A302019 -id:A302019 - cp's OEIS frontend

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.

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.

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

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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