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.

A071109 Expansion of Product_{k>=1} 1/(1+2*x^k).

Original entry on oeis.org

1, -2, 2, -6, 14, -26, 50, -102, 214, -426, 834, -1678, 3398, -6778, 13482, -27022, 54198, -108306, 216346, -432878, 866334, -1732386, 3463626, -6927926, 13858350, -27715378, 55426002, -110855030, 221719582, -443433610, 886848930, -1773709078, 3547455846
Offset: 0

Views

Author

Sharon Sela (sharonsela(AT)hotmail.com), May 27 2002

Keywords

Links

Crossrefs

Cf. A070933, A032302, A000700, A246935, A261562, A261566, A261582.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[Product[1/(1 + 2*x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2015 *)
nmax = 40; CoefficientList[Series[Exp[Sum[(-1)^k*2^k/k*x^k/(1-x^k), {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2015 *)
(O[x]^30 + 3/QPochhammer[-2, x])[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)

Formula

a(n) ~ c * (-2)^n, where c = Product_{j>=1} 1/(1-1/(-2)^j) = 1/QPochhammer[-1/2,-1/2] = 0.8259519860658427384636116224100201356301... . - Vaclav Kotesovec, Aug 25 2015
G.f.: Sum_{i>=0} (-2)^i*x^i/Product_{j=1..i} (1 - x^j). - Ilya Gutkovskiy, Apr 13 2018

Extensions

More terms from Vaclav Kotesovec, Aug 25 2015

A261567 Expansion of Product_{k>=1} (1/(1 + 3*x^k))^k.

Original entry on oeis.org

1, -3, 3, -18, 69, -168, 504, -1578, 4800, -14310, 42396, -128049, 385839, -1154271, 3458847, -10386477, 31173873, -93490386, 280426833, -841384614, 2524300014, -7572585150, 22717270491, -68152872885, 204460229394, -613377236379, 1840126774737, -5520391488054
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 24 2015

Keywords

Comments

In general, for z > 1 or z < -1, if g.f. = Product_{k>=1} (1/(1 - z*x^k))^k, then a(n) ~ c * z^n, where c = Product_{j>=1} 1/(1 - 1/z^j)^(j+1).

Links

Crossrefs

Cf. A255528, A261566, A261582.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[Product[(1/(1 + 3*x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Exp[Sum[(-1)^k*3^k/k*x^k/(1-x^k)^2, {k, 1, nmax}]], {x, 0, nmax}], x]

Formula

a(n) ~ c * (-3)^n, where c = Product_{j>=1} 1/(1 - 1/(-3)^j)^(j+1) = 0.72392917591300902192520561680114697538581509655711959502191898288595312452...

A298988 a(n) = [x^n] Product_{k>=1} 1/(1 + n*x^k)^k.

Original entry on oeis.org

1, -1, 0, -18, 208, -2400, 36504, -663754, 13808320, -324176418, 8487126400, -245122390601, 7741417124880, -265402847130421, 9816338228638872, -389618889514254225, 16518399076342421248, -745025763154442071130, 35619835529954597786208, -1799459812004380374518790, 95780758238408017088795600
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 31 2018

Keywords

Crossrefs

Cf. A255528, A261566, A261567, A266971, A292134, A297326, A298985, A298986, A298987.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[1/(1 + n x^k)^k, {k, 1, n}], {x, 0, n}], {n, 0, 20}]

Formula

a(n) ~ (-1)^n * n^n * (1 - 2/n + 6/n^2 - 14/n^3 + 33/n^4 - 70/n^5 + 149/n^6 - 298/n^7 + 591/n^8 - 1132/n^9 + 2139/n^10 + ...), for coefficients, see A005380. - Vaclav Kotesovec, Aug 21 2018
Showing 1-3 of 3 results.

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