A162429 Main diagonal of table A162424.

1, 2, 6, 33, 171, 1073, 7749, 58846, 489993, 4345078, 40693236, 403876159, 4192624215, 45577188311, 516108618746, 6073914844170, 74083651509143, 934440647979907, 12164742571864976, 163157161537315276

Offset

0,2

Links

Table of n, a(n) for n=0..19.

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

Cf. A162424, A162425, A162426, A162427, A162428; A162435 (variant).

Sequence in context: A083666 A083126 A098960 * A279461 A243324 A018952

Adjacent sequences: A162426 A162427 A162428 * A162430 A162431 A162432

Keyword

nonn

Author

Paul D. Hanna, Jul 03 2009

Status

approved