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

Author

Paul D. Hanna, Nov 21 2013

Keywords

• nonn
• tabl

Examples

			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 +...
		

Crossrefs

Cf. A006456, A224366.

Programs

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(""))

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.