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%I #25 Dec 31 2023 16:51:36
%S 1,1,1,1,2,1,1,3,2,1,4,4,1,1,5,6,2,1,6,9,4,1,7,12,6,1,8,16,10,1,1,9,
%T 20,14,2,1,10,25,20,4,1,11,30,26,6,1,12,36,35,10,1,13,42,44,14,1,14,
%U 49,56,20,1,15,56,68,26,1,16,64,84,36,1,1,17,72,100,46,2,1,18,81,120,60,4,1
%N Triangle read by rows: T(n,k) (n >= 0, 0 <= k <= 1+log_2(floor(n))) giving number of non-squashing partitions of n into k parts.
%C T(n,k) = A181322(n,k) - A181322(n,k-1) for n>0. - _Alois P. Heinz_, Jan 25 2014
%H Alois P. Heinz, <a href="/A089177/b089177.txt">Rows n = 0..1002, flattened</a>
%H 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 Row 0 = {1}, row 1 = {1 1}; for n >=2, row n = row n-1 + (row floor(n/2) shifted one place right).
%F G.f. for column k (k >= 2): x^(2^(k-2))/((1-x)*Product_j=1..k-2} (1-x^(2^j))).
%e Triangle begins:
%e 1;
%e 1, 1;
%e 1, 2, 1;
%e 1, 3, 2;
%e 1, 4, 4, 1;
%e 1, 5, 6, 2;
%e 1, 6, 9, 4;
%e 1, 7, 12, 6;
%e 1, 8, 16, 10, 1;
%p T:= proc(n) option remember;
%p `if`(n=0, 1, zip((x, y)-> x+y, [T(n-1)], [0, T(floor(n/2))], 0)[])
%p end:
%p seq(T(n), n=0..25); # _Alois P. Heinz_, Apr 01 2012
%t row[0] = {1}; row[1] = {1, 1}; row[n_] := row[n] = Plus @@ PadRight[ {row[n-1], Join[{0}, row[Floor[n/2]]]} ]; Table[row[n], {n, 0, 25}] // Flatten (* _Jean-François Alcover_, Jan 31 2014 *)
%Y Cf. A089178. Columns give A002620, A008804, A088932, A088954. Row sums give A000123.
%K nonn,tabf,look,easy
%O 0,5
%A _N. J. A. Sloane_, Dec 08 2003
%E More terms from _Alford Arnold_, May 22 2004
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