A222076
O.g.f.: Sum_{n>=0} n^n*(n+2)^n * exp(-n*(n+2)*x) * x^n / n!.
Original entry on oeis.org
1, 3, 23, 320, 6397, 166467, 5338412, 203578776, 9001795829, 452924585465, 25555585227999, 1598279794889076, 109748572718377660, 8209004345714098500, 664396187060996529528, 57853075421585981420208, 5393119810256349152565573, 535908449308064099732283429, 56548822143306498413322880709
Offset: 0
O.g.f.: A(x) = 1 + 3*x + 23*x^2 + 320*x^3 + 6397*x^4 + 166467*x^5 +...
where
A(x) = 1 + 3*x*exp(-3*x) + 8^2*exp(-8*x)*x^2/2! + 15^3*exp(-15*x)*x^3/3! + 24^4*exp(-24*x)*x^4/4! + 35^5*exp(-35*x)*x^5/5! +...
is a power series in x with integer coefficients.
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Flatten[{1,Table[Sum[Binomial[n,j] * 2^(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+2)^m*x^m*exp(-m*(m+2)*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+2)^k*x^k/(1+k*(k+2)*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+2)^n)}
for(n=0, 20, print1(a(n), ", "))
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), ", "))
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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