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),", "))
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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