cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A380822 Triangle read by rows: T(n,k) is the number of compositions of n with k pairs of equal adjacent parts and all parts in standard order.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 3, 1, 0, 1, 2, 1, 4, 1, 0, 1, 3, 3, 3, 5, 1, 0, 1, 2, 10, 5, 4, 6, 1, 0, 1, 5, 9, 17, 8, 5, 7, 1, 0, 1, 8, 16, 22, 26, 10, 6, 8, 1, 0, 1, 10, 35, 33, 37, 37, 12, 7, 9, 1, 0, 1, 19, 44, 80, 59, 56, 48, 14, 8, 10, 1, 0, 1
Offset: 1

Views

Author

John Tyler Rascoe, May 08 2025

Keywords

Comments

A composition with parts in standard order satisfies the condition that for any part p > 1, the part p - 1 has already appeared. All compositions of this kind have first part 1.

Examples

			Triangle begins:
     k=0   1   2   3   4  5  6  7  8  9
 n=1  [1],
 n=2  [0,  1],
 n=3  [1,  0,  1],
 n=4  [1,  1,  0,  1],
 n=5  [0,  3,  1,  0,  1],
 n=6  [2,  1,  4,  1,  0, 1],
 n=7  [3,  3,  3,  5,  1, 0, 1],
 n=8  [2, 10,  5,  4,  6, 1, 0, 1],
 n=9  [5,  9, 17,  8,  5, 7, 1, 0, 1],
 n=10 [8, 16, 22, 26, 10, 6, 8, 1, 0, 1],
...
Row n = 6 counts:
 T(6,0) = 2: (1,2,1,2), (1,2,3).
 T(6,1) = 1: (1,2,2,1).
 T(6,2) = 4: (1,1,1,2,1), (1,1,2,1,1), (1,1,2,2), (1,2,1,1,1).
 T(6,3) = 1: (1,1,1,1,2).
 T(6,4) = 0: .
 T(6,5) = 1: (1,1,1,1,1,1).
		

Crossrefs

Programs

  • Maple
    b:= proc(n, l, i) option remember; expand(`if`(n=0, 1, add(
          `if`(j=l, x, 1)*b(n-j, j, max(i, j)), j=1..min(n, i+1))))
        end:
    T:= (n, k)-> coeff(b(n, 0$2), x, k):
    seq(seq(T(n, k), k=0..n-1), n=1..12);  # Alois P. Heinz, May 08 2025
  • PARI
    G(k,N) = {my(x='x+O('x^N)); if(k<1,1, G(k-1,N) * x^k * (1 + (1/(1 - sum(j=1,k, x^j/(1-(z-1)*x^j)))) * sum(j=1,k, z^(if(j==k,1,0)) * x^j/(1-(z-1)*x^j))))}
    T_xz(max_row) = {my(N = max_row+1, x='x+O('x^N), h = sum(i=1,N/2+1, G(i,N))); vector(N-1, n, Vecrev(polcoeff(h, n)))}
    T_xz(10)

Formula

G.f.: Sum_{i>0} G(i) where G(k) = G(k-1) * x*k * (1 + 1/(1 - Sum_{j=1..k} ( x^j/(1 - (z-1)*x^j) )) * Sum_{j=1..k} ( z^[j=k] * x^j/(1 - (z-1)*x^j) )) and G(0) = 1.

A383713 Triangle read by rows: T(n,k) is the number of compositions of n with k parts all in standard order.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 3, 1, 0, 0, 0, 1, 3, 4, 1, 0, 0, 0, 0, 4, 6, 5, 1, 0, 0, 0, 0, 2, 10, 10, 6, 1, 0, 0, 0, 0, 1, 9, 20, 15, 7, 1, 0, 0, 0, 0, 1, 7, 25, 35, 21, 8, 1, 0, 0, 0, 0, 0, 7, 26, 55, 56, 28, 9, 1, 0, 0, 0, 0, 0, 4, 29, 71, 105, 84, 36, 10, 1
Offset: 0

Views

Author

John Tyler Rascoe, May 06 2025

Keywords

Comments

A composition with parts in standard order satisfies the condition that for any part p > 1, the part p - 1 has already appeared. All compositions of this kind have first part 1.

