A063730 -id:A063730 - cp's OEIS frontend

A341365 Expansion of (1 / theta_4(x) - 1)^4 / 16.

1, 8, 40, 156, 520, 1552, 4262, 10960, 26716, 62276, 139744, 303412, 640001, 1315832, 2644004, 5204044, 10052182, 19086348, 35672516, 65708116, 119409576, 214289116, 380068582, 666723748, 1157550524, 1990230968, 3390558072, 5726064688, 9590759624, 15938198484, 26289242026

Offset: 4

Ilya Gutkovskiy, Feb 10 2021

nonn

Robert Israel, Table of n, a(n) for n = 4..10000

Cf. A002448, A004405, A014968, A015128, A063730, A284286, A327382, A338223, A341222, A341364, A341366, A341367, A341368, A341369, A341370.

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, 4):
seq(a(n), n=4..34);  # Alois P. Heinz, Feb 10 2021

nmax = 34; CoefficientList[Series[(1/EllipticTheta[4, 0, x] - 1)^4/16, {x, 0, nmax}], x] // Drop[#, 4] &
nmax = 34; CoefficientList[Series[(1/16) (-1 + Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}])^4, {x, 0, nmax}], x] // Drop[#, 4] &

G.f.: (1/16) * (-1 + Product_{k>=1} (1 + x^k) / (1 - x^k))^4.

a(n) ~ A284286(n)/16. - Vaclav Kotesovec, Feb 20 2021

A341423 Number of positive solutions to (x_1)^2 + (x_2)^2 + (x_3)^2 + (x_4)^2 <= n^2.

Original entry on oeis.org

1, 5, 32, 94, 219, 437, 804, 1362, 2177, 3271, 4768, 6708, 9227, 12381, 16254, 20954, 26707, 33461, 41480, 50884, 61703, 74183, 88606, 104862, 123481, 144241, 167604, 193648, 222799, 254731, 290244, 329512, 372545, 419661, 470822, 526646, 587481, 653505

Offset: 2

Ilya Gutkovskiy, Feb 11 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..1000
Index entries for sequences related to sums of squares

Cf. A000122, A001182, A046895, A055403, A055410, A063730, A224213, A253663, A302995, A341424, A341425, A341426, A341427, A341428, 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, 4):
seq(a(n), n=2..39);  # Alois P. Heinz, Feb 11 2021

Table[SeriesCoefficient[(EllipticTheta[3, 0, x] - 1)^4/(16 (1 - x)), {x, 0, n^2}], {n, 2, 39}]

a(n) is the coefficient of x^(n^2) in expansion of (theta_3(x) - 1)^4 / (16 * (1 - x)).

A347711 Number of compositions (ordered partitions) of n into at most 4 squares.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 4, 1, 4, 8, 3, 5, 6, 6, 12, 2, 5, 16, 7, 8, 10, 15, 12, 3, 15, 14, 16, 12, 8, 30, 16, 1, 18, 23, 18, 17, 18, 21, 28, 8, 11, 42, 19, 15, 32, 30, 24, 5, 23, 39, 30, 20, 20, 48, 36, 6, 34, 44, 21, 36, 24, 36, 52, 2, 34, 60, 31, 23, 36, 66, 36

Offset: 0

Ilya Gutkovskiy, Sep 10 2021

nonn

Index entries for sequences related to sums of squares

Cf. A000290, A002654, A006456, A063730, A337165, A347710, A347712, A347713.

a(n) = Sum_{k=0..4} A337165(n,k). - Alois P. Heinz, Sep 10 2021

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A341365 Expansion of (1 / theta_4(x) - 1)^4 / 16.

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A341423 Number of positive solutions to (x_1)^2 + (x_2)^2 + (x_3)^2 + (x_4)^2 <= n^2.

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Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A347711 Number of compositions (ordered partitions) of n into at most 4 squares.

Views

Author

Keywords

Links

Crossrefs

Formula