A045847 -id:A045847 - cp's OEIS frontend

A287617 Main diagonal of A045847.

1, 1, 1, 1, 5, 21, 61, 141, 309, 766, 2191, 6436, 18041, 48348, 128311, 347166, 960197, 2674883, 7413514, 20411986, 56130675, 154907229, 429349768, 1192532452, 3312339849, 9196063371, 25538113056, 70994900341, 197595884071, 550414483911, 1533911418946

Offset: 0

Seiichi Manyama, May 28 2017

nonn

a(n) is the number of representations of n as a sum of n squares of nonnegative integers.

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A045847.

a(n) = [x^n] (1 + theta_3(x))^n/2^n, where theta_() is the Jacobi theta function. - Ilya Gutkovskiy, Jan 17 2018

a(n) ~ c * d^n / sqrt(n), where d = 2.83312434251238... and c = 0.229098651125... - Vaclav Kotesovec, Mar 25 2023

A290054 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Sum_{j>=0} x^(j^3))^k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 0, 0, 1, 4, 3, 0, 0, 0, 1, 5, 6, 1, 0, 0, 0, 1, 6, 10, 4, 0, 0, 0, 0, 1, 7, 15, 10, 1, 0, 0, 0, 0, 1, 8, 21, 20, 5, 0, 0, 0, 1, 0, 1, 9, 28, 35, 15, 1, 0, 0, 2, 0, 0, 1, 10, 36, 56, 35, 6, 0, 0, 3, 2, 0, 0, 1, 11, 45, 84, 70, 21, 1, 0, 4, 6, 0, 0, 0

Offset: 0

Ilya Gutkovskiy, Jul 19 2017

A(n,k) is the number of ways of writing n as a sum of k nonnegative cubes.

			Square array begins:
1,  1,  1,  1,  1,   1,  ...
0,  1,  2,  3,  4,   5,  ...
0,  0,  1,  3,  6,  10,  ...
0,  0,  0,  1,  4,  10,  ...
0,  0,  0,  0,  1,   5,  ...
0,  0,  0,  0,  0,   1,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Index entries for sequences related to sums of cubes

Columns k=0-9 give: A000007, A010057, A173677, A051343, A173678, A173679, A173680, A173676, A173681, A173682.
Main diagonal gives A291700.
Antidiagonal sums give A302019.
Cf. A045847, A122141, A286815.

Table[Function[k, SeriesCoefficient[Sum[x^i^3, {i, 0, n}]^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten

G.f. of column k: (Sum_{j>=0} x^(j^3))^k.

A045849 Number of nonnegative solutions of x1^2 + x2^2 + ... + x7^2 = n.

Original entry on oeis.org

1, 7, 21, 35, 42, 63, 112, 141, 126, 154, 259, 315, 280, 308, 462, 567, 497, 462, 693, 910, 798, 749, 1078, 1281, 1092, 1043, 1407, 1715, 1576, 1449, 1946, 2422, 2016, 1687, 2429, 3045, 2604, 2345, 3066

Offset: 0

N. J. A. Sloane

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..2000 from T. D. Noe)
Index entries for sequences related to sums of squares

Cf. A010052, A038671, A045847.

(1 + EllipticTheta[3, 0, q])^7/128 + O[q]^50 // CoefficientList[#, q]& (* Jean-François Alcover, Aug 26 2019 *)

seq(n)=Vec((sum(k=0, sqrtint(n), x^(k^2)) + O(x*x^n))^7) \\ Andrew Howroyd, Aug 08 2018

def mul(f_ary, b_ary, m)
  s1, s2 = f_ary.size, b_ary.size
  ary = Array.new(s1 + s2 - 1, 0)
  (0..s1 - 1).each{|i|
    (0..s2 - 1).each{|j|
      ary[i + j] += f_ary[i] * b_ary[j]
    }
  }
  ary[0..m]
end
def power(ary, n, m)
  if n == 0
    a = Array.new(m + 1, 0)
    a[0] = 1
    return a
  end
  k = power(ary, n >> 1, m)
  k = mul(k, k, m)
  return k if n & 1 == 0
  return mul(k, ary, m)
end
def A(k, n)
  ary = Array.new(n + 1, 0)
  (0..Math.sqrt(n).to_i).each{|i| ary[i * i] = 1}
  power(ary, k, n)
end
p A(7, 100) # Seiichi Manyama, May 28 2017

Coefficient of q^n in (1 + q + q^4 + q^9 + q^16 + q^25 + q^36 + q^49 + q^64 + ...)^7.

