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A089178 Triangle T(n,k) (n >= 0, 0 <= k <= 1+log_2(floor(n+1))) read by rows: row 0 = {1}, row 1 = {1 1}; for n >=2, row n = row n-1 + (row floor((n-1)/2) shifted one place right).

%I A089178 #21 Dec 31 2023 16:50:55
%S A089178 1,1,1,1,2,1,3,1,1,4,2,1,5,4,1,6,6,1,7,9,1,1,8,12,2,1,9,16,4,1,10,20,
%T A089178 6,1,11,25,10,1,12,30,14,1,13,36,20,1,14,42,26,1,15,49,35,1,1,16,56,
%U A089178 44,2,1,17,64,56,4,1,18,72,68,6,1,19,81,84,10,1,20,90,100,14,1,21,100,120,20
%N A089178 Triangle T(n,k) (n >= 0, 0 <= k <= 1+log_2(floor(n+1))) read by rows: row 0 = {1}, row 1 = {1 1}; for n >=2, row n = row n-1 + (row floor((n-1)/2) shifted one place right).
%H A089178 Alois P. Heinz, <a href="/A089178/b089178.txt">Rows n = 0..1094, flattened</a>
%H A089178 N. J. A. Sloane and J. A. Sellers, <a href="http://arXiv.org/abs/math.CO/0312418">On non-squashing partitions</a>, Discrete Math., 294 (2005), 259-274.
%F A089178 G.f.: (1/(1-x))*(1+Sum(y^(k+1)*x^(2^(k+1)-1)/Product(1-x^(2^j), j=0..k), k=0..infinity)).
%e A089178 Triangle begins:
%e A089178 1;
%e A089178 1, 1;
%e A089178 1, 2;
%e A089178 1, 3, 1;
%e A089178 1, 4, 2;
%e A089178 1, 5, 4;
%e A089178 1, 6, 6;
%e A089178 1, 7, 9, 1;
%p A089178 T:= proc(n) option remember; `if`(n=0, 1,
%p A089178       zip((x, y)-> x+y, [T(n-1)], [0, T(floor((n-1)/2))], 0)[])
%p A089178     end:
%p A089178 seq(T(n), n=0..25);  # _Alois P. Heinz_, Apr 01 2012
%t A089178 row[0] = {1}; row[n_] := row[n] = PadRight[{row[n-1], Join[{0}, row[Floor[(n-1)/2]]]}] // Total; Table[row[n], {n, 0, 25}] // Flatten (* _Jean-François Alcover_, Nov 27 2014 *)
%Y A089178 Also obtained by dividing rows of A089177 by "1 1".
%Y A089178 Row sums give A033485.
%K A089178 nonn,tabf,easy
%O A089178 0,5
%A A089178 _N. J. A. Sloane_, Dec 08 2003
%E A089178 More terms from _Vladeta Jovovic_, Dec 10 2003