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A243148
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Triangle read by rows: T(n,k) = number of partitions of n into k nonzero squares; n >= 0, 0 <= k <= n.
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27
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1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1
(list;
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graph;
refs;
listen;
history;
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internal format)
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OFFSET
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0,216
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LINKS
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Alois P. Heinz, Rows n = 0..200, flattened
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FORMULA
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T(n,k) = [x^n y^k] 1/Product_{j>=1} (1-y*x^A000290(j)).
Sum_{k=1..n} k * T(n,k) = A281541(n).
Sum_{k=1..n} n * T(n,k) = A276559(n).
Sum_{k=0..n} (-1)^k * T(n,k) = A292520(n).
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EXAMPLE
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T(20,5) = 2 = #{ (16,1,1,1,1), (4,4,4,4,4) } since 20 = 4^2 + 4 * 1^2 = 5 * 2^2.
Triangle T(n,k) begins:
1;
0, 1;
0, 0, 1;
0, 0, 0, 1;
0, 1, 0, 0, 1;
0, 0, 1, 0, 0, 1;
0, 0, 0, 1, 0, 0, 1;
0, 0, 0, 0, 1, 0, 0, 1;
0, 0, 1, 0, 0, 1, 0, 0, 1;
0, 1, 0, 1, 0, 0, 1, 0, 0, 1;
0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1;
0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1;
0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1;
(...)
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MAPLE
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b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0),
`if`(i<1 or t<1, 0, b(n, i-1, t)+
`if`(i^2>n, 0, b(n-i^2, i, t-1))))
end:
T:= (n, k)-> b(n, isqrt(n), k):
seq(seq(T(n, k), k=0..n), n=0..14);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1)+(s-> `if`(s>n, 0, expand(x*b(n-s, i))))(i^2)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, isqrt(n))):
seq(T(n), n=0..14); # Alois P. Heinz, Oct 30 2021
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MATHEMATICA
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b[n_, i_, k_, t_] := b[n, i, k, t] = If[n == 0, If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i-1, k, t] + If[i^2 > n, 0, b[n-i^2, i, k, t-1]]]]; T[n_, k_] := b[n, Sqrt[n] // Floor, k, k]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jun 06 2014, after Alois P. Heinz *)
T[n_, k_] := Count[PowersRepresentations[n, k, 2], r_ /; FreeQ[r, 0]]; T[0, 0] = 1; Table[T[n, k], {n, 0, 14}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 19 2016 *)
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PROG
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(PARI) T(n, k, L=n)=if(n>k*L^2, 0, k>n-3, k==n, k<2, issquare(n, &n) && n<=L*k, k>n-6, n-k==3, L=min(L, sqrtint(n-k+1)); sum(r=0, min(n\L^2, k-1), T(n-r*L^2, k-r, L-1), n==k*L^2)) \\ M. F. Hasler, Aug 03 2020
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CROSSREFS
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Columns k = 0..10 give: A000007, A010052 (for n>0), A025426, A025427, A025428, A025429, A025430, A025431, A025432, A025433, A025434.
Row sums give A001156.
T(2n,n) gives A111178.
T(n^2,n) gives A319435.
Cf. A000290, A276559, A281541, A292520, A341040.
T(n,k) = 1 for n in A025284, A025321, A025357, A294675, A295670, A295797 (for k = 2..7, respectively).
Sequence in context: A023531 A320841 A351725 * A089495 A345703 A173857
Adjacent sequences: A243145 A243146 A243147 * A243149 A243150 A243151
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KEYWORD
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nonn,tabl
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AUTHOR
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Alois P. Heinz, May 30 2014
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STATUS
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approved
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