OFFSET
0,2
FORMULA
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).
EXAMPLE
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 main diagonal of the above table forms this sequence.
PROG
(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))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 03 2009
STATUS
approved