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 1-3 of 3 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).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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