cp's OEIS Frontend

A176572 Count the ones in the binary representation of the partitions of n; then add vertically yielding a triangular array T(n,k).

%I A176572 #12 Feb 25 2025 09:00:20
%S A176572 1,2,1,3,1,2,5,1,3,3,7,1,3,4,5,11,1,4,5,7,7,15,1,4,6,8,9,11,22,1,5,7,
%T A176572 11,10,15,15,30,1,5,9,12,13,17,19,22,42,1,6,10,16,15,22,21,29,30,56,1,
%U A176572 6,12,18,19,25,26,32,38,42,77,1,7,14,23,22,33,29,41,42,54,56,101,1,7,16,26,28,37,37,45,52,59,70,77,135,1,8,18,32,33,47,42,58,57,74,76,98,101
%N A176572 Count the ones in the binary representation of the partitions of n; then add vertically yielding a triangular array T(n,k).
%C A176572 Each partition of n is converted into a binary representation with n bits by concatenating binary strings formed from each of the parts p_1(n)+p_2(n)+p_3(n)+..., p_1(n)>=p_2(n)>=p_3(n), larger parts contributing the higher significant bits, the individual part p_i(n) represented by a 1 followed by p_i(n)-1 zeros.
%C A176572 These A000041(n) binary representations are stacked, and the total count of 1's in each column is the n-th row of the triangle.
%H A176572 John Tyler Rascoe, <a href="/A176572/b176572.txt">Rows n = 1..60, flattened</a>
%F A176572 Sum_{k=0..n-1} 2^(n-k)*T(n,k) = A173871(n).
%e A176572 Consider the seven partitions of Five, 5=(10000), 41=(1000)(1), 32=(100)(10), 311=(100)(1)(1), 221=(10)(10)(1), 2111=(10)(1)(1)(1) and 11111=(1)(1)(1)(1)(1),
%e A176572 the corresponding seven concatenated binary representations are
%e A176572  1 0 0 0 0
%e A176572  1 0 0 0 1
%e A176572  1 0 0 1 0
%e A176572  1 0 0 1 1
%e A176572  1 0 1 0 1
%e A176572  1 0 1 1 1
%e A176572  1 1 1 1 1
%e A176572 summing by column yields
%e A176572 7 1 3 4 5 the fifth row of the table.
%e A176572 Triangle begins:
%e A176572   1;
%e A176572   2,1;
%e A176572   3,1,2;
%e A176572   5,1,3,3;
%e A176572   7,1,3,4,5;
%e A176572  11,1,4,5,7,7;
%e A176572  15,1,4,6,8,9,11;
%e A176572  ...
%p A176572 A176572row := proc(n) L := array(1..n,[seq(0,i=1..n)]) ; for pi in combinat[partition](n) do p := sort(pi) ;       p2 := [] ; for i from 1 to nops(p) do p2 := [op(p2),op(convert(2^(op(i,p)-1),base,2))] ; end do: for i from 1 to n do L[i] := L[i]+ op(n-i+1,p2) ; end do: end do: L ; end proc:
%p A176572 for n from 1 to 14 do A176572row(n) ; print(%) ; end do:
%o A176572 (Python)
%o A176572 from sympy .utilities.iterables import ordered_partitions
%o A176572 def A176572(row_n):
%o A176572     p = [i for i in ordered_partitions(row_n)]
%o A176572     A = [[j for k in i[::-1] for j in ([1]+[0]*(k-1))] for i in p]
%o A176572     return [sum(A[i][j] for i in range(len(p))) for j in range(row_n)] # _John Tyler Rascoe_, Feb 24 2025
%Y A176572 Cf. A006128 (row sums), A114994, A130321, A173871.
%K A176572 easy,nonn,tabl,base
%O A176572 1,2
%A A176572 _Alford Arnold_, Apr 22 2010
%E A176572 Edited by _John Tyler Rascoe_, Feb 24 2025