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%I #2 Mar 30 2012 18:37:17
%S 1,2,6,33,171,1073,7749,58846,489993,4345078,40693236,403876159,
%T 4192624215,45577188311,516108618746,6073914844170,74083651509143,
%U 934440647979907,12164742571864976,163157161537315276
%N Main diagonal of table A162424.
%F a(n) = Sum_{m=n(n+1)/2..n(n+1)/2+n} [x^m] S(x)^(n+1) for n>=0 where S(x) = Sum_{n>=0} x^((n+1)(n+2)/2-1).
%e Let coefficients in powers of the series:
%e S = 1 + x^2 + x^5 + x^9 + x^14 + x^20 + x^27 + x^35 + x^44 +...
%e form the following sequences:
%e S^1: [(1),(0,1),(0,0,1),(0,0,0,1),(0,0,0,0,1),...]
%e S^2: [(1),(0,2),(0,1,2),(0,2,0,2),(1,2,0,0,4),...]
%e S^3: [(1),(0,3),(0,3,3),(1,6,0,6),(3,6,3,3,9),...]
%e S^4: [(1),(0,4),(0,6,4),(4,12,1,16),(6,16,12,12,12),...]
%e S^5: [(1),(0,5),(0,10,5),(10,20,5,35),(11,40,30,35,35),...]
%e S^6: [(1),(0,6),(0,15,6),(20,30,15,66),(21,90,61,90,126),...]
%e ...
%e then the sums of the above grouped terms (in parenthesis)
%e form the initial terms of the rows of table A162424:
%e 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...
%e 1,2,3,4,7,7,8,11,13,13,16,15,19,22,21,23,22,...
%e 1,3,6,13,24,34,49,69,94,117,148,174,211,249,...
%e 1,4,10,33,68,123,226,342,547,778,1071,1412,...
%e 1,5,15,70,171,385,836,1485,2630,4271,6590,...
%e 1,6,21,131,388,1073,2674,5634,11173,20379,...
%e 1,7,28,224,806,2709,7749,19055,42770,87773,...
%e 1,8,36,358,1556,6303,20792,58846,150169,346748,...
%e ...
%e The main diagonal of the above table forms this sequence.
%o (PARI) {a(n)=local(S=sum(m=0,n+1,x^((m+1)*(m+2)/2-1))+O(x^((n+1)*(n+2)/2))); sum(m=n*(n+1)/2,n*(n+1)/2+n,polcoeff(S^(n+1),m))}
%Y Cf. A162424, A162425, A162426, A162427, A162428; A162435 (variant).
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jul 03 2009
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