A136505
a(n) = binomial(2^n + 1, n).
Original entry on oeis.org
1, 3, 10, 84, 2380, 237336, 82598880, 99949406400, 422825581068000, 6318976181520699840, 337559127276933693852160, 65182103393445184131620004864, 45946437874792132748338425828443136
Offset: 0
Sequences of the form binomial(2^n +p*n +q, n):
A136556 (0,-1),
A014070 (0,0), this sequence (0,1),
A136506 (0,2),
A060690 (1,-1),
A132683 (1,0),
A132684 (1,1),
A132685 (2,0),
A132686 (2,1),
A132687 (3,-1),
A132688 (3,0),
A132689 (3,1).
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[Binomial(2^n +1, n): n in [0..20]]; // G. C. Greubel, Mar 14 2021
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A136505:= n-> binomial(2^n+1,n); seq(A136505(n), n=0..20); # G. C. Greubel, Mar 14 2021
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Table[Binomial[2^n+1,n], {n,0,15}] (* Vaclav Kotesovec, Jul 02 2016 *)
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{a(n)=polcoeff(sum(i=0,n,(1+2^i*x +x*O(x^n))*log(1+2^i*x +x*O(x^n))^i/i!),n)}
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[binomial(2^n +1, n) for n in (0..20)] # G. C. Greubel, Mar 14 2021
A136506
a(n) = binomial(2^n + 2, n).
Original entry on oeis.org
1, 4, 15, 120, 3060, 278256, 90858768, 105637584000, 436355999662176, 6431591598617108352, 340881559632021623909760, 65533747894341651530074060800, 46081376018330435634530315478453248
Offset: 0
Sequences of the form binomial(2^n +p*n +q, n):
A136556 (0,-1),
A014070 (0,0),
A136505 (0,1), this sequence (0,2),
A060690 (1,-1),
A132683 (1,0),
A132684 (1,1),
A132685 (2,0),
A132686 (2,1),
A132687 (3,-1),
A132688 (3,0),
A132689 (3,1).
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[Binomial(2^n +2, n): n in [0..20]]; // G. C. Greubel, Mar 14 2021
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A136506:= n-> binomial(2^n+2,n); seq(A136506(n), n=0..20); # G. C. Greubel, Mar 14 2021
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Table[Binomial[2^n+2,n],{n,0,20}] (* Harvey P. Dale, Jun 20 2011 *)
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{a(n)=polcoeff(sum(i=0,n,(1+2^i*x +x*O(x^n))^2*log(1+2^i*x +x*O(x^n))^i/i!),n)}
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[binomial(2^n +2, n) for n in (0..20)] # G. C. Greubel, Mar 14 2021
A136508
G.f.: A(x) = Sum_{n>=0} (-1)^n * log(1 - x - 2^n*x^2)^n / n! .
Original entry on oeis.org
1, 1, 3, 7, 23, 81, 361, 1923, 13113, 114433, 1315783, 20286135, 420198791, 12003852369, 464295025509, 25153926114307, 1847231277588405, 191568316434991857, 26902669460380225411, 5357197471644242149975
Offset: 0
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m:=30; R:=PowerSeriesRing(Rationals(), m);
Coefficients(R!( (&+[(-1)^j*Log(1-x-2^j*x^2)^j/Factorial(j) : j in [0..m+2]]) )); // G. C. Greubel, Mar 15 2021
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With[{m = 30}, CoefficientList[Series[Sum[(-1)^j*Log[1 -x -2^j*x^2]^j/j!, {j, 0, m+2}], {x, 0, m}], x]] (* G. C. Greubel, Mar 15 2021 *)
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{a(n)=polcoeff(sum(i=0,n,(-1)^i*log(1-x-2^i*x^2 +x*O(x^n))^i/i!),n)}
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def A136508_list(prec):
P. = PowerSeriesRing(QQ, prec)
return P( sum((-1)^j*log(1-x -2^j*x^2)^j/factorial(j) for j in (0..32)) ).list()
A136508_list(30) # G. C. Greubel, Mar 15 2021
A136509
G.f.: A(x) = Sum_{n>=0} (-1)^n * (1 -x -2^n*x^2)^(-1) * log(1 -x -2^n*x^2)^n / n!.
Original entry on oeis.org
1, 2, 6, 16, 50, 171, 697, 3416, 21126, 169105, 1794683, 25891713, 507686588, 13878639286, 518836271475, 27356839451662, 1968958300103603, 200935638262212462, 27892630019328034846, 5502857784211927305980
Offset: 0
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m:=30; R:=PowerSeriesRing(Rationals(), m);
Coefficients(R!( (&+[(-1)^j*Log(1-x-2^j*x^2)^j/(Factorial(j)*(1 -x -2^j*x^2)) : j in [0..m+2]]) )); // G. C. Greubel, Mar 15 2021
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With[{m=30}, CoefficientList[Series[Sum[(-1)^j*Log[1-x-2^j*x^2]^j/(j!*(1-x -2^j*x^2)), {j,0,m+2}], {x,0,m}], x]] (* G. C. Greubel, Mar 15 2021 *)
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{a(n)=polcoeff(sum(i=0,n,(-1)^i*1/(1-x*(1+2^i*x +x*O(x^n)))*log(1-x-2^i*x^2 +x*O(x^n))^i/i!),n)}
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def A136509_list(prec):
P. = PowerSeriesRing(QQ, prec)
return P( sum((-1)^j*log(1-x -2^j*x^2)^j/(factorial(j)*(1 -x -2^j*x^2)) for j in (0..32)) ).list()
A136509_list(30) # G. C. Greubel, Mar 15 2021
Showing 1-4 of 4 results.