A327549 - OEIS

%I #17 Dec 17 2020 14:47:43

%S 1,0,1,0,2,1,0,4,2,1,0,8,8,2,1,0,16,16,8,2,1,0,32,48,24,8,2,1,0,64,96,

%T 64,24,8,2,1,0,128,256,160,80,24,8,2,1,0,256,512,448,192,80,24,8,2,1,

%U 0,512,1280,1024,576,224,80,24,8,2,1

%N Number T(n,k) of compositions of partitions of n with exactly k compositions; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%H Alois P. Heinz, <a href="/A327549/b327549.txt">Rows n = 0..200, flattened</a>

%F Sum_{k=1..n} k * T(n,k) = A327548(n).

%e T(3,1) = 4: 3, 21, 12, 111.

%e T(3,2) = 2: 2|1, 11|1.

%e T(3,3) = 1: 1|1|1.

%e Triangle T(n,k) begins:

%e   1;

%e   0,   1;

%e   0,   2,    1;

%e   0,   4,    2,    1;

%e   0,   8,    8,    2,   1;

%e   0,  16,   16,    8,   2,   1;

%e   0,  32,   48,   24,   8,   2,  1;

%e   0,  64,   96,   64,  24,   8,  2,  1;

%e   0, 128,  256,  160,  80,  24,  8,  2, 1;

%e   0, 256,  512,  448, 192,  80, 24,  8, 2, 1;

%e   0, 512, 1280, 1024, 576, 224, 80, 24, 8, 2, 1;

%e   ...

%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p       b(n, i-1)+expand(2^(i-1)*x*b(n-i, min(n-i, i)))))

%p     end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):

%p seq(T(n), n=0..12);

%t b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + 2^(i-1) x b[n-i, Min[n-i, i]]]];

%t T[n_] := CoefficientList[b[n, n], x];

%t T /@ Range[0, 12] // Flatten (* _Jean-François Alcover_, Dec 17 2020, after _Alois P. Heinz_ *)

%Y Columns k=0-2 give: A000007, A011782 (for n>0), A134353(n-2) (for n>1).

%Y Row sums give A075900.

%Y T(2n,n) gives A327550.

%Y Cf. A060642, A327548.

%K nonn,tabl

%O 0,5

%A _Alois P. Heinz_, Sep 16 2019