G.f.: (1 + theta_3(q))^7/128, where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 08 2018

A045851 Number of nonnegative solutions of x1^2 + x2^2 + ... + x9^2 = n.

Original entry on oeis.org

1, 9, 36, 84, 135, 198, 336, 540, 675, 766, 1080, 1584, 1857, 1962, 2520, 3480, 3987, 3987, 4656, 6300, 7326, 7182, 8064, 10656, 12063, 11583, 12672, 16116, 18180, 17784, 19104, 23940, 27027, 25554

Offset: 0

N. J. A. Sloane

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..2000 from T. D. Noe)
Index entries for sequences related to sums of squares

Cf. A010052, A045847.

(1 + EllipticTheta[3, 0, q])^9/512 + O[q]^40 // CoefficientList[#, q]& (* Jean-François Alcover, Aug 26 2019 *)

seq(n)=Vec((sum(k=0, sqrtint(n), x^(k^2)) + O(x*x^n))^9) \\ Andrew Howroyd, Aug 08 2018

def mul(f_ary, b_ary, m)
  s1, s2 = f_ary.size, b_ary.size
  ary = Array.new(s1 + s2 - 1, 0)
  (0..s1 - 1).each{|i|
    (0..s2 - 1).each{|j|
      ary[i + j] += f_ary[i] * b_ary[j]
    }
  }
  ary[0..m]
end
def power(ary, n, m)
  if n == 0
    a = Array.new(m + 1, 0)
    a[0] = 1
    return a
  end
  k = power(ary, n >> 1, m)
  k = mul(k, k, m)
  return k if n & 1 == 0
  return mul(k, ary, m)
end
def A(k, n)
  ary = Array.new(n + 1, 0)
  (0..Math.sqrt(n).to_i).each{|i| ary[i * i] = 1}
  power(ary, k, n)
end
p A(9, 100) # Seiichi Manyama, May 28 2017

Coefficient of q^n in (1 + q + q^4 + q^9 + q^16 + q^25 + q^36 + q^49 + q^64 + ...)^9.

G.f.: (1 + theta_3(q))^9/512, where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 08 2018

A045850 Number of nonnegative solutions of x1^2 + x2^2 + ... + x8^2 = n.

Original entry on oeis.org

1, 8, 28, 56, 78, 112, 196, 288, 309, 344, 532, 736, 756, 784, 1092, 1456, 1486, 1400, 1792, 2464, 2562, 2352, 2940, 3872, 3864, 3536, 4256, 5432, 5608, 5216, 6076, 7840, 7933, 6832, 7952, 10472, 10494

Offset: 0

N. J. A. Sloane

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..2000 from T. D. Noe)
Index entries for sequences related to sums of squares

Cf. A010052, A045847.

(1 + EllipticTheta[3, 0, q])^8/256 + O[q]^40 // CoefficientList[#, q]& (* Jean-François Alcover, Aug 26 2019 *)

seq(n)=Vec((sum(k=0, sqrtint(n), x^(k^2)) + O(x*x^n))^8) \\ Andrew Howroyd, Aug 08 2018

def mul(f_ary, b_ary, m)
  s1, s2 = f_ary.size, b_ary.size
  ary = Array.new(s1 + s2 - 1, 0)
  (0..s1 - 1).each{|i|
    (0..s2 - 1).each{|j|
      ary[i + j] += f_ary[i] * b_ary[j]
    }
  }
  ary[0..m]
end
def power(ary, n, m)
  if n == 0
    a = Array.new(m + 1, 0)
    a[0] = 1
    return a
  end
  k = power(ary, n >> 1, m)
  k = mul(k, k, m)
  return k if n & 1 == 0
  return mul(k, ary, m)
end
def A(k, n)
  ary = Array.new(n + 1, 0)
  (0..Math.sqrt(n).to_i).each{|i| ary[i * i] = 1}
  power(ary, k, n)
end
p A(8, 100) # Seiichi Manyama, May 28 2017

Coefficient of q^n in (1 + q + q^4 + q^9 + q^16 + q^25 + q^36 + q^49 + q^64 + ...)^8.

G.f.: (1 + theta_3(q))^8/256, where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 08 2018

A045852 Number of nonnegative solutions of x1^2 + x2^2 + ... + x10^2 = n.

