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.

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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).

A281591 First differences of A281590.

Original entry on oeis.org

3, 5, 6, 5, 7, 6, 8, 7, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 10, 11, 12, 11, 11, 12, 11, 12, 12, 12, 12, 13, 12, 13, 12, 13, 13, 13, 13, 13, 13, 13, 14, 13, 14, 13, 14, 14, 14, 14, 14, 14, 15, 14, 15, 14, 15, 14, 15, 15, 15, 15, 15, 15, 16
Offset: 1

Views

Author

Vaclav Kotesovec, Apr 14 2017

Keywords

Links

Crossrefs

Cf. A284896, A281590.

Programs

  • Mathematica
    nmax = 1000; A284896 = Rest[CoefficientList[Series[Product[1/(1 + x^k)^(k^2), {k, 1, nmax}], {x, 0, nmax}], x]]; csign = {1}; Do[If[(A284896[[n]] < 0 && A284896[[n+1]] >= 0) || (A284896[[n]] >= 0 && A284896[[n+1]] < 0), csign = Flatten[{csign, n + 1}]], {n, 1, Length[A284896] - 1}]; Differences[csign]

A284993 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1+x^j)^(j^k) in powers of x.

Original entry on oeis.org

1, 1, -1, 1, -1, 0, 1, -1, -1, -1, 1, -1, -3, -2, 1, 1, -1, -7, -6, 1, -1, 1, -1, -15, -20, 0, 0, 1, 1, -1, -31, -66, -8, 11, 4, -1, 1, -1, -63, -212, -54, 99, 42, 2, 2, 1, -1, -127, -666, -284, 725, 455, 63, 8, -2, 1, -1, -255, -2060, -1350, 4935, 4580, 958, 73
Offset: 0

Views

Author

Seiichi Manyama, Apr 07 2017

Keywords

Examples

			Square array begins:
   1,  1,  1,   1,   1,    1, ...
  -1, -1, -1,  -1,  -1,   -1, ...
   0, -1, -3,  -7, -15,  -31, ...
  -1, -2, -6, -20, -66, -212, ...
   1,  1,  0,  -8, -54, -284, ...

Links

Crossrefs

Columns k=0-5 give A081362, A255528, A284896, A284897, A284898, A284899.
Cf. A144048, A284992.

Formula

G.f. of column k: Product_{j>=1} 1/(1+x^j)^(j^k).
Previous Showing 11-13 of 13 results.

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

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