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A327549 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.

%I A327549 #17 Dec 17 2020 14:47:43
%S A327549 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 A327549 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 A327549 0,512,1280,1024,576,224,80,24,8,2,1
%N A327549 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 A327549 Alois P. Heinz, <a href="/A327549/b327549.txt">Rows n = 0..200, flattened</a>
%F A327549 Sum_{k=1..n} k * T(n,k) = A327548(n).
%e A327549 T(3,1) = 4: 3, 21, 12, 111.
%e A327549 T(3,2) = 2: 2|1, 11|1.
%e A327549 T(3,3) = 1: 1|1|1.
%e A327549 Triangle T(n,k) begins:
%e A327549   1;
%e A327549   0,   1;
%e A327549   0,   2,    1;
%e A327549   0,   4,    2,    1;
%e A327549   0,   8,    8,    2,   1;
%e A327549   0,  16,   16,    8,   2,   1;
%e A327549   0,  32,   48,   24,   8,   2,  1;
%e A327549   0,  64,   96,   64,  24,   8,  2,  1;
%e A327549   0, 128,  256,  160,  80,  24,  8,  2, 1;
%e A327549   0, 256,  512,  448, 192,  80, 24,  8, 2, 1;
%e A327549   0, 512, 1280, 1024, 576, 224, 80, 24, 8, 2, 1;
%e A327549   ...
%p A327549 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A327549       b(n, i-1)+expand(2^(i-1)*x*b(n-i, min(n-i, i)))))
%p A327549     end:
%p A327549 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
%p A327549 seq(T(n), n=0..12);
%t A327549 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 A327549 T[n_] := CoefficientList[b[n, n], x];
%t A327549 T /@ Range[0, 12] // Flatten (* _Jean-François Alcover_, Dec 17 2020, after _Alois P. Heinz_ *)
%Y A327549 Columns k=0-2 give: A000007, A011782 (for n>0), A134353(n-2) (for n>1).
%Y A327549 Row sums give A075900.
%Y A327549 T(2n,n) gives A327550.
%Y A327549 Cf. A060642, A327548.
%K A327549 nonn,tabl
%O A327549 0,5
%A A327549 _Alois P. Heinz_, Sep 16 2019