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A235998 Triangle read by rows: T(n,k) is the number of compositions of n having k distinct parts (n>=1, 1<=k<=floor((sqrt(1+8*n)-1)/2)).

%I A235998 #54 Oct 18 2021 08:18:36
%S A235998 1,2,2,2,3,5,2,14,4,22,6,2,44,18,4,68,56,3,107,146,4,172,312,24,2,261,
%T A235998 677,84,6,396,1358,288,2,606,2666,822,4,950,5012,2226,4,1414,9542,
%U A235998 5304,120,5,2238,17531,12514,480,2,3418,32412,27904,1800,6,5411,58995,61080,5580
%N A235998 Triangle read by rows: T(n,k) is the number of compositions of n having k distinct parts (n>=1, 1<=k<=floor((sqrt(1+8*n)-1)/2)).
%C A235998 Row n has length A003056(n) hence the first element of column k is in row A000217(k).
%C A235998 The equivalent sequence for partitions is A116608.
%C A235998 For the number of compositions of n see A011782.
%C A235998 For the connection to overcompositions see A235999.
%C A235998 Row sums give A011782(n), n >= 1.
%C A235998 First column is A000005, second column is A131661.
%C A235998 T(k*(k+1)/2,k) = T(A000217(k),k) = A000142(k) = k!. - _Alois P. Heinz_, Jan 20 2014
%H A235998 Alois P. Heinz, <a href="/A235998/b235998.txt">Rows n = 1..500, flattened</a>
%e A235998 Triangle begins:
%e A235998   1;
%e A235998   2;
%e A235998   2,    2;
%e A235998   3,    5;
%e A235998   2,   14;
%e A235998   4,   22,     6;
%e A235998   2,   44,    18;
%e A235998   4,   68,    56;
%e A235998   3,  107,   146;
%e A235998   4,  172,   312,    24;
%e A235998   2,  261,   677,    84;
%e A235998   6,  396,  1358,   288;
%e A235998   2,  606,  2666,   822;
%e A235998   4,  950,  5012,  2226;
%e A235998   4, 1414,  9542,  5304,  120;
%e A235998   5, 2238, 17531, 12514,  480;
%e A235998   2, 3418, 32412, 27904, 1800;
%e A235998   6, 5411, 58995, 61080, 5580;
%e A235998   ...
%p A235998 b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
%p A235998       expand(add(b(n-i*j, i-1, p+j)/j!*`if`(j=0, 1, x), j=0..n/i))))
%p A235998     end:
%p A235998 T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2, 0)):
%p A235998 seq(T(n), n=1..25); # _Alois P. Heinz_, Jan 20 2014, revised May 25 2014
%t A235998 b[n_, i_, p_] := b[n, i, p] = If[n==0, p!, If[i<1, 0, Sum[b[n-i*j, i-1, p+ j]/j!*If[j==0, 1, x], {j, 0, n/i}]]]; T[n_] := Function[p, Table[ Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][b[n, n, 0]]; Table[T[n], {n, 1, 25}] // Flatten (* _Jean-François Alcover_, Dec 10 2015, after _Alois P. Heinz_ *)
%Y A235998 Cf. A003056, A116608, A235790, A235999, A236002.
%K A235998 nonn,tabf,nice,look
%O A235998 1,2
%A A235998 _Omar E. Pol_, Jan 19 2014
%E A235998 More terms from _Alois P. Heinz_, Jan 19 2014