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 1-2 of 2 results.

A261647 Expansion of Product_{k>=0} ((1+x^(2*k+1))/(1-x^(2*k+1)))^3.

Original entry on oeis.org

1, 6, 18, 44, 102, 216, 428, 816, 1494, 2650, 4584, 7740, 12804, 20808, 33264, 52400, 81462, 125100, 189966, 285516, 425016, 627040, 917436, 1331856, 1919332, 2746926, 3905784, 5519352, 7754064, 10833192, 15055216, 20817600, 28647414, 39241336, 53517060
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 28 2015

Keywords

Links

Crossrefs

Cf. A015128, A156616.
Cf. A080054, A007096, A014969, A261648, A014970, A014972, A103261.
Cf. A261610, A261649, A261651.
Cf. A261611, A261650, A261652.

Programs

  • Mathematica
    nmax=60; CoefficientList[Series[Product[((1+x^(2*k+1))/(1-x^(2*k+1)))^3,{k,0,nmax}],{x,0,nmax}],x]

Formula

a(n) ~ exp(Pi*sqrt(3*n/2)) * 3^(1/4) / (8 * 2^(1/4) * n^(3/4)).

A261652 Expansion of Product_{k>=0} ((1+x^(4*k+1))/(1-x^(4*k+1)))^3.

Original entry on oeis.org

1, 6, 18, 38, 66, 108, 182, 306, 486, 728, 1068, 1578, 2318, 3312, 4614, 6388, 8862, 12192, 16488, 22038, 29400, 39156, 51702, 67554, 87810, 113982, 147384, 189200, 241446, 307356, 390408, 493662, 621006, 778712, 974628, 1216284, 1511756, 1872840, 2315538
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 28 2015

Keywords

Comments

In general, if j > 0, a > 0, b > 0, GCD(a,b) = 1 and g.f. = Product_{k>=0} ((1 + x^(a*k+b))/(1 - x^(a*k+b)))^j, then a(n) ~ Gamma(b/a)^j * 2^(j/2 - 3/2 - 2*b*j/a) * a^(-j/4 - 1/4 + b*j/(2*a)) * exp(Pi*sqrt(j*n/a)) * j^(1/4 - j/4 + b*j/(2*a)) * Pi^(b*j/a - j) * n^(j/4 - 3/4 - b*j/(2*a)).

Crossrefs

Cf. A015128 (a=1, b=1, j=1), A156616.
Cf. A080054 (a=2, b=1, j=1), A007096 (a=2, b=1, j=2), A261647 (a=2, b=1, j=3), A014969 (a=2, b=1, j=4), A261648 (a=2, b=1, j=5), A014970 (a=2, b=1, j=6), A014972 (a=2, b=1, j=8), A103261 (a=2, b=1, j=10).
Cf. A261610 (a=3, b=1, j=1), A261649 (a=3, b=1, j=2), A261651 (a=3, b=1, j=3).
Cf. A261611 (a=4, b=1, j=1), A261650 (a=4, b=1, j=2), A261652 (a=4, b=1, j=3).

Programs

  • Mathematica
    nmax=60; CoefficientList[Series[Product[((1+x^(4*k+1))/(1-x^(4*k+1)))^3,{k,0,nmax}],{x,0,nmax}],x]

Formula

a(n) ~ exp(Pi*sqrt(3*n)/2) * 2^(1/4) * Gamma(1/4)^3 / (8 * 3^(1/8) * Pi^(9/4) * n^(3/8)).
Showing 1-2 of 2 results.

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

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