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
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).
-
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
-
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)
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
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).
-
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)
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
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).
-
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
-
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)
Showing 1-3 of 3 results.
Comments