A375100 - OEIS

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.

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

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OFFSET

1,2

LINKS

Table of n, a(n) for n=1..66.

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).

Cf. A003242, A372015, A374925.

Sequence in context: A104766 A361681 A105084 * A126125 A221876 A128514

Adjacent sequences: A375097 A375098 A375099 * A375101 A375102 A375103

KEYWORD

nonn,easy,tabl

AUTHOR

John Tyler Rascoe, Jul 29 2024

STATUS

approved