A103265 -id:A103265 - cp's OEIS frontend

A161091 Number of partitions of n into squares where every part appears at least 3 times.

0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 2, 2, 3, 4, 3, 3, 4, 5, 4, 4, 6, 6, 5, 6, 7, 8, 7, 7, 8, 10, 8, 8, 11, 11, 10, 11, 13, 13, 13, 13, 15, 18, 15, 16, 20, 21, 19, 21, 23, 24, 24, 24, 27, 30, 28, 28, 33, 36, 33, 35, 39, 42, 41, 42, 45, 49, 47, 48, 55, 56, 54, 58, 63, 67, 65, 66, 72, 78

Offset: 1

R. H. Hardin, Jun 02 2009

nonn

			a(23)=4 because we have (4^5)(1^3), (4^4)(1^7), (4^3)(1^11), and (1^23). - _Emeric Deutsch_, Jun 24 2009

R. H. Hardin, Table of n, a(n) for n = 1..1000

Cf. A001156, A103265.

g := product(1+x^(3*j^2)/(1-x^(j^2)), j = 1 .. 20): gser := series(g, x = 0, 90): seq(coeff(gser, x, n), n = 2 .. 84); # Emeric Deutsch, Jun 24 2009

nmax = 100; Rest[CoefficientList[Series[Product[(1 + x^(3*k^2)/(1-x^(k^2))), {k, 1, Sqrt[nmax]+1}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 15 2025 *)

G.f.: Product_{j>=1} (1 + x^(3j^2)/(1-x^(j^2))). - Emeric Deutsch, Jun 24 2009

A300414 Expansion of Product_{k>=2} (1 + x^Fibonacci(k))/(1 - x^Fibonacci(k)).

Original entry on oeis.org

1, 2, 4, 8, 12, 20, 30, 42, 62, 84, 114, 154, 198, 260, 332, 418, 530, 654, 810, 994, 1202, 1462, 1752, 2094, 2500, 2948, 3486, 4092, 4776, 5582, 6468, 7490, 8650, 9928, 11406, 13036, 14862, 16934, 19196, 21758, 24592, 27706, 31216, 35038, 39284, 43990, 49100, 54798, 61008, 67798

Offset: 0

Ilya Gutkovskiy, Mar 05 2018

nonn

Convolution of the sequences A000119 and A003107.

Index entries for related partition-counting sequences

Cf. A000045, A000119, A003107, A015128, A080054, A103265, A280263, A280366, A300413, A300415.

nmax = 49; CoefficientList[Series[Product[(1 + x^Fibonacci[k])/(1 - x^Fibonacci[k]), {k, 2, 20}], {x, 0, nmax}], x]

G.f.: Product_{k>=2} (1 + x^A000045(k))/(1 - x^A000045(k)).

A319243 Expansion of Product_{k>0} (1 - x^(k^2))/(1 + x^(k^2)).

Original entry on oeis.org

1, -2, 2, -2, 0, 2, -2, 2, 0, -4, 6, -6, 4, 2, -6, 6, -6, 2, 4, -6, 8, -6, -2, 6, -8, 8, -2, -4, 8, -10, 6, 2, -6, 6, -2, -6, 8, -6, 2, 6, -8, 6, -2, -6, 8, -4, -2, 10, -14, 8, 0, -6, 16, -14, 4, 2, -16, 18, -2, -10, 24, -30, 14, 4, -20, 34, -26, 2, 16, -30, 26, 2, -20

Offset: 0

Seiichi Manyama, Sep 15 2018

sign

Product_{k>0} (1 - x^(k^m))/(1 + x^(k^m)): A002448 (m=1), this sequence (m=2), A319244 (m=3).

Cf. A103265.

N=99; x='x+O('x^N); Vec(prod(k=1, sqrtint(N), (1-x^(k^2))/(1+x^(k^2))))

Convolution inverse of A103265.

A361805 Expansion of Product_{j=1..n, k=1..n} (1 + x^(k^j)) / (1 - x^(k^j)).

Original entry on oeis.org

1, 2, 10, 52, 278, 1508, 8262, 45604, 253186, 1412196, 7906866, 44411420, 250124308, 1411963200, 7986664250, 45255888828, 256840959728, 1459686175768, 8306130772008, 47318321533008, 269839722667800, 1540242835509060, 8799238591245006, 50308756959106988

Offset: 0

Vaclav Kotesovec, Jan 28 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A369577, A369578.
Cf. A015128, A103265, A280263.
Cf. A000041, A001156, A003108.
Cf. A000009, A033461, A279329.

Table[SeriesCoefficient[Product[Product[(1+x^(k^j))/(1-x^(k^j)), {k, 1, n^(1/j)}], {j, 1, n}], {x, 0, n}], {n, 0, 40}]

a(n) ~ c * (1 + sqrt(2))^(2*n) / sqrt(n), where c = 0.6431307610999754935775134585988078560575016233514072350040712130921818...

cp's OEIS Frontend

A161091 Number of partitions of n into squares where every part appears at least 3 times.

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A300414 Expansion of Product_{k>=2} (1 + x^Fibonacci(k))/(1 - x^Fibonacci(k)).

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A319243 Expansion of Product_{k>0} (1 - x^(k^2))/(1 + x^(k^2)).

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A361805 Expansion of Product_{j=1..n, k=1..n} (1 + x^(k^j)) / (1 - x^(k^j)).

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