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%I #2 Mar 30 2012 18:37:17
%S 1,1,1,2,1,1,3,3,1,1,4,6,4,1,1,5,10,13,7,1,1,6,15,33,24,7,1,1,7,21,70,
%T 68,34,8,1,1,8,28,131,171,123,49,11,1,1,9,36,224,388,385,226,69,13,1,
%U 1,10,45,358,806,1073,836,342,94,13,1,1,11,55,543,1556,2709,2674,1485,547
%N Square table defined by T(n,k) = Sum_{m=k(k+1)/2..k(k+1)/2+k} [x^m] S(x)^n for n>=1, k>=0, where S(x) = Sum_{n>=0} x^((n+1)(n+2)/2-1), as read by antidiagonals.
%e This table begins:
%e 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...
%e 1,2,3,4,7,7,8,11,13,13,16,15,19,22,21,23,22,29,27,31,30,29,39,34,...
%e 1,3,6,13,24,34,49,69,94,117,148,174,211,249,298,331,366,439,498,...
%e 1,4,10,33,68,123,226,342,547,778,1071,1412,1901,2392,2997,3762,...
%e 1,5,15,70,171,385,836,1485,2630,4271,6590,9536,14030,19286,25995,...
%e 1,6,21,131,388,1073,2674,5634,11173,20379,35305,56757,89734,...
%e 1,7,28,224,806,2709,7749,19055,42770,87773,168778,302008,519407,...
%e 1,8,36,358,1556,6303,20792,58846,150169,346748,737442,1464203,...
%e 1,9,45,543,2829,13711,52221,168912,489993,1269145,2990595,...
%e 1,10,55,790,4897,28191,123717,455882,1499600,4345078,11360122,...
%e 1,11,66,1111,8140,55231,278334,1166066,4335837,14032832,...
%e 1,12,78,1519,13080,103742,598217,2843920,11916721,43041924,...
%e ...
%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),(0,0,0,0,0,1),...]
%e S^2: [(1),(0,2),(0,1,2),(0,2,0,2),(1,2,0,0,4),(0,2,0,1,2,2),...]
%e S^3: [(1),(0,3),(0,3,3),(1,6,0,6),(3,6,3,3,9),(1,12,0,6,9,6),...]
%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 S^7: [(1),(0,7),(0,21,7),(35,42,35,112),(42,182,112,210),...]
%e S^8: [(1),(0,8),(0,28,8),(56,56,70,176),(84,336,196,448),...]
%e S^9: [(1),(0,9),(0,36,9),(84,72,126,261),(162,576,336,882),...]
%e S^10:[(1),(0,10),(0,45,10),(120,90,210,370),(297,930,570,1620),...]
%e ...
%e then the sums of the above grouped terms (enclosed in parenthesis)
%e form the initial terms of the rows of this table:
%e T(3,4) = (3+6+3+3+9) = 24 ;
%e T(4,3) = (4+12+1+16) = 33 ;
%e T(5,3) = (10+20+5+35) = 70.
%e Summing the coefficients of S^n in this way generates all the rows of this table.
%o (PARI) {T(n,k)=local(S=sum(m=0,k+1,x^((m+1)*(m+2)/2-1))+O(x^((k+2)*(k+3)/2))); sum(m=k*(k+1)/2,k*(k+1)/2+k,polcoeff(S^n,m))}
%Y Cf. rows: A162425, A162426, A162427.
%Y Cf. A162428 (antidiagonal sums), A162429 (main diagonal).
%Y Cf. A162430 (variant).
%K nonn,tabl
%O 1,4
%A _Paul D. Hanna_, Jul 03 2009
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