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 A162428 Antidiagonal sums of table A162424. 6

%I #2 Mar 30 2012 18:37:17

%S 1,2,4,8,16,37,87,210,523,1352,3579,9715,26956,76367,220408,648044,

%T 1940191,5907256,18279257,57449874,183261022,592970009,1945136089,

%U 6465415945,21764630304,74168652771,255759582378,892121078978

%N Antidiagonal sums of table A162424.

%F a(n) = Sum_{k=0..n} Sum_{m=k*(k+1)/2..k*(k+1)/2+k} [x^m] S(x)^(n-k) for n>=1 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 antidiagonal sums 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(k=0,n,sum(m=k*(k+1)/2,k*(k+1)/2+k,polcoeff(S^(n-k),m)))}

%Y Cf. A162424, A162425, A162426, A162427, A162429, A162434 (variant).

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jul 03 2009

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