A207651
G.f.: Sum_{n>=0} Product_{k=1..n} (1 - (1-x)^k)/(1 - x^k).
Original entry on oeis.org
1, 1, 3, 8, 25, 83, 323, 1410, 7062, 39660, 248287, 1709505, 12843315, 104446836, 913968191, 8560027375, 85427505885, 904899664970, 10139054456975, 119802780498730, 1488769376468607, 19409525611304801, 264890181139521141, 3776619220990535910
Offset: 0
G.f.: A(x) = 1 + x + 3*x^2 + 8*x^3 + 25*x^4 + 83*x^5 + 323*x^6 +...
such that, by definition,
A(x) = 1 + (1-(1-x))/(1-x) + (1-(1-x))*(1-(1-x)^2)/((1-x)*(1-x^2)) + (1-(1-x))*(1-(1-x)^2)*(1-(1-x)^3)/((1-x)*(1-x^2)*(1-x^3)) +...
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{a(n)=polcoeff(sum(m=0,n,prod(k=1,m,(1-(1-x)^k)/(1-x^k +x*O(x^n)) )),n)}
for(n=0,25,print1(a(n),", "))
A207653
G.f.: Sum_{n>=0} Product_{k=1..n} (1 - (1-x)^(2*k-1))/(1 - x^(2*k-1)).
Original entry on oeis.org
1, 1, 4, 16, 77, 460, 3287, 27561, 265307, 2880875, 34821316, 463543454, 6737545832, 106158368798, 1802204594518, 32793160634292, 636683459975767, 13137118248246982, 287070448575006268, 6622644707103106925, 160846900060253917905, 4102379491083664461080
Offset: 0
G.f.: A(x) = 1 + x + 4*x^2 + 16*x^3 + 77*x^4 + 460*x^5 + 3287*x^6 +...
such that, by definition,
A(x) = 1 + (1-(1-x))/(1-x) + (1-(1-x))*(1-(1-x)^3)/((1-x)*(1-x^3)) + (1-(1-x))*(1-(1-x)^3)*(1-(1-x)^5)/((1-x)*(1-x^3)*(1-x^5)) +...
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{a(n)=polcoeff(sum(m=0,n,prod(k=1,m,(1-(1-x)^(2*k-1))/(1-x^(2*k-1) +x*O(x^n)) )),n)}
for(n=0,40,print1(a(n),", "))
A207654
G.f.: Sum_{n>=0} Product_{k=1..n} ((1+x)^(2*k-1) - 1)/(1 - x^(2*k-1)).
Original entry on oeis.org
1, 1, 4, 22, 173, 1816, 23659, 367573, 6622465, 135637477, 3111148862, 78984029782, 2198423489832, 66562555228478, 2177861372888738, 76571625673934064, 2878937040339348981, 115260759545001030638, 4895471242828376133806, 219853190410155476470763
Offset: 0
G.f.: A(x) = 1 + x + 4*x^2 + 22*x^3 + 173*x^4 + 1816*x^5 + 23659*x^6 +...
such that, by definition,
A(x) = 1 + ((1+x)-1)/(1-x) + ((1+x)-1)*((1+x)^3-1)/((1-x)*(1-x^3)) + ((1+x)-1)*((1+x)^3-1)*((1+x)^5-1)/((1-x)*(1-x^3)*(1-x^5)) +...
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With[{nn=20},CoefficientList[Series[Sum[Product[((1+x)^(2k-1)-1)/(1- x^(2k-1)),{k,n}],{n,0,nn}],{x,0,nn}],x]] (* Harvey P. Dale, Sep 06 2015 *)
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{a(n)=polcoeff(sum(m=0,n,prod(k=1,m,((1+x)^(2*k-1)-1)/(1-x^(2*k-1) +x*O(x^n)) )),n)}
for(n=0,25,print1(a(n),", "))
Showing 1-3 of 3 results.