A339419 -id:A339419 - cp's OEIS frontend

A339418 Number of compositions (ordered partitions) of n into an even number of squares.

1, 0, 1, 0, 1, 2, 1, 4, 2, 6, 9, 8, 20, 16, 35, 44, 55, 102, 105, 196, 242, 344, 540, 652, 1084, 1380, 2037, 2964, 3912, 6042, 7976, 11776, 16634, 22968, 33963, 46156, 67457, 94510, 133180, 192316, 266514, 385338, 540138, 767008, 1094576, 1534704, 2200821, 3094248

Offset: 0

Ilya Gutkovskiy, Dec 03 2020

nonn

			a(9) = 6 because we have [4, 1, 1, 1, 1, 1], [1, 4, 1, 1, 1, 1], [1, 1, 4, 1, 1, 1], [1, 1, 1, 4, 1, 1], [1, 1, 1, 1, 4, 1] and [1, 1, 1, 1, 1, 4].

Cf. A000290, A006456, A034008, A317665, A339364, A339416, A339419.

b:= proc(n, t) option remember; local r, f, g;
      if n=0 then t else r, f, g:=$0..2; while f<=n
      do r, f, g:= r+b(n-f, 1-t), f+2*g-1, g+1 od; r fi
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..50);  # Alois P. Heinz, Dec 03 2020

nmax = 47; CoefficientList[Series[4/(3 + 2 EllipticTheta[3, 0, x] - EllipticTheta[3, 0, x]^2), {x, 0, nmax}], x]

G.f.: 4 / (3 + 2 * theta_3(x) - theta_3(x)^2), where theta_3() is the Jacobi theta function.
a(n) = (A006456(n) + A317665(n)) / 2.
a(n) = Sum_{k=0..n} A006456(k) * A317665(n-k).

A339421 Number of compositions (ordered partitions) of n into an odd number of cubes.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 3, 1, 5, 1, 7, 1, 9, 4, 11, 11, 13, 22, 15, 37, 18, 56, 29, 80, 56, 109, 107, 142, 190, 184, 313, 255, 490, 391, 731, 644, 1045, 1082, 1458, 1792, 2044, 2895, 2957, 4531, 4463, 6863, 6972, 10126, 11090, 14739, 17691, 21484, 27954, 31741

Offset: 0

Ilya Gutkovskiy, Dec 03 2020

nonn

			a(10) = 3 because we have [8, 1, 1], [1, 8, 1] and [1, 1, 8].

Cf. A000578, A023358, A166444, A323633, A339369, A339419, A339420.

b:= proc(n, t) option remember; local r, f, g;
      if n=0 then t else r, f, g:=$0..2; while f<=n
      do r, f, g:= r+b(n-f, 1-t), f+3*g*(g-1)+1, g+1 od; r fi
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 03 2020

nmax = 57; CoefficientList[Series[(1/2) (1/(1 - Sum[x^(k^3), {k, 1, Floor[nmax^(1/3)] + 1}]) - 1/Sum[x^(k^3), {k, 0, Floor[nmax^(1/3)] + 1}]), {x, 0, nmax}], x]

G.f.: (1/2) * (1 / (1 - Sum_{k>=1} x^(k^3)) - 1 / Sum_{k>=0} x^(k^3)).
a(n) = (A023358(n) - A323633(n)) / 2.
a(n) = -Sum_{k=0..n-1} A023358(k) * A323633(n-k).

A339431 Number of compositions (ordered partitions) of n into an odd number of distinct squares.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 6, 0, 1, 0, 0, 0, 0, 6, 0, 0, 0, 1, 6, 0, 0, 6, 6, 0, 0, 0, 0, 6, 1, 0, 6, 0, 0, 6, 6, 0, 0, 6, 6, 0, 0, 7, 6, 0, 0, 6, 6, 120, 6, 0, 0, 6, 0, 6, 12, 0, 1, 6, 126, 0, 0, 12, 6, 0, 0, 0, 12, 126, 0, 12, 6, 120, 0, 7

Offset: 0

Ilya Gutkovskiy, Dec 04 2020

			a(55) = 120 because we have [25, 16, 9, 4, 1] (120 permutations).

Cf. A000290, A331844, A332304, A339367, A339419, A339430.

b:= proc(n, i, p) option remember; `if`(n=0, irem(p, 2)*p!,
     (s-> `if`(s>n, 0, b(n, i+1, p)+b(n-s, i+1, p+1)))(i^2))
    end:
a:= n-> b(n, 1, 0):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 04 2020

b[n_, i_, p_] := b[n, i, p] = If[n == 0, Mod[p, 2]*p!, With[{s = i^2}, If[s > n, 0, b[n, i + 1, p] + b[n - s, i + 1, p + 1]]]];
a[n_] := b[n, 1, 0];
a /@ Range[0, 100] (* Jean-François Alcover, Mar 14 2021, after Alois P. Heinz *)

cp's OEIS Frontend

A339418 Number of compositions (ordered partitions) of n into an even number of squares.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A339421 Number of compositions (ordered partitions) of n into an odd number of cubes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A339431 Number of compositions (ordered partitions) of n into an odd number of distinct squares.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica