A190590 Expansion of series reversion of x/(1 + x + 2*x^4).
1, 1, 1, 1, 3, 11, 31, 71, 157, 397, 1141, 3301, 9087, 24311, 66067, 185771, 532121, 1520889, 4316233, 12255913, 35079739, 101232419, 293236615, 849895311, 2465119669, 7167636741, 20909386941, 61162159501, 179214613111, 525803297743, 1544899158331
Offset: 1
Examples
g.f.: x + x^2 + x^3 + x^4 + 3*x^5 + 11*x^6 + 31*x^7 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 1..1000 (terms 1..165 from Vincenzo Librandi)
- Vladimir Kruchinin, The method for obtaining expressions for coefficients of reverse generating functions, arXiv:1211.3244 [math.CO], 2012
Programs
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Mathematica
Rest[CoefficientList[InverseSeries[Series[x/(1+x+2*x^4),{x,0,20}],x],x]] (* Vaclav Kotesovec, Aug 20 2013 *) -
Maxima
a(n):=sum(binomial(j, -3*n+4*j-1)*2^(n-j)*binomial(n, j), j, floor((3*n+1)/4), n)/n;
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PARI
x='x+O('x^66); /* that many terms */ Vec(serreverse(x/(1+x+2*x^4))) /* show terms */ /* Joerg Arndt, May 27 2011 */
Formula
a(n) = Sum_{j=floor((3*n+1)/4)..n} binomial(j,-3*n+4*j-1)*2^(n-j)*binomial(n,j)/n.
Recurrence: 3*(n-1)*(3*n-7)*(3*n+1)*a(n) = 3*(2*n-3)*(18*n^2 - 54*n + 29)*a(n-1) - 3*(n-2)*(54*n^2 - 216*n + 209)*a(n-2) + 54*(n-3)*(n-2)*(2*n-5)*a(n-3) + 485*(n-4)*(n-3)*(n-2)*a(n-4). - Vaclav Kotesovec, Aug 20 2013
a(n) ~ 6^(1/4)*sqrt(2*6^(3/4)+16)*(1+4/3*6^(1/4))^n/(24*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 20 2013