Original entry on oeis.org

1, 10, 45, 120, 220, 342, 570, 960, 1350, 1640, 2191, 3240, 4200, 4720, 5760, 7920, 9865, 10620, 11965, 15600, 19332, 20550, 22200, 28080, 34200, 35582, 37395, 45720, 54600, 56970, 59460, 71040, 84330, 87090, 88195, 104040, 123200, 125710, 126540, 148560

Offset: 0

N. J. A. Sloane

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..2000 from T. D. Noe)

Cf. A010052, A045847.

b:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,
      b(n, k-1)+add(b(n-j^2, k-1), j=1..isqrt(n))))
    end:
a:= b(n, 10):
seq(a(n), n=0..40);  # Alois P. Heinz, Feb 10 2021

Take[CoefficientList[Expand[(Total[x^Range[0,5]^2])^10],x],50] (* Harvey P. Dale, May 20 2011 *)

def mul(f_ary, b_ary, m)
  s1, s2 = f_ary.size, b_ary.size
  ary = Array.new(s1 + s2 - 1, 0)
  (0..s1 - 1).each{|i|
    (0..s2 - 1).each{|j|
      ary[i + j] += f_ary[i] * b_ary[j]
    }
  }
  ary[0..m]
end
def power(ary, n, m)
  if n == 0
    a = Array.new(m + 1, 0)
    a[0] = 1
    return a
  end
  k = power(ary, n >> 1, m)
  k = mul(k, k, m)
  return k if n & 1 == 0
  return mul(k, ary, m)
end
def A(k, n)
  ary = Array.new(n + 1, 0)
  (0..Math.sqrt(n).to_i).each{|i| ary[i * i] = 1}
  power(ary, k, n)
end
p A(10, 100) # Seiichi Manyama, May 28 2017

Coefficient of q^n in (1 + q + q^4 + q^9 + q^16 + q^25 + q^36 + q^49 + q^64 + ...)^10.

G.f.: ((1 + theta_3(x)) / 2)^10. - Ilya Gutkovskiy, Feb 10 2021

A363778 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/(Sum_{j>=0} x^(j^2))^k.

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, 1, 0, 1, -3, 3, -1, 0, 1, -4, 6, -4, 0, 0, 1, -5, 10, -10, 3, 1, 0, 1, -6, 15, -20, 12, 0, -2, 0, 1, -7, 21, -35, 31, -9, -5, 3, 0, 1, -8, 28, -56, 65, -36, -2, 12, -3, 0, 1, -9, 36, -84, 120, -96, 24, 24, -18, 1, 0, 1, -10, 45, -120, 203, -210, 105, 20, -54, 18, 2, 0

Offset: 0

Seiichi Manyama, Jun 21 2023

			Square array begins:
  1,  1,  1,   1,   1,   1,    1, ...
  0, -1, -2,  -3,  -4,  -5,   -6, ...
  0,  1,  3,   6,  10,  15,   21, ...
  0, -1, -4, -10, -20, -35,  -56, ...
  0,  0,  3,  12,  31,  65,  120, ...
  0,  1,  0,  -9, -36, -96, -210, ...
  0, -2, -5,  -2,  24, 105,  294, ...

Columns k=0..3 give A000007, A317665, A363774, A363775.
Main diagonal gives A363780.
Cf. A045847, A162552, A363779.

T(0,k) = 1; T(n,k) = -(k/n) * Sum_{j=1..n} A162552(j) * T(n-j,k).

A290429 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Sum_{j>=0} x^(j(j+1)(j+2)/6))^k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 0, 0, 1, 4, 3, 0, 1, 0, 1, 5, 6, 1, 2, 0, 0, 1, 6, 10, 4, 3, 2, 0, 0, 1, 7, 15, 10, 5, 6, 0, 0, 0, 1, 8, 21, 20, 10, 12, 3, 0, 0, 0, 1, 9, 28, 35, 21, 21, 12, 0, 1, 0, 0, 1, 10, 36, 56, 42, 36, 30, 4, 3, 0, 1, 0, 1, 11, 45, 84, 78, 63, 61, 20, 6, 3, 2, 0, 0, 1, 12, 55, 120, 135, 112, 112, 60, 15, 12, 3, 2, 0, 0

Offset: 0

Ilya Gutkovskiy, Jul 31 2017

A(n,k) is the number of ways of writing n as a sum of k tetrahedral (or triangular pyramidal) numbers (A000292).

			Square array begins:
1,  1,  1,  1,   1,   1,  ...
0,  1,  2,  3,   4,   5,  ...
0,  0,  1,  3,   6,  10,  ...
0,  0,  0,  1,   4,  10,  ...
0,  1,  2,  3,   5,  10,  ...
0,  0,  2,  6,  12,  21,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Eric Weisstein's World of Mathematics, Tetrahedral Number
Index to sequences related to pyramidal numbers

Cf. A000292, A104246, A286180, A290054, A290430.
Cf. A000007 (column 0), A023533 (column 1), A282172 (column 5).
Main diagonal gives A303170.
Similar to, but different from, A045847.

