A232266 Triangle where T(n,k) = number of compositions of n^2 - k^2 into sums of squares for k=0..n, n>=0, as read by rows.

1, 1, 1, 2, 1, 1, 11, 7, 3, 1, 124, 88, 30, 5, 1, 2870, 2024, 710, 124, 11, 1, 133462, 94137, 33033, 5767, 502, 22, 1, 12477207, 8800750, 3088365, 539192, 46832, 2024, 43, 1, 2344649612, 1653790807, 580347968, 101321507, 8800750, 380315, 8176, 88, 1, 885591183971, 624648802700, 219201637352, 38269865019, 3324109524, 143647802, 3088365, 33033, 175, 1

Offset

0,4

Links

Table of n, a(n) for n=0..54.

Formula

T(n,k) = A006456(n^2-k^2).

T(n,k) = [x^(n^2-k^2)] 1/(1 - Sum_{j>=1} x^(j^2)).

T(n,0) = Sum_{k=1..n} T(n,k) for n>=1.

Example

Triangle begins:
1;
1, 1;
2, 1, 1;
11, 7, 3, 1;
124, 88, 30, 5, 1;
2870, 2024, 710, 124, 11, 1;
133462, 94137, 33033, 5767, 502, 22, 1;
12477207, 8800750, 3088365, 539192, 46832, 2024, 43, 1;
2344649612, 1653790807, 580347968, 101321507, 8800750, 380315, 8176, 88, 1;
885591183971, 624648802700, 219201637352, 38269865019, 3324109524, 143647802, 3088365, 33033, 175, 1; ...
where T(n,k) = coefficient of x^(n^2-k^2) in the series:
1/(1 - x - x^4 - x^9 - x^16 - x^25 - x^36 -...- x^(n^2) -...) = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 5*x^7 + 7*x^8 + 11*x^9 + 16*x^10 + 22*x^11 + 30*x^12 + 43*x^13 + 62*x^14 + 88*x^15 + 124*x^16 + 175*x^17 + 249*x^18 + 354*x^19 + 502*x^20 + 710*x^21 + 1006*x^22 + 1427*x^23 + 2024*x^24 + 2870*x^25 +...

Prog

(PARI) {T(n, k)=polcoeff(1/(1-sum(m=1, n+1, x^(m^2))+x*O(x^(n^2-k^2))), n^2-k^2)}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. A006456, A224366.

Sequence in context: A205447 A297544 A297802 * A234013 A158202 A176307

Adjacent sequences: A232263 A232264 A232265 * A232267 A232268 A232269

Keyword

nonn,tabl

Author

Paul D. Hanna, Nov 21 2013

Status

approved