 Number A(n,k) of compositions of n where differences between neighboring parts are in {-k,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A214248 Number A(n,k) of compositions of n where differences between neighboring parts are in {-k,...,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A214249 Number A(n,k) of compositions of n where differences between neighboring parts are in {-k,...,k} \ {0}; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A214246 Number A(n,k) of compositions of n where differences between neighboring parts are in {-k,0,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A214257 Number A(n,k) of compositions of n where the difference between largest and smallest parts is <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A214268 Number A(n,k) of compositions of n where the difference between largest and smallest parts is <= k and adjacent parts are unequal; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A323111 Number of compositions of 2n where the difference between largest and smallest parts equals n.

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