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Triangle read by rows in which row n lists the partition numbers of the first n positive integers.
3

%I #13 Apr 26 2012 11:22:01

%S 1,1,2,1,2,3,1,2,3,5,1,2,3,5,7,1,2,3,5,7,11,1,2,3,5,7,11,15,1,2,3,5,7,

%T 11,15,22,1,2,3,5,7,11,15,22,30,1,2,3,5,7,11,15,22,30,42,1,2,3,5,7,11,

%U 15,22,30,42,56,1,2,3,5,7,11,15,22,30,42,56,77,1,2,3,5,7,11,15,22,30,42

%N Triangle read by rows in which row n lists the partition numbers of the first n positive integers.

%C Sequence of row lengths of array A092078.

%C Interpreted as a sequence a(n) = p(m) = A000041(m) (number of partitions of m) with m := n-L, where L is the largest number from A000217 (triangular numbers) which is less than n.

%C Row sums give A026905. - _Omar E. Pol_, Apr 23 2012

%F T(n,k) = A000041(k), k >= 1. - _Omar E. Pol_, Apr 23 2012

%e a(17)= p(2)= 2 because 17 = (15=A000217(5)) + 2.

%e a(10)= p(4)= 5 because 10 = (6=A000217(3)) + 4 (note that T < n, not <=n, is asked for).

%e Contribution from _Omar E. Pol_, Apr 24 2012 (Start):

%e Triangle begins:

%e 1;

%e 1, 2;

%e 1, 2, 3;

%e 1, 2, 3, 5;

%e 1, 2, 3, 5, 7;

%e 1, 2, 3, 5, 7, 11;

%e 1, 2, 3, 5, 7, 11, 15;

%e 1, 2, 3, 5, 7, 11, 15, 22;

%e 1, 2, 3, 5, 7, 11, 15, 22, 30;

%e 1, 2, 3, 5, 7, 11, 15, 22, 30, 42;

%e (End)

%Y Cf. A000041.

%K nonn,tabl,easy

%O 1,3

%A _Wolfdieter Lang_, Mar 19 2004

%E Better definition from _Omar E. Pol_, Apr 24 2012