A162424 Square table defined by T(n,k) = Sum_{m=k(k+1)/2..k(k+1)/2+k} [x^m] S(x)^n for n>=1, k>=0, where S(x) = Sum_{n>=0} x^((n+1)(n+2)/2-1), as read by antidiagonals.

1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 13, 7, 1, 1, 6, 15, 33, 24, 7, 1, 1, 7, 21, 70, 68, 34, 8, 1, 1, 8, 28, 131, 171, 123, 49, 11, 1, 1, 9, 36, 224, 388, 385, 226, 69, 13, 1, 1, 10, 45, 358, 806, 1073, 836, 342, 94, 13, 1, 1, 11, 55, 543, 1556, 2709, 2674, 1485, 547

Offset

1,4

Links

Table of n, a(n) for n=1..74.

Example

This table begins:
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...
1,2,3,4,7,7,8,11,13,13,16,15,19,22,21,23,22,29,27,31,30,29,39,34,...
1,3,6,13,24,34,49,69,94,117,148,174,211,249,298,331,366,439,498,...
1,4,10,33,68,123,226,342,547,778,1071,1412,1901,2392,2997,3762,...
1,5,15,70,171,385,836,1485,2630,4271,6590,9536,14030,19286,25995,...
1,6,21,131,388,1073,2674,5634,11173,20379,35305,56757,89734,...
1,7,28,224,806,2709,7749,19055,42770,87773,168778,302008,519407,...
1,8,36,358,1556,6303,20792,58846,150169,346748,737442,1464203,...
1,9,45,543,2829,13711,52221,168912,489993,1269145,2990595,...
1,10,55,790,4897,28191,123717,455882,1499600,4345078,11360122,...
1,11,66,1111,8140,55231,278334,1166066,4335837,14032832,...
1,12,78,1519,13080,103742,598217,2843920,11916721,43041924,...
...
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),(0,0,0,0,0,1),...]
S^2: [(1),(0,2),(0,1,2),(0,2,0,2),(1,2,0,0,4),(0,2,0,1,2,2),...]
S^3: [(1),(0,3),(0,3,3),(1,6,0,6),(3,6,3,3,9),(1,12,0,6,9,6),...]
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),...]
S^7: [(1),(0,7),(0,21,7),(35,42,35,112),(42,182,112,210),...]
S^8: [(1),(0,8),(0,28,8),(56,56,70,176),(84,336,196,448),...]
S^9: [(1),(0,9),(0,36,9),(84,72,126,261),(162,576,336,882),...]
S^10:[(1),(0,10),(0,45,10),(120,90,210,370),(297,930,570,1620),...]
...
then the sums of the above grouped terms (enclosed in parenthesis)
form the initial terms of the rows of this table:
T(3,4) = (3+6+3+3+9) = 24 ;
T(4,3) = (4+12+1+16) = 33 ;
T(5,3) = (10+20+5+35) = 70.
Summing the coefficients of S^n in this way generates all the rows of this table.

Prog

(PARI) {T(n, k)=local(S=sum(m=0, k+1, x^((m+1)*(m+2)/2-1))+O(x^((k+2)*(k+3)/2))); sum(m=k*(k+1)/2, k*(k+1)/2+k, polcoeff(S^n, m))}

Crossrefs

Cf. rows: A162425, A162426, A162427.

Cf. A162428 (antidiagonal sums), A162429 (main diagonal).

Cf. A162430 (variant).

Sequence in context: A197061 A230861 A119724 * A302998 A303484 A008571

Adjacent sequences: A162421 A162422 A162423 * A162425 A162426 A162427

Keyword

nonn,tabl

Author

Paul D. Hanna, Jul 03 2009

Status

approved