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A375100
Triangle read by rows: T(n,k) is the number of n-color compositions of n with k pairs of adjacent parts that are the same color.
0
1, 2, 1, 5, 2, 1, 11, 6, 3, 1, 24, 18, 8, 4, 1, 53, 47, 26, 12, 5, 1, 118, 118, 79, 38, 17, 6, 1, 261, 297, 220, 122, 56, 23, 7, 1, 577, 740, 593, 370, 185, 80, 30, 8, 1, 1276, 1816, 1583, 1068, 589, 274, 111, 38, 9, 1, 2823, 4408, 4166, 3008, 1795, 908, 395, 150, 47, 10, 1
OFFSET
1,2
FORMULA
G.f.: A(x,y) = 1/(1 - Sum_{i>0} (x^i)/(1 - (y-1)*x^i - x)).
EXAMPLE
Triangle begins:
k=0 1 2 3 4 5 6 7 8
n=1: 1;
n=2: 2, 1;
n=3: 5, 2, 1;
n=4: 11, 6, 3, 1;
n=5: 24, 18, 8, 4, 1;
n=6: 53, 47, 26, 12, 5, 1;
n=7: 118, 118, 79, 38, 17, 6, 1;
n=8: 261, 297, 220, 122, 56, 23, 7, 1;
n=9: 577, 740, 593, 370, 185, 80, 30, 8, 1;
...
Row n = 3 counts:
T(3,0) = 5: (1,2_2), (2_2,1), (3_1), (3_2), (3_3).
T(3,1) = 2: (1,2_1), (2_1,1).
T(3,2) = 1: (1,1,1).
PROG
(PARI)
T_xy(max_row) = {my(N=max_row+1, x='x+O('x^N), h= 1/(1-sum(i=1, N, x^i/(1-(x^i)*(y-1)-x)))); for(n=1, N-1, print(Vecrev(polcoeff(h, n))))}
T_xy(10)
CROSSREFS
Cf. A088305 (row sums), A242551 (column k=0).
Sequence in context: A104766 A361681 A105084 * A126125 A221876 A128514
KEYWORD
nonn,easy,tabl
AUTHOR
John Tyler Rascoe, Jul 29 2024
STATUS
approved