Examples

			Triangle begins:
     k=0  1  2  3  4   5   6   7   8  9 10
 n=0  [1],
 n=1  [0, 1],
 n=2  [0, 0, 1],
 n=3  [0, 0, 1, 1],
 n=4  [0, 0, 0, 2, 1],
 n=5  [0, 0, 0, 1, 3,  1],
 n=6  [0, 0, 0, 1, 3,  4,  1],
 n=7  [0, 0, 0, 0, 4,  6,  5,  1],
 n=8  [0, 0, 0, 0, 2, 10, 10,  6,  1],
 n=9  [0, 0, 0, 0, 1,  9, 20, 15,  7, 1],
 n=10 [0, 0, 0, 0, 1,  7, 25, 35, 21, 8, 1],
 ...
Row n = 6 counts:
 T(6,3) = 1: (1,2,3).
 T(6,4) = 3: (1,1,2,2), (1,2,1,2), (1,2,2,1).
 T(6,5) = 4: (1,1,1,1,2), (1,1,1,2,1), (1,1,2,1,1), (1,2,1,1,1).
 T(6,6) = 1: (1,1,1,1,1,1).
		

Crossrefs

Cf. A000110 (column sums), A047998, A107429, A126347 (triangle transposed with no zeros), A278984, A383253 (row sums).

Programs

  • PARI
    T_xy(max_row) = {my(N = max_row+1, x='x+O('x^N), h = 1 + sum(i=1,1+(N/2), y^i * x^(i*(i+1)/2)/prod(j=1,i, 1 - y*(x-x^(j+1))/(1-x)))); vector(N, n, Vecrev(polcoeff(h, n-1)))}
    T_xy(10)

Formula

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

A383751 Number of Carlitz compositions of n with parts in standard order.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 2, 3, 2, 5, 8, 10, 19, 31, 44, 73, 123, 193, 315, 524, 847, 1392, 2317, 3810, 6303, 10506, 17451, 29066, 48603, 81223, 135965, 228153, 383014, 643756, 1083693, 1825640, 3078574, 5197246, 8780823, 14847669, 25128385, 42558687, 72131730, 122343844
Offset: 0

Views

Author

John Tyler Rascoe, May 08 2025

Keywords

Comments

A composition with parts in standard order satisfies the condition that for any part p > 1, the part p - 1 has already appeared. All compositions of this kind have first part 1.

Examples

			a(9) = 5 counts: (1,2,1,2,1,2), (1,2,1,2,3), (1,2,1,3,2), (1,2,3,1,2), (1,2,3,2,1).
		

Crossrefs

Programs

  • Maple
    b:= proc(n, l, i) option remember; `if`(n=0, 1, add(
          `if`(j=l, 0, b(n-j, j, max(i, j))), j=1..min(n, i+1)))
        end:
    a:= n-> b(n, 0$2):
    seq(a(n), n=0..43);  # Alois P. Heinz, May 09 2025
  • PARI
    G(k,N) = {my(x='x+O('x^N)); if(k<1,1, G(k-1,N) * x^k * (1 + (1/(1 - sum(j=1,k, x^j/(1+x^j)))) * sum(j=1,k-1, x^j/(1+x^j))))}
    A_x(N) = {my(x='x+O('x^N)); Vec(sum(i=0,N/2+1, G(i,N+1)))}
    A_x(50)

Formula

G.f.: Sum_{i>=0} G(i) where G(k) = G(k-1) * x*k * (1 + 1/(1 - Sum_{j=1..k} ( x^j/(1+x^j) )) * Sum_{j=1..k-1} ( x^j/(1+x^j) )) and G(0) = 1.
Showing 1-3 of 3 results.