A340946 -id:A340946 - 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).

A340905 Number of ways to write n as an ordered sum of 6 squares of positive integers.

Original entry on oeis.org

1, 0, 0, 6, 0, 0, 15, 0, 6, 20, 0, 30, 15, 0, 60, 12, 15, 60, 31, 60, 30, 60, 90, 36, 86, 60, 120, 120, 15, 180, 141, 60, 165, 140, 180, 186, 120, 180, 285, 156, 126, 360, 255, 216, 270, 260, 390, 240, 262, 420, 426, 360, 210, 540, 530, 216, 540, 540, 480, 600, 300, 600, 825, 312, 576, 840

Offset: 6

Ilya Gutkovskiy, Jan 31 2021

Alois P. Heinz, Table of n, a(n) for n = 6..20000
Index entries for sequences related to sums of squares

Cf. A000141, A000290, A010052, A025430, A045848, A063691, A063725, A063730, A340481, A340906, A340915, A340946, A340947.

Column k=6 of A337165.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add((s->
      `if`(s>n, 0, b(n-s, t-1)))(j^2), j=1..isqrt(n))))
    end:
a:= n-> b(n, 6):
seq(a(n), n=6..71);  # Alois P. Heinz, Jan 31 2021

nmax = 71; CoefficientList[Series[(EllipticTheta[3, 0, x] - 1)^6/64, {x, 0, nmax}], x] // Drop[#, 6] &

G.f.: (theta_3(x) - 1)^6 / 64, where theta_3() is the Jacobi theta function.

A340481 Number of ways to write n as an ordered sum of 5 squares of positive integers.

Original entry on oeis.org

1, 0, 0, 5, 0, 0, 10, 0, 5, 10, 0, 20, 5, 0, 30, 6, 10, 20, 20, 30, 5, 30, 30, 20, 35, 10, 60, 45, 0, 60, 50, 30, 45, 50, 60, 70, 35, 30, 110, 50, 31, 110, 80, 80, 50, 70, 120, 70, 75, 90, 140, 110, 20, 140, 160, 60, 135, 120, 120, 180, 40, 130, 230, 80, 120, 170, 200, 155, 85, 200, 190

Offset: 5

Ilya Gutkovskiy, Jan 31 2021

Alois P. Heinz, Table of n, a(n) for n = 5..20000
Index entries for sequences related to sums of squares

Cf. A000132, A000290, A010052, A025429, A038671, A063691, A063725, A063730, A340905, A340906, A340915, A340946, A340947.

Column k=5 of A337165.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add((s->
      `if`(s>n, 0, b(n-s, t-1)))(j^2), j=1..isqrt(n))))
    end:
a:= n-> b(n, 5):
seq(a(n), n=5..75);  # Alois P. Heinz, Jan 31 2021

nmax = 75; CoefficientList[Series[(EllipticTheta[3, 0, x] - 1)^5/32, {x, 0, nmax}], x] // Drop[#, 5] &

G.f.: (theta_3(x) - 1)^5 / 32, where theta_3() is the Jacobi theta function.

A340906 Number of ways to write n as an ordered sum of 7 squares of positive integers.

Original entry on oeis.org

1, 0, 0, 7, 0, 0, 21, 0, 7, 35, 0, 42, 35, 0, 105, 28, 21, 140, 49, 105, 105, 106, 210, 84, 182, 210, 217, 287, 105, 420, 378, 126, 497, 392, 420, 532, 350, 630, 714, 434, 546, 980, 742, 609, 980, 896, 1071, 882, 875, 1470, 1239, 1099, 1155, 1722, 1652, 882, 1933, 1995, 1554, 2072, 1505

Offset: 7

Ilya Gutkovskiy, Jan 31 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..20000
Index entries for sequences related to sums of squares

Cf. A000290, A008451, A010052, A025431, A045849, A063691, A063725, A063730, A340481, A340905, A340915, A340946, A340947.

