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A168018 Triangle read by rows in which row n lists the number of partitions of n into parts divisible by d, where d is a divisor of n.

%I A168018 #12 Apr 02 2017 03:20:01
%S A168018 1,2,1,3,1,5,2,1,7,1,11,3,2,1,15,1,22,5,2,1,30,3,1,42,7,2,1,56,1,77,
%T A168018 11,5,3,2,1,101,1,135,15,2,1,176,7,3,1,231,22,5,2,1,297,1,385,30,11,3,
%U A168018 2,1,490,1,627,42,7,5,2,1,792,15,3,1,1002,56,2,1,1255,1,1575,77,22,11,5,3,2
%N A168018 Triangle read by rows in which row n lists the number of partitions of n into parts divisible by d, where d is a divisor of n.
%C A168018 Positive values of triangle A168021.
%C A168018 Note that column 1 lists the numbers of partitions A000041(n).
%C A168018 Row n has A000005(n) terms.
%C A168018 Also, it appears that row n lists the partition numbers of the divisors of n, in decreasing order. [_Omar E. Pol_, Nov 23 2009]
%H A168018 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa3dt.jpg">Illustration of the partitions of n, for n = 1 .. 9</a>
%e A168018 For example:
%e A168018 Consider row 8: (22, 5, 2, 1). The divisors of 8 are 1, 2, 4, 8 (see A027750). Also, there are 22 partitions of 8 into parts divisible by 1 (A000041(8)=22); 5 partitions of 8 into parts divisible by 2: {(8),(6+2),(4+4),(4+2+2),(2+2+2+2)}; 2 partitions of 8 into parts divisible by 4: {(8),(4+4)}; and 1 partition of 8 into parts divisible by 8. Then row 8 is formed by 22, 5, 2, 1.
%e A168018 Triangle begins:
%e A168018 1;
%e A168018 2, 1;
%e A168018 3, 1;
%e A168018 5, 2, 1;
%e A168018 7, 1;
%e A168018 11, 3, 2, 1;
%e A168018 15, 1;
%e A168018 22, 5, 2, 1;
%e A168018 30, 3, 1;
%e A168018 42, 7, 2, 1;
%e A168018 56, 1;
%e A168018 77, 11, 5, 3, 2, 1;
%p A168018 A168018 := proc(n) local dvs,p,i,d,a,pp,divs,par; dvs := sort(convert(numtheory[divisors](n),list)) ; p := combinat[partition](n) ; for i from 1 to nops(dvs) do d := op(i,dvs) ; a := 0 ; for pp in p do divs := true; for par in pp do if par mod d <> 0 then divs := false; end if; end do ; if divs then a := a+1 ; end if; end do ; printf("%d,",a) ; end do ; end proc: for n from 1 to 40 do A168018(n) ; end do : # _R. J. Mathar_, Feb 05 2010
%Y A168018 Row sums give A047968.
%Y A168018 Cf. A000005, A000041, A027750, A135010, A138121, A168016, A168017, A168019, A168020, A168021.
%K A168018 nonn,tabf
%O A168018 1,2
%A A168018 _Omar E. Pol_, Nov 22 2009
%E A168018 Terms beyond row 12 from _R. J. Mathar_, Feb 05 2010