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Search: a340988 -id:a340988
Displaying 1-6 of 6 results found.
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A341040
Number T(n,k) of partitions of n into k distinct nonzero squares; triangle T(n,k), n>=0, 0<=k<=A248509(n), read by rows.
+10
20
1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1
(list; graph; refs; listen; history; text; internal format)
OFFSET
0
COMMENTS
T(n,k) is defined for n, k >= 0. The triangle contains only the terms with 0 <= k <= A248509(n). T(n,k) = 0 for k > A248509(n).
LINKS
Alois P. Heinz, Rows n = 0..4000, flattened
FORMULA
T(n,k) = [x^n*y^k] Product_{j>=1} (1 + y*x^(j^2)).
T(A000330(n),n) = 1.
Row n = [0] <=> n in { A001422 }.
Sum_{k>=0} 2^k * T(n,k) = A279360(n).
Sum_{k>=0} k * T(n,k) = A281542(n).
Sum_{k>=0} (-1)^k * T(n,k) = A276516(n).
EXAMPLE
T(62,3) = 2 is the first term > 1 and counts partitions [49,9,4] and [36,25,1].
Triangle T(n,k) begins:
1;
0, 1;
0;
0;
0, 1;
0, 0, 1;
0;
0;
0;
0, 1;
0, 0, 1;
0;
0;
0, 0, 1;
0, 0, 0, 1;
...
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1)+`if`(i^2>n, 0, expand(b(n-i^2, i-1)*x))))
end:
T:= n->(p->seq(coeff(p, x, i), i=0..max(0, degree(p))))(b(n, isqrt(n))):
seq(T(n), n=0..45);
MATHEMATICA
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
b[n, i - 1] + If[i^2 > n, 0, Expand[b[n - i^2, i - 1]*x]]]];
T[n_] := CoefficientList[b[n, Floor@Sqrt[n]], x] /. {} -> {0};
T /@ Range[0, 45] // Flatten (* Jean-François Alcover, Feb 15 2021, after Alois P. Heinz *)
CROSSREFS
Columns k=0-10 give: A000007, A010052 (for n>0), A025441, A025442, A025443, A025444, A340988, A340998, A340999, A341000, A341001.
Row sums give A033461.
Cf. A000290, A000330, A001422, A243148, A248509, A276516, A279360, A281542, A337165.
KEYWORD
nonn,look,tabf
AUTHOR
Alois P. Heinz, Feb 03 2021
STATUS
approved
A025444
Number of partitions of n into 5 distinct nonzero squares.
+10
8
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 1, 0
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,104
LINKS
David A. Corneth, Table of n, a(n) for n = 0..9999
Index entries for sequences related to sums of squares
FORMULA
a(n) = [x^n y^5] Product_{k>=1} (1 + y*x^(k^2)). - Ilya Gutkovskiy, Apr 22 2019
EXAMPLE
a(111) = 2 via 1 + 4 + 9 + 16 + 81 = 1 + 9 + 16 + 36 + 49. - David A. Corneth, Feb 02 2021
MAPLE
From R. J. Mathar, Oct 18 2010: (Start)
A025444aux := proc(n, m, nmax) local a, m, upn, lv ; if m = 1 then if issqr(n) and nmax^2 >= n and n >= 1 then return 1; else return 0; end if; else a := 0 ; for upn from 1 to nmax do lv := n-upn^2 ; if lv <0 then break; end if; a := a + procname(lv, m-1, upn-1) ; end do: return a; end if; end proc:
A025444 := proc(n) A025444aux(n, 5, n) ; end proc: (End)
CROSSREFS
Cf. A000290, A008452, A010052, A025433, A025441, A025442, A025443, A025444, A045851, A340946, A340988, A340998, A340999, A341000, A341001.
Column k=5 of A341040.
KEYWORD
nonn,look
AUTHOR
David W. Wilson
STATUS
approved
A341000
Number of partitions of n into 9 distinct nonzero squares.
+10
7
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 2, 0, 0, 0, 1, 2, 0, 0, 2
(list; graph; refs; listen; history; text; internal format)
OFFSET
285,92
LINKS
David A. Corneth, Table of n, a(n) for n = 285..5000
Index entries for sequences related to sums of squares
EXAMPLE
a(381) = 2 via 1 + 4 + 9 + 16 + 36 + 49 + 64 + 81 + 121 = 1 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100. - David A. Corneth, Feb 02 2021
CROSSREFS
Cf. A000290, A008452, A010052, A025433, A025441, A025442, A025443, A025444, A045851, A340946, A340988, A340998, A340999, A341001.
Column k=9 of A341040.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Feb 02 2021
STATUS
approved
A341001
Number of partitions of n into 10 distinct nonzero squares.
+10
7
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 1
(list; graph; refs; listen; history; text; internal format)
OFFSET
385,97
LINKS
Alois P. Heinz, Table of n, a(n) for n = 385..20000
Index entries for sequences related to sums of squares
CROSSREFS
Cf. A000144, A000290, A010052, A025434, A025441, A025442, A025443, A025444, A045852, A340947, A340988, A340998, A340999, A341000.
Column k=10 of A341040.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Feb 02 2021
STATUS
approved
A340998
Number of partitions of n into 7 distinct nonzero squares.
+10
6
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 2, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 2, 0, 0, 1, 0, 1, 0, 1, 2, 0, 0, 0, 2, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 2, 3, 0, 0, 2, 2
(list; graph; refs; listen; history; text; internal format)
OFFSET
140,61
LINKS
Table of n, a(n) for n=140..240.
Index entries for sequences related to sums of squares
CROSSREFS
Cf. A000290, A008451, A010052, A025431, A025441, A025442, A025443, A025444, A045849, A224982, A340906, A340988, A340999, A341000, A341001.
Column k=7 of A341040.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Feb 02 2021
STATUS
approved
A340999
Number of partitions of n into 8 distinct nonzero squares.
+10
6
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 0, 1, 0, 0, 2, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 3, 0, 0, 1
(list; graph; refs; listen; history; text; internal format)
OFFSET
204,73
LINKS
Table of n, a(n) for n=204..303.
Index entries for sequences related to sums of squares
CROSSREFS
Cf. A000143, A000290, A010052, A025432, A025441, A025442, A025443, A025444, A045850, A224983, A340915, A340988, A340998, A341000, A341001.
Column k=8 of A341040.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Feb 02 2021
STATUS
approved
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