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),", "))
A218686
O.g.f.: Sum_{n>=0} n^n * (1+n^2*x)^n * x^n/n! * exp(-n*(1+n^2*x)*x).
Original entry on oeis.org
1, 1, 2, 15, 107, 1164, 13932, 207527, 3424441, 65365273, 1366815507, 31899555046, 806153628997, 22260455705106, 659196741236329, 21028295211402871, 713819243969142111, 25836118882427921161, 988875977638287049631, 40043648314495526922945
Offset: 0
O.g.f: A(x) = 1 + x + 2*x^2 + 15*x^3 + 107*x^4 + 1164*x^5 + 13932*x^6 +...
where
A(x) = 1 + (1+x)*x*exp(-(1+x)*x) + 2^2*(1+2^2*x)^2*x^2/2!*exp(-2*(1+2^2*x)*x) + 3^3*(1+3^2*x)^3*x^3/3!*exp(-3*(1+3^2*x)*x) + 4^4*(1+4^2*x)^4*x^4/4!*exp(-4*(1+4^2*x)*x) + 5^5*(1+5^2*x)^5*x^5/5!*exp(-5*(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,k^k*(1+k^2*x)^k*x^k/k!*exp(-k*x*(1+k^2*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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