Column k=7 of A337165.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add((s->
      `if`(s>n, 0, b(n-s, t-1)))(j^2), j=1..isqrt(n))))
    end:
a:= n-> b(n, 7):
seq(a(n), n=7..67);  # Alois P. Heinz, Jan 31 2021

nmax = 67; CoefficientList[Series[(EllipticTheta[3, 0, x] - 1)^7/128, {x, 0, nmax}], x] // Drop[#, 7] &

G.f.: (theta_3(x) - 1)^7 / 128, where theta_3() is the Jacobi theta function.

A340915 Number of ways to write n as an ordered sum of 8 squares of positive integers.

Original entry on oeis.org

1, 0, 0, 8, 0, 0, 28, 0, 8, 56, 0, 56, 70, 0, 168, 64, 28, 280, 84, 168, 280, 176, 420, 224, 345, 560, 392, 616, 420, 848, 924, 336, 1246, 1064, 868, 1464, 988, 1680, 1820, 1120, 1904, 2464, 1932, 1904, 2870, 2752, 2772, 2912, 2892, 4256, 3640, 3248, 4480, 5040, 4760, 3696, 6120

Offset: 8

Ilya Gutkovskiy, Jan 31 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..20000
Index entries for sequences related to sums of squares

Cf. A000143, A000290, A010052, A025432, A045850, A063691, A063725, A063730, A340481, A340905, A340906, A340946, A340947.

Column k=8 of A337165.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add((s->
      `if`(s>n, 0, b(n-s, t-1)))(j^2), j=1..isqrt(n))))
    end:
a:= n-> b(n, 8):
seq(a(n), n=8..64);  # Alois P. Heinz, Jan 31 2021

nmax = 64; CoefficientList[Series[(EllipticTheta[3, 0, x] - 1)^8/256, {x, 0, nmax}], x] // Drop[#, 8] &

G.f.: (theta_3(x) - 1)^8 / 256, where theta_3() is the Jacobi theta function.

A340947 Number of ways to write n as an ordered sum of 10 squares of positive integers.

Original entry on oeis.org

1, 0, 0, 10, 0, 0, 45, 0, 10, 120, 0, 90, 210, 0, 360, 262, 45, 840, 300, 360, 1260, 480, 1260, 1350, 1015, 2520, 1560, 2200, 3150, 2880, 4186, 2880, 5430, 6240, 3780, 8300, 7080, 7920, 11160, 7320, 13257, 14640, 10600, 16470, 18570, 18240, 19620, 22230, 25135, 27720, 28020, 28480, 38160

Offset: 10

Ilya Gutkovskiy, Jan 31 2021

nonn

David A. Corneth, Table of n, a(n) for n = 10..10009
Index entries for sequences related to sums of squares

Cf. A000144, A000290, A010052, A025434, A045852, A063691, A063725, A063730, A340481, A340905, A340906, A340915, A340946.

Column k=10 of A337165.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add((s->
      `if`(s>n, 0, b(n-s, t-1)))(j^2), j=1..isqrt(n))))
    end:
a:= n-> b(n, 10):
seq(a(n), n=10..62);  # Alois P. Heinz, Jan 31 2021

nmax = 62; CoefficientList[Series[(EllipticTheta[3, 0, x] - 1)^10/1024, {x, 0, nmax}], x] // Drop[#, 10] &

G.f.: (theta_3(x) - 1)^10 / 1024, where theta_3() is the Jacobi theta function.

A025444 Number of partitions of n into 5 distinct nonzero squares.

Original entry on oeis.org

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

Offset: 0

David W. Wilson

			a(111) = 2 via 1 + 4 + 9 + 16 + 81 = 1 + 9 + 16 + 36 + 49. - _David A. Corneth_, Feb 02 2021

David A. Corneth, Table of n, a(n) for n = 0..9999
Index entries for sequences related to sums of squares

Cf. A000290, A008452, A010052, A025433, A025441, A025442, A025443, A025444, A045851, A340946, A340988, A340998, A340999, A341000, A341001.

Column k=5 of A341040.

