A338464 -id:A338464 - cp's OEIS frontend

A337165 Number T(n,k) of compositions of n into k nonzero squares; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 0, 3, 0, 0, 1, 0, 0, 0, 0, 4, 0, 0, 1, 0, 0, 1, 0, 0, 5, 0, 0, 1, 0, 1, 0, 3, 0, 0, 6, 0, 0, 1, 0, 0, 2, 0, 6, 0, 0, 7, 0, 0, 1, 0, 0, 0, 3, 0, 10, 0, 0, 8, 0, 0, 1, 0, 0, 0, 1, 4, 0, 15, 0, 0, 9, 0, 0, 1

Offset: 0

Alois P. Heinz, Feb 03 2021

			Triangle T(n,k) begins:
  1;
  0, 1;
  0, 0, 1;
  0, 0, 0, 1;
  0, 1, 0, 0, 1;
  0, 0, 2, 0, 0,  1;
  0, 0, 0, 3, 0,  0,  1;
  0, 0, 0, 0, 4,  0,  0, 1;
  0, 0, 1, 0, 0,  5,  0, 0, 1;
  0, 1, 0, 3, 0,  0,  6, 0, 0, 1;
  0, 0, 2, 0, 6,  0,  0, 7, 0, 0, 1;
  0, 0, 0, 3, 0, 10,  0, 0, 8, 0, 0, 1;
  0, 0, 0, 1, 4,  0, 15, 0, 0, 9, 0, 0, 1;
  ...

Alois P. Heinz, Rows n = 0..350, flattened

Columns k=0-10 give: A000007, A010052, A063725, A063691, A063730, A340481, A340905, A340906, A340915, A340946, A340947.
Row sums give A006456.
T(2n,n) gives A338464.
Main diagonal gives A000012.
Cf. A000290, A281704, A317665, A341040.

b:= proc(n) option remember; `if`(n=0, 1, add((s->
     `if`(s>n, 0, expand(x*b(n-s))))(j^2), j=1..isqrt(n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):
seq(T(n), n=0..14);

b[n_] := b[n] = If[n == 0, 1, Sum[With[{s = j^2},
     If[s>n, 0, Expand[x*b[n - s]]]], {j, 1, Sqrt[n]}]];
T[n_] := CoefficientList[b[n], x];
T /@ Range[0, 14] // Flatten (* Jean-François Alcover, Feb 07 2021, after Alois P. Heinz *)

G.f. of column k: (Sum_{j>=1} x^(j^2))^k.
Sum_{k=0..n} k * T(n,k) = A281704(n).
Sum_{k=0..n} (-1)^k * T(n,k) = A317665(n).

A341071 Number of compositions of 2n into n squarefree parts.

Original entry on oeis.org

1, 1, 3, 7, 23, 76, 267, 932, 3263, 11410, 40028, 140955, 498467, 1769249, 6299282, 22485487, 80435919, 288275461, 1034862558, 3720499175, 13393720428, 48275699654, 174194970601, 629187941167, 2274710068067, 8230764250326, 29805342037721, 108009638665648

Offset: 0

Alois P. Heinz, Feb 04 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A005117, A333047, A338464, A338465, A341072.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
      `if`(numtheory[issqrfree](j), b(n-j, t-1), 0), j=1..n)))
    end:
a:= n-> b(2*n, n):
seq(a(n), n=0..35);

b[n_, t_] := b[n, t] =
  If[n == 0, If[t == 0, 1, 0], If[t < 1, 0, Sum[
  If[SquareFreeQ[j], b[n - j, t - 1], 0], {j, 1, n}]]];
a[n_] := b[2n, n];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, May 26 2023, from Maple program *)

a(n) ~ c * d^n / sqrt(n), where d = 3.6921332109291502908930783402282652076342401140592914... and c = 0.27220098720735952890181498005903942865018992276116... - Vaclav Kotesovec, Feb 14 2021

cp's OEIS Frontend

A337165 Number T(n,k) of compositions of n into k nonzero squares; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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