A218684
O.g.f.: Sum_{n>=0} (1+n^2*x)^n * x^n/n! * exp(-(1+n^2*x)*x).
Original entry on oeis.org
1, 0, 1, 2, 7, 18, 96, 260, 1851, 5270, 46515, 137942, 1447202, 4433772, 53787706, 169169912, 2326986783, 7477418982, 114916173009, 375898894514, 6380455164161, 21185872231238, 393499602818322, 1323362744628080, 26691270481453228, 90755667374332324
Offset: 0
O.g.f: A(x) = 1 + x^2 + 2*x^3 + 7*x^4 + 18*x^5 + 96*x^6 + 260*x^7 +...
where
A(x) = exp(-x) + (1+x)*x*exp(-(1+x)*x) + (1+2^2*x)^2*x^2/2!*exp(-(1+2^2*x)*x) + (1+3^2*x)^3*x^3/3!*exp(-(1+3^2*x)*x) + (1+4^2*x)^4*x^4/4!*exp(-(1+4^2*x)*x) + (1+5^2*x)^5*x^5/5!*exp(-(1+5^2*x)*x) +...
simplifies to a power series in x with integer coefficients.
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{a(n)=polcoeff(sum(k=0,n,(1+k^2*x)^k*x^k/k!*exp(-x*(1+k^2*x)+x*O(x^n))),n)}
for(n=0,30,print1(a(n),", "))
A218685
O.g.f.: Sum_{n>=0} (1+n^3*x)^n * x^n/n! * exp(-(1+n^3*x)*x).
Original entry on oeis.org
1, 0, 1, 6, 34, 270, 3415, 31230, 681026, 6949920, 230637870, 2546120514, 119281951006, 1394371349490, 87612425583018, 1069010047029672, 86763885548985810, 1094149501538197236, 111443560982774811439, 1442387644419293694144, 180179254059921915232864
Offset: 0
O.g.f: A(x) = 1 + x^2 + 6*x^3 + 34*x^4 + 270*x^5 + 3415*x^6 +...
where
A(x) = exp(-x) + (1+x)*x*exp(-(1+x)*x) + (1+2^3*x)^2*x^2/2!*exp(-(1+2^3*x)*x) + (1+3^3*x)^3*x^3/3!*exp(-(1+3^3*x)*x) + (1+4^3*x)^4*x^4/4!*exp(-(1+4^3*x)*x) + (1+5^3*x)^5*x^5/5!*exp(-(1+5^3*x)*x) +...
simplifies to a power series in x with integer coefficients.
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{a(n)=polcoeff(sum(k=0,n,(1+k^3*x)^k*x^k/k!*exp(-x*(1+k^3*x)+x*O(x^n))),n)}
for(n=0,30,print1(a(n),", "))
A218687
O.g.f.: Sum_{n>=0} n^n * (1+n^3*x)^n * x^n/n! * exp(-n*(1+n^3*x)*x).
Original entry on oeis.org
1, 1, 2, 31, 398, 10476, 296407, 12613297, 592445192, 36797742660, 2524966492661, 212912151736648, 19819138754732997, 2155966497948737905, 259256365067737582615, 35050667748654756208069, 5257919606219599751747894, 858816581875175776426876930
Offset: 0
O.g.f: A(x) = 1 + x + 2*x^2 + 31*x^3 + 398*x^4 + 10476*x^5 + 296407*x^6 +...
where
A(x) = 1 + (1+x)*x*exp(-(1+x)*x) + 2^2*(1+2^3*x)^2*x^2/2!*exp(-2*(1+2^3*x)*x) + 3^3*(1+3^3*x)^3*x^3/3!*exp(-3*(1+3^3*x)*x) + 4^4*(1+4^3*x)^4*x^4/4!*exp(-4*(1+4^3*x)*x) + 5^5*(1+5^3*x)^5*x^5/5!*exp(-5*(1+5^3*x)*x) +...
simplifies to a power series in x with integer coefficients.
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{a(n)=polcoeff(sum(k=0,n,k^k*(1+k^3*x)^k*x^k/k!*exp(-k*x*(1+k^3*x)+x*O(x^n))),n)}
for(n=0,30,print1(a(n),", "))
Showing 1-3 of 3 results.
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