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A195011 Triangle read by rows: T(n,m) = number of vector partitions of n in S with crank m counted according to the weight w_1.

%I A195011 #27 Sep 29 2013 11:15:36
%S A195011 1,1,0,1,1,0,1,1,1,0,2,2,1,1,0,2,2,2,1,1,0,4,4,3,2,1,1,0,5,4,4,3,2,1,
%T A195011 1,0,7,7,6,5,3,2,1,1,0,10,9,8,6,5,3,2,1,1,0,13,13,11,10,7,5,3,2,1,1,0,
%U A195011 17,16,15,12,10,7,5,3,2,1,1
%N A195011 Triangle read by rows: T(n,m) = number of vector partitions of n in S with crank m counted according to the weight w_1.
%C A195011 This is the essential triangle in the square array of the Andrews-Garvan-Liang paper, see chapter 7 and page 3 for a precise definition.
%C A195011 Apparently columns (their positive terms) converge to A000041.
%H A195011 G. E. Andrews, <a href="http://www.math.psu.edu/vstein/alg/antheory/preprint/andrews/17.pdf">The number of smallest parts in the partitions of n</a>
%H A195011 G. E. Andrews, F. G. Garvan, and J. Liang, <a href="http://www.math.psu.edu/andrews/pdf/287.pdf">Combinatorial interpretation of congruences for the spt-function</a>
%H A195011 Frank Garvan, <a href="http://www.combinatorics.net/conf/A75/Slides/02_03_Garvan.pdf">Dyson's rank function and Andrews's SPT-function</a>, slide 17.
%e A195011 Triangle begins:
%e A195011 1;
%e A195011 1, 0;
%e A195011 1, 1, 0;
%e A195011 1, 1, 1, 0;
%e A195011 2, 2, 1, 1, 0;
%e A195011 2, 2, 2, 1, 1, 0;
%e A195011 4, 4, 3, 2, 1, 1, 0;
%e A195011 5, 4, 4, 3, 2, 1, 1, 0;
%e A195011 7, 7, 6, 5, 3, 2, 1, 1, 0;
%e A195011 10,9, 8, 6, 5, 3, 2, 1, 1, 0;
%Y A195011 Cf. A092269, A195012, A195820.
%K A195011 nonn,tabl
%O A195011 0,11
%A A195011 _Omar E. Pol_, Jan 10 2012