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A243148 Triangle read by rows: T(n,k) = number of partitions of n into k nonzero squares; n >= 0, 0 <= k <= n. 26
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 = #{ (16,1,1,1,1), (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;

  (...)

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 *)

PROG

(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

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, A341040.

T(n,k) = 1 for n in A025284, A025321, A025357, A294675, A295670, A295797 (for k = 2..7, respectively).

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 March 5 06:18 EST 2021. Contains 341816 sequences. (Running on oeis4.)