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A243148 Number T(n,k) of partitions of n into k nonzero squares; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 18
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; table; graph; refs; listen; history; text; internal format)
OFFSET

0

LINKS

Alois P. Heinz, Rows n = 0..200, flattened

FORMULA

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).

EXAMPLE

T(20,5) = 2: (16)1111, 44444.

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;

MAPLE

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);

MATHEMATICA

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 *)

CROSSREFS

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.

Sequence in context: A134286 A023531 A320841 * A089495 A173857 A114482

Adjacent sequences:  A243145 A243146 A243147 * A243149 A243150 A243151

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, May 30 2014

STATUS

approved

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Last modified January 17 05:26 EST 2019. Contains 319207 sequences. (Running on oeis4.)