A222077
O.g.f.: Sum_{n>=0} n^n*(n+3)^n * exp(-n*(n+3)*x) * x^n / n!.
Original entry on oeis.org
1, 4, 34, 504, 10572, 285408, 9419440, 367571200, 16562241744, 846509123520, 48401180913824, 3061687935718272, 212316590908782336, 16018267935253721088, 1306322033185206970368, 114519518777575592865792, 10740222055670467832259840, 1073051903942317493993088000
Offset: 0
O.g.f.: A(x) = 1 + 4*x + 34*x^2 + 504*x^3 + 10572*x^4 + 285408*x^5 +...
where
A(x) = 1 + 4*x*exp(-4*x) + 10^2*exp(-10*x)*x^2/2! + 18^3*exp(-18*x)*x^3/3! + 28^4*exp(-28*x)*x^4/4! + 40^5*exp(-40*x)*x^5/5! +...
is a power series in x with integer coefficients.
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Flatten[{1,Table[Sum[Binomial[n,j] * 3^(n-j) * StirlingS2[n+j,n],{j,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, May 22 2014 *)
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{a(n)=polcoeff(sum(m=0, n, m^m*(m+3)^m*x^m*exp(-m*(m+3)*x+x*O(x^n))/m!), n)}
for(n=0, 20, print1(a(n), ", "))
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{a(n)=(1/n!)*polcoeff(sum(k=0, n, k^k*(k+3)^k*x^k/(1+k*(k+3)*x +x*O(x^n))^(k+1)), n)}
for(n=0, 20, print1(a(n), ", "))
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{a(n)=1/n!*sum(k=0, n, (-1)^(n-k)*binomial(n, k)*k^n*(k+3)^n)}
for(n=0, 20, print1(a(n), ", "))
A222078
O.g.f.: Sum_{n>=0} n^n*(n+4)^n * exp(-n*(n+4)*x) * x^n / n!.
Original entry on oeis.org
1, 5, 47, 742, 16357, 459369, 15651260, 626935936, 28872594389, 1503262704775, 87328047029511, 5600639046765690, 393092088068927860, 29974039720132943036, 2467669218502361588472, 218168186315818183909344, 20617367868151866462395205, 2074120178028300166990286691
Offset: 0
O.g.f.: A(x) = 1 + 5*x + 47*x^2 + 742*x^3 + 16357*x^4 + 459369*x^5 +...
where
A(x) = 1 + 5*x*exp(-5*x) + 12^2*exp(-12*x)*x^2/2! + 21^3*exp(-21*x)*x^3/3! + 32^4*exp(-32*x)*x^4/4! + 45^5*exp(-45*x)*x^5/5! +...
is a power series in x with integer coefficients.
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Flatten[{1,Table[Sum[Binomial[n,j] * 4^(n-j) * StirlingS2[n+j,n],{j,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, May 22 2014 *)
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{a(n)=polcoeff(sum(m=0, n, m^m*(m+4)^m*x^m*exp(-m*(m+4)*x+x*O(x^n))/m!), n)}
for(n=0, 20, print1(a(n), ", "))
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{a(n)=(1/n!)*polcoeff(sum(k=0, n, k^k*(k+4)^k*x^k/(1+k*(k+4)*x +x*O(x^n))^(k+1)), n)}
for(n=0, 20, print1(a(n), ", "))
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{a(n)=1/n!*sum(k=0, n, (-1)^(n-k)*binomial(n, k)*k^n*(k+4)^n)}
for(n=0, 20, print1(a(n), ", "))
A222079
O.g.f.: Sum_{n>=0} n^n*(n+5)^n * exp(-n*(n+5)*x) * x^n / n!.
Original entry on oeis.org
1, 6, 62, 1040, 24076, 703800, 24786512, 1020779520, 48130232528, 2557117300640, 151180506557280, 9846055638729216, 700523098562671360, 54066239308284456960, 4499576117943522662400, 401709919258066014720000, 38299206898825369235170560, 3883999501445283274005895680
Offset: 0
O.g.f.: A(x) = 1 + 6*x + 62*x^2 + 1040*x^3 + 24076*x^4 + 703800*x^5 +...
where
A(x) = 1 + 6*x*exp(-6*x) + 14^2*exp(-14*x)*x^2/2! + 24^3*exp(-24*x)*x^3/3! + 36^4*exp(-36*x)*x^4/4! + 50^5*exp(-50*x)*x^5/5! +...
is a power series in x with integer coefficients.
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Flatten[{1,Table[Sum[Binomial[n,j] * 5^(n-j) * StirlingS2[n+j,n],{j,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, May 22 2014 *)
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{a(n)=polcoeff(sum(m=0, n, m^m*(m+5)^m*x^m*exp(-m*(m+5)*x+x*O(x^n))/m!), n)}
for(n=0, 20, print1(a(n), ", "))
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{a(n)=(1/n!)*polcoeff(sum(k=0, n, k^k*(k+5)^k*x^k/(1+k*(k+5)*x +x*O(x^n))^(k+1)), n)}
for(n=0, 20, print1(a(n), ", "))
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{a(n)=1/n!*sum(k=0, n, (-1)^(n-k)*binomial(n, k)*k^n*(k+5)^n)}
for(n=0, 20, print1(a(n), ", "))
Showing 1-3 of 3 results.
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