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)

a(n) = [x^n y^5] Product_{k>=1} (1 + y*x^(k^2)). - Ilya Gutkovskiy, Apr 22 2019

A341370 Expansion of (1 / theta_4(x) - 1)^9 / 512.

Original entry on oeis.org

1, 18, 180, 1311, 7740, 39204, 176388, 721530, 2728053, 9651056, 32246892, 102515508, 311923386, 912771468, 2579132196, 7060677537, 18781247700, 48660380190, 123061973176, 304351869708, 737293187286, 1752035386188, 4089222211212, 9384936015492, 21201250825554

Offset: 9

Ilya Gutkovskiy, Feb 10 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 9..10000

Cf. A002448, A004410, A014968, A015128, A327387, A338223, A340946, A341228, A341364, A341365, A341366, A341367, A341368, A341369.

g:= proc(n, i) option remember; `if`(n=0, 1/2, `if`(i=1, 0,
      g(n, i-1))+add(2*g(n-i*j, i-1), j=`if`(i=1, n, 1)..n/i))
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0,
      g(n$2)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 9):
seq(a(n), n=9..33);  # Alois P. Heinz, Feb 10 2021

nmax = 33; CoefficientList[Series[(1/EllipticTheta[4, 0, x] - 1)^9/512, {x, 0, nmax}], x] // Drop[#, 9] &
nmax = 33; CoefficientList[Series[(1/512) (-1 + Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}])^9, {x, 0, nmax}], x] // Drop[#, 9] &

G.f.: (1/512) * (-1 + Product_{k>=1} (1 + x^k) / (1 - x^k))^9.

A341000 Number of partitions of n into 9 distinct nonzero squares.

Original entry on oeis.org

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

Offset: 285

Ilya Gutkovskiy, Feb 02 2021

nonn

			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

David A. Corneth, Table of n, a(n) for n = 285..5000
Index entries for sequences related to sums of squares

Cf. A000290, A008452, A010052, A025433, A025441, A025442, A025443, A025444, A045851, A340946, A340988, A340998, A340999, A341001.

Column k=9 of A341040.

A341428 Number of positive solutions to (x_1)^2 + (x_2)^2 + ... + (x_9)^2 <= n^2.

Original entry on oeis.org

1, 46, 760, 7751, 43910, 186098, 652710, 1943742, 5178030, 12411211, 27773308, 57798904, 114152429, 214399664, 387571706, 673189698, 1135916808, 1857320784, 2966816950, 4623984661, 7066527283, 10577150039, 15589368584, 22580091614, 32256768126

Offset: 3

Ilya Gutkovskiy, Feb 11 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..1000
Index entries for sequences related to sums of squares

Cf. A000122, A001182, A055408, A055415, A253663, A302995, A340946, A341398, A341404, A341423, A341424, A341425, A341426, A341427, A341429.

b:= proc(n, k) option remember; `if`(k=0, 1, `if`(n=0, 0,
      add((s->`if`(s>n, 0, b(n-s, k-1)))(j^2), j=1..isqrt(n))))
    end:
a:= n-> b(n^2, 9):
seq(a(n), n=3..27);  # Alois P. Heinz, Feb 11 2021

Table[SeriesCoefficient[(EllipticTheta[3, 0, x] - 1)^9/(512 (1 - x)), {x, 0, n^2}], {n, 3, 27}]

a(n) is the coefficient of x^(n^2) in expansion of (theta_3(x) - 1)^9 / (512 * (1 - x)).

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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A340905 Number of ways to write n as an ordered sum of 6 squares of positive integers.

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A340481 Number of ways to write n as an ordered sum of 5 squares of positive integers.

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A340906 Number of ways to write n as an ordered sum of 7 squares of positive integers.

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A340915 Number of ways to write n as an ordered sum of 8 squares of positive integers.

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A340947 Number of ways to write n as an ordered sum of 10 squares of positive integers.

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A025444 Number of partitions of n into 5 distinct nonzero squares.

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A341370 Expansion of (1 / theta_4(x) - 1)^9 / 512.

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