A218670
O.g.f.: Sum_{n>=0} n^n * (1+n*x)^n * x^n/n! * exp(-n*x*(1+n*x)).
Original entry on oeis.org
1, 1, 2, 7, 26, 116, 556, 2927, 16388, 97666, 612136, 4023878, 27579410, 196537134, 1451102836, 11074811191, 87160086800, 706055915318, 5876662642720, 50182337830986, 439036984440316, 3930618736372336, 35970734643745496, 336153100655220126, 3205000520319374116
Offset: 0
O.g.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 26*x^4 + 116*x^5 + 556*x^6 + 2927*x^7 +...
where
A(x) = 1 + (1+x)*x*exp(-x*(1+x)) + 2^2*(1+2*x)^2*x^2/2!*exp(-2*x*(1+2*x)) + 3^3*(1+3*x)^3*x^3/3!*exp(-3*x*(1+3*x)) + 4^4*(1+4*x)^4*x^4/4!*exp(-4*x*(1+4*x)) + 5^5*(1+5*x)^5*x^5/5!*exp(-5*x*(1+5*x)) +...
simplifies to a power series in x with integer coefficients.
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{a(n)=local(A=1+x);A=sum(k=0,n,k^k*(1+k*x)^k*x^k/k!*exp(-k*x*(1+k*x)+x*O(x^n)));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
A218678
O.g.f.: Sum_{n>=0} n^n * (1+n*x)^(3*n) * x^n/n! * exp(-n*x*(1+n*x)^3).
Original entry on oeis.org
1, 1, 4, 22, 161, 1321, 12541, 130383, 1482875, 18153076, 237430711, 3295833146, 48274094584, 742868875984, 11963384310515, 200974595790271, 3511980095379727, 63682377891348689, 1195661594431548085, 23199930176668566579, 464421513762097397125, 9576744471125816269165
Offset: 0
O.g.f.: A(x) = 1 + x + 4*x^2 + 22*x^3 + 161*x^4 + 1321*x^5 + 12541*x^6 +...
where
A(x) = 1 + (1+x)^3*x*exp(-x*(1+x)^3) + 2^2*(1+2*x)^6*x^2/2!*exp(-2*x*(1+2*x)^3) + 3^3*(1+3*x)^9*x^3/3!*exp(-3*x*(1+3*x)^3) + 4^4*(1+4*x)^12*x^4/4!*exp(-4*x*(1+4*x)^3) + 5^5*(1+5*x)^15*x^5/5!*exp(-5*x*(1+5*x)^3) +...
simplifies to a power series in x with integer coefficients.
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{a(n)=local(A=1+x);A=sum(k=0,n,k^k*(1+k*x)^(3*k)*x^k/k!*exp(-k*x*(1+k*x)^3+x*O(x^n)));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
A218679
O.g.f.: Sum_{n>=0} n^n * (1+n*x)^(4*n) * x^n/n! * exp(-n*x*(1+n*x)^4).
Original entry on oeis.org
1, 1, 5, 31, 273, 2652, 30071, 375628, 5135649, 75945388, 1202006514, 20243446719, 360517872287, 6758311053521, 132833835618576, 2728019848249377, 58370987166092073, 1297916560174624569, 29924140267551540116, 713934350929955200551, 17594768127940813003452
Offset: 0
O.g.f.: A(x) = 1 + x + 5*x^2 + 31*x^3 + 273*x^4 + 2652*x^5 + 30071*x^6 +...
where
A(x) = 1 + (1+x)^4*x*exp(-x*(1+x)^4) + 2^2*(1+2*x)^8*x^2/2!*exp(-2*x*(1+2*x)^4) + 3^3*(1+3*x)^12*x^3/3!*exp(-3*x*(1+3*x)^4) + 4^4*(1+4*x)^16*x^4/4!*exp(-4*x*(1+4*x)^4) + 5^5*(1+5*x)^20*x^5/5!*exp(-5*x*(1+5*x)^4) +...
simplifies to a power series in x with integer coefficients.
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{a(n)=local(A=1+x);A=sum(k=0,n,k^k*(1+k*x)^(4*k)*x^k/k!*exp(-k*x*(1+k*x)^4+x*O(x^n)));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
Showing 1-3 of 3 results.
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