0,2

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).

Let coefficients in powers of the series:

S = 1 + x^2 + x^5 + x^9 + x^14 + x^20 + x^27 + x^35 + x^44 +...

form the following sequences:

S^1: [(1),(0,1),(0,0,1),(0,0,0,1),(0,0,0,0,1),...]

S^2: [(1),(0,2),(0,1,2),(0,2,0,2),(1,2,0,0,4),...]

S^3: [(1),(0,3),(0,3,3),(1,6,0,6),(3,6,3,3,9),...]

S^4: [(1),(0,4),(0,6,4),(4,12,1,16),(6,16,12,12,12),...]

S^5: [(1),(0,5),(0,10,5),(10,20,5,35),(11,40,30,35,35),...]

S^6: [(1),(0,6),(0,15,6),(20,30,15,66),(21,90,61,90,126),...]

...

then the sums of the above grouped terms (in parenthesis)

form the initial terms of the rows of table A162424:

1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...

1,2,3,4,7,7,8,11,13,13,16,15,19,22,21,23,22,...

1,3,6,13,24,34,49,69,94,117,148,174,211,249,...

1,4,10,33,68,123,226,342,547,778,1071,1412,...

1,5,15,70,171,385,836,1485,2630,4271,6590,...

1,6,21,131,388,1073,2674,5634,11173,20379,...

1,7,28,224,806,2709,7749,19055,42770,87773,...

1,8,36,358,1556,6303,20792,58846,150169,346748,...

...

The antidiagonal sums of the above table forms this sequence.

(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)))}

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

Sequence in context: A002876 A095236 A018536 this_sequence A028497 A018575 A013116

Adjacent sequences: A162425 A162426 A162427 this_sequence A162429 A162430 A162431

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Jul 03 2009