cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A295086 Expansion of Product_{k>=1} 1/(1 + x^k)^(k*(3*k-1)/2).

Original entry on oeis.org

1, -1, -4, -8, 1, 24, 78, 111, 75, -249, -876, -1847, -2251, -871, 5170, 17052, 34742, 47176, 34576, -44016, -224561, -530104, -875149, -1030871, -475480, 1488315, 5658668, 12109163, 19411024, 22693048, 12926630, -24000623, -102605376, -230257606, -386964449
Offset: 0

Views

Author

Seiichi Manyama, Nov 15 2017

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = n*(3*n-1)/2, g(n) = -1.

Links

Crossrefs

Cf. A294846 (b=3), A284896 (b=4), this sequence (b=5), A295121 (b=6), A295122 (b=7), A295123 (b=8).

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(1/prod(k=1, N, (1+x^k)^(k*(3*k-1)/2)))

Formula

Convolution inverse of A294102.
G.f.: Product_{k>=1} 1/(1 + x^k)^A000326(k).
a(0) = 1 and a(n) = (1/(2*n)) * Sum_{k=1..n} b(k)*a(n-k) where b(n) = Sum_{d|n} d^2*(3*d-1)*(-1)^(n/d).

A295121 Expansion of Product_{k>=1} 1/(1 + x^k)^(k*(2*k-1)).

Original entry on oeis.org

1, -1, -5, -10, 3, 42, 124, 160, 15, -677, -1941, -3425, -2807, 3488, 21004, 49547, 77879, 63395, -65104, -406091, -988889, -1655508, -1779329, -145347, 5087175, 15405270, 30158849, 42617486, 36116136, -19457047, -161973496, -418712896, -759063566
Offset: 0

Views

Author

Seiichi Manyama, Nov 15 2017

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = n*(2*n-1), g(n) = -1.

Links

Crossrefs

Cf. A294846 (b=3), A284896 (b=4), A295086 (b=5), this sequence (b=6), A295122 (b=7), A295123 (b=8).

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(1/prod(k=1, N, (1+x^k)^(k*(2*k-1))))

Formula

Convolution inverse of A294836.
G.f.: Product_{k>=1} 1/(1 + x^k)^A000384(k).
a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} b(k)*a(n-k) where b(n) = Sum_{d|n} d^2*(2*d-1)*(-1)^(n/d).

A295122 Expansion of Product_{k>=1} 1/(1 + x^k)^(k*(5*k-3)/2).

Original entry on oeis.org

1, -1, -6, -12, 6, 65, 179, 202, -137, -1392, -3492, -5135, -1325, 15437, 52934, 101787, 116827, -16945, -462603, -1350732, -2475989, -2889620, -343236, 8559858, 26972213, 53099230, 72521956, 47535918, -86985043, -409729146, -952305325, -1577038736
Offset: 0

Views

Author

Seiichi Manyama, Nov 15 2017

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = n*(5*n-3)/2, g(n) = -1.

Links

Crossrefs

Cf. A294846 (b=3), A284896 (b=4), A295086 (b=5), A295121 (b=6), this sequence (b=7), A295123 (b=8).

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(1/prod(k=1, N, (1+x^k)^(k*(5*k-3)/2)))

Formula

Convolution inverse of A294837.
G.f.: Product_{k>=1} 1/(1 + x^k)^A000566(k).
a(0) = 1 and a(n) = (1/(2*n)) * Sum_{k=1..n} b(k)*a(n-k) where b(n) = Sum_{d|n} d^2*(5*d-3)*(-1)^(n/d).

A295123 Expansion of Product_{k>=1} 1/(1 + x^k)^(k*(3*k-2)).

Original entry on oeis.org

1, -1, -7, -14, 10, 93, 242, 229, -410, -2446, -5500, -6458, 4062, 38899, 104715, 165843, 103045, -327200, -1393131, -3075317, -4305200, -2069461, 9129361, 35219829, 75832840, 109569915, 74818084, -143480059, -686408279, -1607860793, -2614721006, -2674073316
Offset: 0

Views

Author

Seiichi Manyama, Nov 15 2017

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = n*(3*n-2), g(n) = -1.

Links

Crossrefs

Cf. A294846 (b=3), A284896 (b=4), A295086 (b=5), A295121 (b=6), A295122 (b=7), this sequence (b=8).

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(1/prod(k=1, N, (1+x^k)^(k*(3*k-2))))

Formula

Convolution inverse of A294838.
G.f.: Product_{k>=1} 1/(1 + x^k)^A000567(k).
a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} b(k)*a(n-k) where b(n) = Sum_{d|n} d^2*(3*d-2)*(-1)^(n/d).

A305255 a(n) = [x^n] exp(Sum_{k>=1} (-1)^k*x^k/(k*(1 - x^k)^n)).

Original entry on oeis.org

1, -1, -1, -4, -3, 14, 240, 1686, 9479, 36761, 3412, -1951731, -27296124, -268495319, -2093667873, -11586874946, -3788945531, 1127535019748, 21900095232973, 297591401221473, 3270627818325128, 28116733997044842, 129815302615081267, -1568168714539146596, -59839621829784309343
Offset: 0

Views

Author

Ilya Gutkovskiy, May 28 2018

Keywords

Links

Crossrefs

Cf. A255528, A293554, A294846, A305205, A305206.

Programs

  • Mathematica
    Table[SeriesCoefficient[Exp[Sum[(-1)^k x^k/(k (1 - x^k)^n), {k, 1, n}]], {x, 0, n}], {n, 0, 24}]
Table[SeriesCoefficient[Product[1/(1 + x^k)^Binomial[n + k - 2, n - 1], {k, 1, n}], {x, 0, n}], {n, 0, 24}]

Formula

a(n) = [x^n] Product_{k>=1} 1/(1 + x^k)^binomial(n+k-2,n-1).
Showing 1-5 of 5 results.

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