A045851 -id:A045851 - 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

A340946 Number of ways to write n as an ordered sum of 9 squares of positive integers.

Original entry on oeis.org

1, 0, 0, 9, 0, 0, 36, 0, 9, 84, 0, 72, 126, 0, 252, 135, 36, 504, 156, 252, 630, 288, 756, 576, 606, 1260, 756, 1207, 1260, 1584, 2052, 1008, 2727, 2688, 1764, 3663, 2718, 3816, 4608, 2853, 5418, 6048, 4620, 5868, 7506, 7464, 7308, 8442, 8958, 11088, 10404, 9684, 13986, 14184, 13020

Offset: 9

Ilya Gutkovskiy, Jan 31 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 9..20000
Index entries for sequences related to sums of squares

Cf. A000290, A008452, A010052, A025433, A045851, A063691, A063725, A063730, A340481, A340905, A340906, A340915, A340947.

Column k=9 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, 9):
seq(a(n), n=9..63);  # Alois P. Heinz, Jan 31 2021

nmax = 63; CoefficientList[Series[(EllipticTheta[3, 0, x] - 1)^9/512, {x, 0, nmax}], x] // Drop[#, 9] &

G.f.: (theta_3(x) - 1)^9 / 512, 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

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.

A341404 Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_9)^2 <= n.

Original entry on oeis.org

1, 10, 46, 130, 265, 463, 799, 1339, 2014, 2780, 3860, 5444, 7301, 9263, 11783, 15263, 19250, 23237, 27893, 34193, 41519, 48701, 56765, 67421, 79484, 91067, 103739, 119855, 138035, 155819, 174923, 198863, 225890, 251444, 277976, 311492, 349122, 384420, 421284

Offset: 0

Ilya Gutkovskiy, Feb 10 2021

nonn

Partial sums of A045851.

Index entries for sequences related to sums of squares

Cf. A000122, A000606, A003059, A045851, A055408, A055415, A224212, A224213, A302862, A341400, A341401, A341402, A341403, A341405.

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, 9)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..38);  # Alois P. Heinz, Feb 10 2021

nmax = 38; CoefficientList[Series[(1 + EllipticTheta[3, 0, x])^9/(512 (1 - x)), {x, 0, nmax}], x]

G.f.: (1 + theta_3(x))^9 / (512 * (1 - x)).

a(n^2) = A055408(n).

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.

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A340946 Number of ways to write n as an ordered sum of 9 squares of positive integers.

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

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

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A341404 Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_9)^2 <= n.

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