A144690
Limit of the coefficient of x^(2^m+n) in B(x)^(n+1) as m grows, where B(x) = Sum_{k>=0} x^(2^k).
Original entry on oeis.org
1, 2, 6, 16, 130, 636, 5712, 34336, 811458, 7151380, 113034746, 1049982792, 25276020640, 293841338896, 5712436923000, 68827002466176, 3739997267623490, 60752008945662372, 1718332635327516238, 26832922324005759560, 1099199814287516279394
Offset: 0
-
{ a(n) = local(m=n+log(n+.5)\log(2), B=sum(k=0,m,x^(2^k)));if(n<0, 0, polcoeff((B+O(x^(2^m+n+1)))^(n+1),2^m+n)) }
a(14), a(15) corrected and a(16)-a(23) added by
Max Alekseyev, May 03 2011
A144691
Limit of the coefficient of x^(2^m+n) in B(x)^(n+1)/(n+1) as m grows, where B(x) = Sum_{k>=0} x^(2^k).
Original entry on oeis.org
1, 1, 2, 4, 26, 106, 816, 4292, 90162, 715138, 10275886, 87498566, 1944309280, 20988667064, 380829128200, 4301687654136, 219999839271970, 3375111608092354, 90438559754079802, 1341646116200287978, 52342848299405537114, 921821277222438350170
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 26*x^4 + 106*x^5 + 816*x^6 +...
A(x/G(x)) = G(x) = x/Series_Reversion[x*A(x)], where
G(x) = 1 + x + x^2 + 17*x^4 + 408*x^6 + 69473*x^8 + 6018928*x^10 +...
and G(x) appears to continue with only even powers of x (cf. A144692).
The inverse binomial transform forms the g.f. of A202582:
A(x/(1+x))/(1+x) = 1 + x^2 + 19*x^4 + 515*x^6 + 74383*x^8 + 6816465*x^10 +...+ A202582(n)*x^n +...
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{ a(n) = local(m=n+log(n+.5)\log(2), B=sum(k=0,m,x^(2^k))); if(n<0, 0, polcoeff((B+O(x^(2^m+n+1)))^(n+1)/(n+1),2^m+n)) }
a(14), a(15) corrected and a(16)-a(23) added by
Max Alekseyev, May 03 2011
A277042
G.f.: A(x) = x/Series_Reversion[x*G(x)] where A(x*G(x)) = G(x) = g.f. of A277041.
Original entry on oeis.org
1, 1, 0, 1, 0, 0, 27, 0, 0, 10666, 0, 0, 6174792, 0, 0
Offset: 0
G.f.: A(x) = 1 + x + x^3 + 27*x^6 + 10666*x^9 + 6174792*x^12 +...
such that: A(x*G(x)) = G(x) = g.f. of A277041 where
G(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 11*x^5 + 51*x^6 + 246*x^7 + 897*x^8 + 13526*x^9 + 115631*x^10 + 614681*x^11 + 8739556*x^12 + 89877217*x^13 + 596072842*x^14 +...+ A277041(n)*x^n +...
A202582
Inverse binomial transform of A144691.
Original entry on oeis.org
1, 0, 1, 0, 19, 0, 515, 0, 74383, 0, 6816465, 0, 1457117673, 0, 241183200687, 0, 188350353304919, 0, 60855583632497865, 0, 39858196864723826583, 0, 17024263169695049621551, 0, 20817292362271689177123509, 0, 13408255577123563666760376685, 0
Offset: 0
G.f.: A(x) = 1 + x^2 + 19*x^4 + 515*x^6 + 74383*x^8 + 6816465*x^10 +...
where
x/Series_Reversion(x*A(x)) = 1 + x^2 + 17*x^4 + 408*x^6 + 69473*x^8 + 6018928*x^10 +...+ A144692(n)*x^n +...
The g.f. G(x) of A144692 begins:
G(x) = 1 + x + x^2 + 17*x^4 + 408*x^6 + 69473*x^8 + 6018928*x^10 +...
where G(x) satisfies: A(x) = G(x*A(x))/(1+x) and G(x) = A(x/(G(x)-x)) + x.
Showing 1-4 of 4 results.
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