Table[Function[k, SeriesCoefficient[Sum[x^(i (i + 1) (i + 2)/6), {i, 0, n}]^k, {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten

G.f. of column k: (Sum_{j>=0} x^(j*(j+1)*(j+2)/6))^k.

A290430 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Sum_{j>=0} x^(j(j+1)(2*j+1)/6))^k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 0, 0, 1, 4, 3, 0, 0, 0, 1, 5, 6, 1, 0, 1, 0, 1, 6, 10, 4, 0, 2, 0, 0, 1, 7, 15, 10, 1, 3, 2, 0, 0, 1, 8, 21, 20, 5, 4, 6, 0, 0, 0, 1, 9, 28, 35, 15, 6, 12, 3, 0, 0, 0, 1, 10, 36, 56, 35, 12, 20, 12, 0, 0, 0, 0, 1, 11, 45, 84, 70, 28, 31, 30, 4, 0, 1, 0, 0, 1, 12, 55, 120, 126, 64, 49, 60, 20, 0, 3, 0, 0, 0

Offset: 0

Ilya Gutkovskiy, Jul 31 2017

A(n,k) is the number of ways of writing n as a sum of k square pyramidal numbers (A000330).

			Square array begins:
1,  1,  1,  1,  1,   1,  ...
0,  1,  2,  3,  4,   5,  ...
0,  0,  1,  3,  6,  10,  ...
0,  0,  0,  1,  4,  10,  ...
0,  0,  0,  0,  1,   5,  ...
0,  1,  2,  3,  4,   6,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Eric Weisstein's World of Mathematics, Square Pyramidal Number
Index to sequences related to pyramidal numbers

Cf. A000330, A045847, A122141, A286815, A290429.
Cf. A000007 (column 0), A253903 (column 1), A282173 (column 6).
Main diagonal gives A303172.

Table[Function[k, SeriesCoefficient[Sum[x^(i (i + 1) (2 i + 1)/6), {i, 0, n}]^k, {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten

G.f. of column k: (Sum_{j>=0} x^(j*(j+1)*(2*j+1)/6))^k.

A302018 Expansion of 1/(1 - x*(1 + theta_3(x))/2), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 1, 2, 3, 5, 9, 15, 26, 44, 75, 129, 220, 377, 644, 1101, 1883, 3219, 5506, 9414, 16098, 27527, 47069, 80488, 137630, 235343, 402427, 688134, 1176685, 2012085, 3440591, 5883279, 10060183, 17202533, 29415676, 50299693, 86010564, 147074801, 251492331, 430042340, 735356089, 1257431006

Offset: 0

Ilya Gutkovskiy, Mar 30 2018

nonn

N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Jacobi Theta Functions
Index entries for sequences related to sums of squares

Antidiagonal sums of A045847.

Cf. A010052, A032803, A181649, A302019.

nmax = 40; CoefficientList[Series[1/(1 - x (1 + EllipticTheta[3, 0, x])/2), {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[1/(1 - x Sum[x^k^2, {k, 0, nmax}]), {x, 0, nmax}], x]

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

a(0) = 1; a(n) = Sum_{k=1..n} A010052(k-1)*a(n-k).

cp's OEIS Frontend

A287617 Main diagonal of A045847.

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A290054 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Sum_{j>=0} x^(j^3))^k.

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Mathematica

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A045849 Number of nonnegative solutions of x1^2 + x2^2 + ... + x7^2 = n.

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Mathematica

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A045851 Number of nonnegative solutions of x1^2 + x2^2 + ... + x9^2 = n.

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Mathematica

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A045850 Number of nonnegative solutions of x1^2 + x2^2 + ... + x8^2 = n.

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Mathematica

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Ruby

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A045852 Number of nonnegative solutions of x1^2 + x2^2 + ... + x10^2 = n.

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Maple

Mathematica

Ruby

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A363778 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/(Sum_{j>=0} x^(j^2))^k.

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A290429 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Sum_{j>=0} x^(j*(j+1)*(j+2)/6))^k.

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A290429 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Sum_{j>=0} x^(j(j+1)(j+2)/6))^k.

A290430 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Sum_{j>=0} x^(j(j+1)(2*j+1)/6))^k.