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A089177 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.

%I A089177 #38 Jul 03 2025 18:39:18
%S A089177 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 A089177 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 A089177 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 A089177 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 A089177 T(n,k) = A181322(n,k) - A181322(n,k-1) for n>0. - _Alois P. Heinz_, Jan 25 2014
%H A089177 Alois P. Heinz, <a href="/A089177/b089177.txt">Rows n = 0..1002, flattened</a>
%H A089177 N. J. A. Sloane and J. A. Sellers, <a href="https://arxiv.org/abs/math/0312418">On non-squashing partitions</a>, arXiv:math/0312418 [math.CO], 2003; Discrete Math., 294 (2005), 259-274.
%F A089177 Row 0 = {1}, row 1 = {1 1}; for n >=2, row n = row n-1 + (row floor(n/2) shifted one place right).
%F A089177 G.f. for column k (k >= 2): x^(2^(k-2))/((1-x)*Product_{j=0..k-2} (1-x^(2^j))). [corrected by _Jason Yuen_, Jan 12 2025]
%F A089177 Conjecture: let R(n,x) be the n-th reversed row polynomial, then R(n,x) = Sum_{k=0..A000523(A053645(n)) + 1} T(A053645(n),k)*R(2^(A000523(n)-k),x) for n > 0, n != 2^m with R(0,x) = 1 where R(2^m,x) is the (m+1)-th row polynomial of A078121. - _Mikhail Kurkov_, Jun 28 2025
%e A089177 Triangle begins:
%e A089177   1;
%e A089177   1, 1;
%e A089177   1, 2,  1;
%e A089177   1, 3,  2;
%e A089177   1, 4,  4,  1;
%e A089177   1, 5,  6,  2;
%e A089177   1, 6,  9,  4;
%e A089177   1, 7, 12,  6;
%e A089177   1, 8, 16, 10,  1;
%p A089177 T:= proc(n) option remember;
%p A089177      `if`(n=0, 1, zip((x, y)-> x+y, [T(n-1)], [0, T(floor(n/2))], 0)[])
%p A089177     end:
%p A089177 seq(T(n), n=0..25);  # _Alois P. Heinz_, Apr 01 2012
%t A089177 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 A089177 Cf. A078121, A089178. Columns give A002620, A008804, A088932, A088954. Row sums give A000123.
%K A089177 nonn,tabf,look,easy
%O A089177 0,5
%A A089177 _N. J. A. Sloane_, Dec 08 2003
%E A089177 More terms from _Alford Arnold_, May 22 2004