%I A053632
%S A053632 1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,2,2,2,2,2,1,1,1,1,1,1,2,2,3,3,3,3,3,
%T A053632 3,2,2,1,1,1,1,1,1,2,2,3,4,4,4,5,5,5,5,4,4,4,3,2,2,1,1,1,1,1,1,2,2,3,4,
%U A053632 5,5,6,7,7,8,8,8,8,8,7,7,6,5,5,4,3,2,2,1,1,1,1,1,1,2,2,3,4
%N A053632 Array giving coefficients in expansion of Product_{k=1..n} (1+x^k).
%C A053632 Or, triangle T(n,k) read by rows, giving number of subsets of {1,2,...,
               n} with sum k. - Roger CUCULIERE (cuculier(AT)imaginet.fr), Nov 19 
               2000
%C A053632 Row n consists of A000124[n] terms. These are also the successive vectors 
               (their nonzero elements) when one starts with the infinite vector 
               (of zeros) with 1 inserted somewhere and then shifts it one step 
               (right or left) and adds to the original, then shifts the result 
               two steps and adds, three steps and adds, et cetera.
%C A053632 T(n,k) = number of partitions of k into distinct parts <= n. Triangle 
               of distribution of Wilcoxon's signed rank statistic. - Mitch Harris 
               (Harris.Mitchell(AT)mgh.harvard.edu), Mar 23 2006
%C A053632 T(n,k)=number of binary words of length n in which the sum of the positions 
               of the 0's is k. Example: T(4,5)=2 because we have 0110 (sum of the 
               positions of the 0's is 1+4=5) and 1001 (sum of the positions of 
               the 0's is 2+3=5). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 
               23 2006
%D A053632 Wilcoxon, F., Individual Comparisons by Ranking Methods, Biometrics Bulletin, 
               v. 1, no. 6 (1945), p. 80-83.
%H A053632 S. R. Finch, <a href="http://algo.inria.fr/bsolve/">Signum equations 
               and extremal coefficients</a>.
%F A053632 T(n,k) = T(n-1, k) + T(n-1, k-n), T(0,0)=1, T(0,k) = 0, T(n,k) = 0 if 
               k < 0 or k > (n+1 choose 2). g.f. = (1+x)(1+x^2)...(1+x^n). - Mitch 
               Harris (Harris.Mitchell(AT)mgh.harvard.edu), Mar 23 2006
%e A053632 1; 1,1; 1,1,1,1; 1,1,1,2,1,1,1; 1,1,1,2,2,2,2,2,1,1,1; 1,1,1,2,2,3,3,
               3,3,3,3,2,2,1,1,1; ...
%p A053632 with(gfun,seriestolist); map(op,[seq(seriestolist(series(mul(1+(z^i),
               i=1..n),z,binomial(n+1,2)+1)), n=0..10)]);
%Y A053632 Cf. A053633, A068009. The rows interpreted as binary numbers: A068052, 
               A068053. The rows converge towards A000009.
%Y A053632 Sequence in context: A031262 A047072 A053258 this_sequence A124060 A140194 
               A159923
%Y A053632 Adjacent sequences: A053629 A053630 A053631 this_sequence A053633 A053634 
               A053635
%K A053632 tabf,nonn,easy,nice
%O A053632 0,11
%A A053632 N. J. A. Sloane (njas(AT)research.att.com), Mar 22 2000
%E A053632 Comments and Maple code from Antti Karttunen, Feb 13 2002