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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.

%I A375100 #12 Jul 31 2024 09:11:07
%S A375100 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,
%T A375100 1,261,297,220,122,56,23,7,1,577,740,593,370,185,80,30,8,1,1276,1816,
%U A375100 1583,1068,589,274,111,38,9,1,2823,4408,4166,3008,1795,908,395,150,47,10,1
%N 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.
%F A375100 G.f.: A(x,y) = 1/(1 - Sum_{i>0} (x^i)/(1 - (y-1)*x^i - x)).
%e A375100 Triangle begins:
%e A375100       k=0    1    2    3    4   5   6   7   8
%e A375100  n=1:   1;
%e A375100  n=2:   2,   1;
%e A375100  n=3:   5,   2,   1;
%e A375100  n=4:  11,   6,   3,   1;
%e A375100  n=5:  24,  18,   8,   4,   1;
%e A375100  n=6:  53,  47,  26,  12,   5,  1;
%e A375100  n=7: 118, 118,  79,  38,  17,  6,  1;
%e A375100  n=8: 261, 297, 220, 122,  56, 23,  7,  1;
%e A375100  n=9: 577, 740, 593, 370, 185, 80, 30,  8,  1;
%e A375100  ...
%e A375100 Row n = 3 counts:
%e A375100 T(3,0) = 5: (1,2_2), (2_2,1), (3_1), (3_2), (3_3).
%e A375100 T(3,1) = 2: (1,2_1), (2_1,1).
%e A375100 T(3,2) = 1: (1,1,1).
%o A375100 (PARI)
%o A375100 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))))}
%o A375100 T_xy(10)
%Y A375100 Cf. A088305 (row sums), A242551 (column k=0).
%Y A375100 Cf. A003242, A372015, A374925.
%K A375100 nonn,easy,tabl
%O A375100 1,2
%A A375100 _John Tyler Rascoe_, Jul 29 2024