A293526 -id:A293526 - cp's OEIS frontend

A293507 Expansion of e.g.f. exp(x/(1 - x^4)).

1, 1, 1, 1, 1, 121, 721, 2521, 6721, 378001, 5473441, 39972241, 199679041, 7005552841, 176899522801, 2186722497961, 17454339826561, 459473703430561, 16503993702423361, 306140370496394401, 3555223271216311681, 80917223353652470681, 3568770455830785208081

Offset: 0

Seiichi Manyama, Oct 10 2017

nonn

E.g.f.: exp(x/(1 - x^m)): A000262 (m=1), A088009 (m=2), A293493 (m=3), this sequence (m=4).

Cf. A293526.

CoefficientList[Series[E^(x/(1 - x^4)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Oct 11 2017 *)

N=66; x='x+O('x^N); Vec(serlaplace(exp(x/(1-x^4))))

N=66; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, exp(x^(4*k-3)))))

E.g.f.: Product_{k>0} exp(x^(4*k-3)).
a(n) ~ exp(1/4 + sqrt(n) - n) * n^(n-1/4) / 2. - Vaclav Kotesovec, Oct 11 2017
a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/4)} binomial(n-1,4*k) * (4*k+1)! * a(n-4*k-1). - Ilya Gutkovskiy, Feb 24 2022
a(n) = n! * Sum_{k=0..floor(n/4)} binomial(n-3*k-1,k)/(n-4*k)!. - Seiichi Manyama, Jun 08 2024

A293568 E.g.f.: exp(x^3/(x^4 - 1)).

Original entry on oeis.org

1, 0, 0, -6, 0, 0, 360, -5040, 0, -60480, 3628800, -39916800, 19958400, -3113510400, 130767436800, -1318571654400, 3487131648000, -355687428096000, 12813639597158400, -126713646259200000, 1013709170073600000, -85161707377883136000, 2819368492175499264000

Offset: 0

Seiichi Manyama, Oct 12 2017

sign

E.g.f.: Product_{k>0} exp(x^(-(m*k-1))): A293532 (m=2), A293567 (m=3), this sequence (m=4).

Cf. A293526.

With[{nn=30},CoefficientList[Series[Exp[x^3/(x^4-1)],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Dec 31 2020 *)

N=66; x='x+O('x^N); Vec(serlaplace(exp(x^3/(x^4-1))))

N=66; x='x+O('x^N); Vec(serlaplace(1/prod(k=1, N, exp(x^(4*k-1)))))

E.g.f.: Product_{k>0} exp(x^(-(4*k-1))).

cp's OEIS Frontend

A293507 Expansion of e.g.f. exp(x/(1 - x^4)).

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A293568 E.g.f.: exp(x^3/(x^4 - 1)).

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Mathematica

PARI

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