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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.

%I A232266 #11 Apr 26 2016 12:43:24
%S A232266 1,1,1,2,1,1,11,7,3,1,124,88,30,5,1,2870,2024,710,124,11,1,133462,
%T A232266 94137,33033,5767,502,22,1,12477207,8800750,3088365,539192,46832,2024,
%U A232266 43,1,2344649612,1653790807,580347968,101321507,8800750,380315,8176,88,1,885591183971,624648802700,219201637352,38269865019,3324109524,143647802,3088365,33033,175,1
%N 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.
%F A232266 T(n,k) = A006456(n^2-k^2).
%F A232266 T(n,k) = [x^(n^2-k^2)] 1/(1 - Sum_{j>=1} x^(j^2)).
%F A232266 T(n,0) = Sum_{k=1..n} T(n,k) for n>=1.
%e A232266 Triangle begins:
%e A232266 1;
%e A232266 1, 1;
%e A232266 2, 1, 1;
%e A232266 11, 7, 3, 1;
%e A232266 124, 88, 30, 5, 1;
%e A232266 2870, 2024, 710, 124, 11, 1;
%e A232266 133462, 94137, 33033, 5767, 502, 22, 1;
%e A232266 12477207, 8800750, 3088365, 539192, 46832, 2024, 43, 1;
%e A232266 2344649612, 1653790807, 580347968, 101321507, 8800750, 380315, 8176, 88, 1;
%e A232266 885591183971, 624648802700, 219201637352, 38269865019, 3324109524, 143647802, 3088365, 33033, 175, 1; ...
%e A232266 where T(n,k) = coefficient of x^(n^2-k^2) in the series:
%e A232266 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 +...
%o A232266 (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)}
%o A232266 for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
%Y A232266 Cf. A006456, A224366.
%K A232266 nonn,tabl
%O A232266 0,4
%A A232266 _Paul D. Hanna_, Nov 21 2013