A202943
G.f.: [ Sum_{n>=0} (n+1)*(n+2)/2 * 2^(n*(n-1)) * x^n ]^(1/3).
Original entry on oeis.org
1, 1, 7, 199, 20026, 7296946, 10006653574, 52756427071846, 1080758244198360481, 86574556540356639703921, 27234507698931717202501389871, 33749875110161915818408975272861391, 165150307912136693948216143106251788630208
Offset: 0
G.f.: A(x) = 1 + x + 7*x^2 + 199*x^3 + 20026*x^4 + 7296946*x^5 +...
where
A(x)^3 = 1 + 3*x + 6*2^2*x^2 + 10*2^6*x^3 + 15*2^12*x^4 + 21*2^20*x^5 +...
more explicitly,
A(x)^3 = 1 + 3*x + 24*x^2 + 640*x^3 + 61440*x^4 + 22020096*x^5 +...+ A202944(n)*x^n +...
The residues modulo 2 of this sequence begin:
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,...];
which forms the characteristic function:
(1+x)*(1+x^2)*(1+x^8)*(1 + x^32 + x^128 + x^160 + x^512 + x^544 + x^640 + x^672 +...+ x^(32*A000695(n)) +...).
A202944
G.f.: Sum_{n>=0} (n+1)*(n+2)/2 * 2^(n*(n-1)) * x^n.
Original entry on oeis.org
1, 3, 24, 640, 61440, 22020096, 30064771072, 158329674399744, 3242591731706757120, 259730156557830486753280, 81704042592835098143342198784, 101249788741429138756344678419791872, 495451126236886402802673428420654515879936
Offset: 0
G.f.: A(x) = 1 + 3*x + 24*x^2 + 640*x^3 + 61440*x^4 + 22020096*x^5 +...
where
A(x) = 1 + 3*x + 6*2^2*x^2 + 10*2^6*x^3 + 15*2^12*x^4 + 21*2^20*x^5 +...
Note that the cube root of the g.f. is an integer series:
A(x)^(1/3) = 1 + x + 7*x^2 + 199*x^3 + 20026*x^4 + 7296946*x^5 +...+ A202943(n)*x^n +...
A202947
G.f.: [ Sum_{n>=0} (n+1) * 2^(n^2) * x^n ]^(1/2).
Original entry on oeis.org
1, 2, 22, 980, 161638, 100318460, 240313495420, 2251316821283048, 83005840299778004614, 12089092134684999622076396, 6972054121242613685463168904468, 15950722005044706228925521886595357720, 144954811888851643278920459489891540357638876
Offset: 0
G.f.: A(x) = 1 + 2*x + 22*x^2 + 980*x^3 + 161638*x^4 + 100318460*x^5 +...
where
A(x)^2 = 1 + 2*2*x + 3*2^4*x^2 + 4*2^9*x^3 + 5*2^16*x^4 + 6*2^25*x^5 +...
more explicitly,
A(x)^2 = 1 + 4*x + 48*x^2 + 2048*x^3 + 327680*x^4 + 201326592*x^5 +...+ A197927(n+1)*x^n +...
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{a(n)=polcoeff(sum(m=0,n,(m+1)*2^(m^2)*x^m+x*O(x^n))^(1/2),n)}
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{a(n)=if(n==0,1,(n+1)*2^(n^2-1)-sum(k=1,n-1,a(n-k)*a(k)/2))}
A210310
G.f.: [ Sum_{n>=0} (n+1)*(n+2)/2 * 4^(n^2) * x^n ]^(1/3).
Original entry on oeis.org
1, 4, 496, 869824, 21467623936, 7881126729140224, 44075357435370071351296, 3802951448073847111253622882304, 5104235473390420925196874786915866443776, 107176786696765659714361271737312271270497663320064
Offset: 0
G.f.: A(x) = 1 + 4*x + 496*x^2 + 869824*x^3 + 21467623936*x^4 +...
where
A(x)^3 = 1 + 3*4*x + 6*4^4*x^2 + 10*4^9*x^3 + 15*4^16*x^4 + 21*4^25*x^5 +...
-
{a(n)=polcoeff(sum(m=0, n, (m+1)*(m+2)/2*4^(m^2)*x^m+x*O(x^n))^(1/3), n)}
for(n=0,20,print1(a(n),", "))
Showing 1-4 of 4 results.
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