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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. 1
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified March 23 14:17 EDT 2019. Contains 321431 sequences. (Running on oeis4.)