A086331
Expansion of e.g.f. exp(x)/(1 + LambertW(-x)).
Original entry on oeis.org
1, 2, 7, 43, 393, 4721, 69853, 1225757, 24866481, 572410513, 14738647221, 419682895325, 13094075689225, 444198818128313, 16278315877572141, 640854237634448101, 26973655480577228769, 1208724395795734172705, 57453178877303382607717, 2887169565412587866031533
Offset: 0
a(2) = 7 because {}->{}, 1->1, 2->2, and the four functions from {1,2} into {1,2}. Note A000169(2) = 9 because it counts these 7 and 1->2, 2->1.
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a:= n-> add(binomial(n,k)*k^k, k=0..n):
seq(a(n), n=0..25); # Alois P. Heinz, Dec 30 2021
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nn=10;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[Series[Exp[x]/(1-t),{x,0,nn}],x] (* Geoffrey Critzer, Dec 19 2011 *)
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a(n) = sum(k=0,n, binomial(n, k)*k^k ); \\ Joerg Arndt, May 10 2013
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^k/(1-x)^(k+1))) \\ Seiichi Manyama, Jul 04 2022
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my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x)*sum(k=0, N, (k*x)^k/k!))) \\ Seiichi Manyama, Jul 04 2022
A323280
a(n) = Sum_{k=0..n} binomial(n,k) * k^(2*k).
Original entry on oeis.org
1, 2, 19, 781, 68553, 10100761, 2236373953, 693667946945, 286962262702657, 152652510206521921, 101513694573289791441, 82511051259976074269425, 80480313356721971865934369, 92773167329045961244649105633, 124768226258051318899374299271601
Offset: 0
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Table[1 + Sum[Binomial[n, k]*k^(2*k), {k, 1, n}], {n, 0, 15}] (* Vaclav Kotesovec, May 31 2019 *)
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a(n) = sum(k=0, n, binomial(n, k)*k^(2*k));
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k^2*x)^k/(1-x)^(k+1))) \\ Seiichi Manyama, Jul 04 2022
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my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x)*sum(k=0, N, (k^2*x)^k/k!))) \\ Seiichi Manyama, Jul 04 2022
A277454
a(n) = 1 + Sum_{k=1..n} binomial(n,k) * 2^k * k^k.
Original entry on oeis.org
1, 3, 21, 271, 5065, 122811, 3651997, 128566663, 5227782161, 241072839667, 12430169195941, 708612945554559, 44253858433505497, 3004570398043291819, 220341964157226260525, 17357760973540312138231, 1461813975265547356467745, 131061164660246579394042339
Offset: 0
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Table[1+Sum[Binomial[n, k]*2^k*k^k, {k, 1, n}], {n, 0, 20}]
CoefficientList[Series[E^x/(1+LambertW[-2*x]), {x, 0, 20}], x] * Range[0, 20]!
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{a(n) = sum(k=0, n, binomial(n, k)*(2*k)^k)} \\ Seiichi Manyama, Jan 12 2019
A277457
E.g.f.: exp(2*x)/(1+LambertW(-x)).
Original entry on oeis.org
1, 3, 12, 71, 616, 7197, 105052, 1829291, 36922928, 846851993, 21744781684, 617832652527, 19242299657896, 651815827343189, 23857403245171724, 938247816632341043, 39455261828928309088, 1766645684585351990961, 83913998998426051745764, 4214295288128637488870327, 223120214856875472660345176
Offset: 0
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CoefficientList[Series[Exp[2*x]/(1+LambertW[-x]), {x, 0, 20}], x]*Range[0, 20]!
Table[1 + Sum[Binomial[n, m]*(1 + Sum[Binomial[m, k]*k^k, {k, 1, m}]), {m, 1, n}], {n, 0, 20}]
Table[2^n + Sum[Binomial[n, k]*2^(n-k)*k^k, {k, 1, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 28 2016 *)
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x='x+O('x^50); Vec(serlaplace(exp(2*x)/(1 + lambertw(-x)))) \\ G. C. Greubel, Nov 07 2017
A356827
Expansion of e.g.f. exp(x * exp(3*x)).
Original entry on oeis.org
1, 1, 7, 46, 361, 3436, 37729, 463366, 6280369, 93015352, 1491337441, 25684077706, 472217487625, 9221588527204, 190441412508481, 4143470377262806, 94663498086222049, 2264440394856702832, 56570146384760433217, 1472545685988162638722
Offset: 0
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A356827 := proc(n)
add((3*k)^(n-k) * binomial(n,k),k=0..n) ;
end proc:
seq(A356827(n),n=0..70) ; # R. J. Mathar, Dec 04 2023
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my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x*exp(3*x))))
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-3*k*x)^(k+1)))
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a(n) = sum(k=0, n, (3*k)^(n-k)*binomial(n, k));
Showing 1-5 of 5 results.
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