A162428 Antidiagonal sums of table A162424.

1, 2, 4, 8, 16, 37, 87, 210, 523, 1352, 3579, 9715, 26956, 76367, 220408, 648044, 1940191, 5907256, 18279257, 57449874, 183261022, 592970009, 1945136089, 6465415945, 21764630304, 74168652771, 255759582378, 892121078978

Offset

0,2

Links

Table of n, a(n) for n=0..27.

Formula

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).

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 antidiagonal sums 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(k=0, n, sum(m=k*(k+1)/2, k*(k+1)/2+k, polcoeff(S^(n-k), m)))}

Crossrefs

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

Sequence in context: A348847 A018536 A337673 * A344491 A028497 A197244

Adjacent sequences: A162425 A162426 A162427 * A162429 A162430 A162431

Keyword

nonn

Author

Paul D. Hanna, Jul 03 2009

Status

approved