A038671 -id:A038671 - cp's OEIS frontend

A045847 Matrix whose (i,j)-th entry is number of representations of j as a sum of i squares of nonnegative integers; read by diagonals.

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, 1, 0, 1, 10, 36, 56, 42, 36, 30, 4, 3, 2, 0, 0, 1, 11, 45, 84, 78, 63, 61, 20, 6, 6, 2, 0, 0

Offset: 0

N. J. A. Sloane

			Rows are
1,0,0,..;
1,1,0,0,1,0..;
1,2,1,0,2,2,..;
1,3,3,1,...

Seiichi Manyama, Ascending antidiagonals n = 0..139, flattened
H. Wilf, A combinatorial determinant, arXiv:math/9809120 [math.CO], 1998.

Rows k=0-10 give: A000007, A010052, A000925, A002102, A014110, A038671, A045848, A045849, A045850, A045851, A045852.
Diagonal gives A287617.
Antidiagonal sums give A302018.

i-th row is expansion of (1+x+x^4+x^9+...)^i.

More terms from Erich Friedman

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.

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

A341400 Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_5)^2 <= n.

Original entry on oeis.org

1, 6, 16, 26, 36, 57, 87, 107, 122, 157, 207, 247, 277, 322, 392, 452, 482, 537, 637, 717, 773, 863, 973, 1053, 1113, 1203, 1343, 1473, 1553, 1668, 1858, 1998, 2053, 2173, 2373, 2543, 2673, 2818, 3018, 3218, 3338, 3483, 3753, 3973, 4113, 4344, 4634, 4834, 4944, 5139, 5449

Offset: 0

Ilya Gutkovskiy, Feb 10 2021

nonn

Partial sums of A038671.

Index entries for sequences related to sums of squares

Cf. A000122, A000606, A003059, A038671, A055404, A055411, A175360, A224212, A224213, A302862, A341401, A341402.

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:= proc(n) option remember; b(n, 5)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..50);  # Alois P. Heinz, Feb 10 2021

nmax = 50; CoefficientList[Series[(1 + EllipticTheta[3, 0, x])^5/(32 (1 - x)), {x, 0, nmax}], x]

G.f.: (1 + theta_3(x))^5 / (32 * (1 - x)).

a(n^2) = A055404(n).

A280718 Expansion of (Sum_{k>=0} x^(k(3k-1)/2))^5.

Original entry on oeis.org

1, 5, 10, 10, 5, 6, 20, 30, 20, 5, 10, 30, 35, 30, 30, 30, 25, 30, 60, 60, 25, 5, 35, 80, 70, 51, 35, 50, 80, 90, 80, 30, 35, 60, 80, 95, 90, 90, 50, 75, 140, 140, 85, 20, 70, 120, 130, 120, 95, 115, 100, 115, 140, 155, 110, 40, 80, 200, 230, 140, 81, 120, 200, 190, 180, 120, 80, 100, 160, 240, 200, 155, 120, 140, 245, 260, 230

Offset: 0

Ilya Gutkovskiy, Feb 10 2017

nonn

Number of ways to write n as an ordered sum of 5 pentagonal numbers (A000326).
a(n) > 0 for all n >= 0.
Every number is the sum of at most 5 pentagonal numbers.
Every number is the sum of at most k k-gonal numbers (Fermat's polygonal number theorem).

			a(5) = 6 because we have:
[5, 0, 0, 0, 0]
[0, 5, 0, 0, 0]
[0, 0, 5, 0, 0]
[0, 0, 0, 5, 0]
[0, 0, 0, 0, 5]
[1, 1, 1, 1, 1]

Ilya Gutkovskiy, Extended graphical example
Eric Weisstein's World of Mathematics, Pentagonal Number
Index to sequences related to polygonal numbers

Cf. A000326, A003679, A008439, A038671, A280719, A282248.

nmax = 76; CoefficientList[Series[Sum[x^(k (3 k - 1)/2), {k, 0, nmax}]^5, {x, 0, nmax}], x]

G.f.: (Sum_{k>=0} x^(k*(3*k-1)/2))^5.

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A045847 Matrix whose (i,j)-th entry is number of representations of j as a sum of i squares of nonnegative integers; read by diagonals.

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

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A341400 Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_5)^2 <= n.

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Maple

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A280718 Expansion of (Sum_{k>=0} x^(k*(3*k-1)/2))^5.

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Mathematica

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A280718 Expansion of (Sum_{k>=0} x^(k(3k-1)/2))^5.