A277463
E.g.f.: sinh(x)/(1+LambertW(-x)).
Original entry on oeis.org
0, 1, 2, 13, 112, 1321, 19296, 335637, 6764864, 154946449, 3973820800, 112789880413, 3509627281920, 118790978349369, 4344883388878592, 170767066282574821, 7177162988688031744, 321206181612447781921, 15250250261039350358016, 765586309042945067185581
Offset: 0
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CoefficientList[Series[Sinh[x]/(1+LambertW[-x]), {x, 0, 25}], x] * Range[0, 25]!
Table[(1-(-1)^n + Sum[(1-(-1)^(n-k)) * Binomial[n,k] * k^k, {k, 1, n}])/2, {n, 0, 25}]
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x='x+O('x^50); concat([0], Vec(serlaplace(sinh(x)/(1 + lambertw(-x))))) \\ G. C. Greubel, Nov 05 2017
A277462
E.g.f.: cos(x)/(1 + LambertW(-x)).
Original entry on oeis.org
1, 1, 3, 24, 233, 2860, 42875, 758856, 15488657, 358164432, 9254769459, 264273873600, 8264362186489, 280896392748608, 10310601442639147, 406479520869636480, 17129450693008029729, 768404013933189112064, 36557893891263190204259, 1838650651518153170939904
Offset: 0
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CoefficientList[Series[Cos[x]/(1+LambertW[-x]), {x, 0, 25}], x] * Range[0, 25]!
Table[Cos[Pi*n/2] + Sum[Binomial[n, k] * Cos[Pi*(n-k)/2] * k^k, {k, 1, n}], {n, 0, 25}] (* Vaclav Kotesovec, Oct 28 2016 *)
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x='x+O('x^50); Vec(serlaplace(cos(x)/(1 + lambertw(-x)))) \\ G. C. Greubel, Nov 07 2017
A277478
E.g.f.: -cosh(x)*LambertW(-x).
Original entry on oeis.org
0, 1, 2, 12, 76, 720, 8766, 131096, 2319416, 47361600, 1096018330, 28344108672, 810053677764, 25352185339520, 862335856149782, 31674845755622400, 1249527587684729584, 52687201308036059136, 2364751154207006978994, 112562199977955164819456
Offset: 0
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CoefficientList[Series[-Cosh[x]*LambertW[-x], {x, 0, 20}], x] * Range[0, 20]!
Table[Sum[(1 + (-1)^(n-k)) * Binomial[n, k] * k^(k-1)/2, {k, 1, n}], {n, 0, 20}]
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x='x+O('x^50); concat([0], Vec(serlaplace(-cosh(x)*lambertw(-x)))) \\ G. C. Greubel, Nov 07 2017
A218296
Expansion of e.g.f. Sum_{n>=0} n^n * cosh(n*x) * x^n/n!.
Original entry on oeis.org
1, 1, 4, 30, 352, 5560, 109056, 2540720, 68401152, 2087897472, 71236526080, 2686375597312, 110951893303296, 4980913763830784, 241491517062512640, 12575483733378816000, 700015678015053758464, 41480146826887546372096, 2606901492484549499682816
Offset: 0
E.g.f.: A(x) = 1 + x + 4*x^2/2! + 30*x^3/3! + 352*x^4/4! + 5560*x^5/5! +...
where
A(x) = 1 + 1^1*x*cosh(1*x) + 2^2*cosh(2*x)*x^2/2! + 3^3*cosh(3*x)*x^3/3! + 4^4*cosh(4*x)*x^4/4! + 5^5*cosh(5*x)*x^5/5! +...
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CoefficientList[Series[1 + 1/2*x/(1-x) - 1/2*LambertW[-x*E^x]/(1 + LambertW[-x*E^x]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jul 08 2013 *)
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a(n)=n!*polcoeff(sum(k=0, n, k^k*cosh(k*x +x*O(x^n))*x^k/k!), n)
for(n=0, 30, print1(a(n), ", "))
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LambertW(x,N)=sum(n=1,N,(-n)^(n-1)*x^n/n!)
{a(n)=local(X=x+x*O(x^n));n!*polcoeff(1 + (1/2)*x/(1-X) - (1/2)*LambertW(-x*exp(X),n)/(1 + LambertW(-x*exp(X),n)),n)}
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Jul 08 2013
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a(n) = sum(k=0, n\2, (n-2*k)^n*binomial(n, 2*k)); \\ Seiichi Manyama, Feb 15 2023
A195509
Expansion of e.g.f. (exp(x*exp(x)) + exp(x/exp(x)))/2.
Original entry on oeis.org
1, 1, 1, 4, 25, 96, 481, 3368, 20721, 141760, 1146721, 9098112, 77652169, 726208640, 6891125697, 69344336896, 738718169569, 8076031881216, 92647353941569, 1106883171037184, 13616813607795321, 174298975125127168, 2304515271134124577
Offset: 0
E.g.f.: A(x) = 1 + x + x^2/2! + 4*x^3/3! + 25*x^4/4! + 96*x^5/5! + 481*x^6/6! +...
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a := proc(n) local m: add(binomial(n, 2*m)*(n - 2*m)^(2*m), m = 0 .. floor(1/2*n - 1/2)): end proc:
seq(a(n), n = 1..30); # Petros Hadjicostas, May 06 2020 (for n >= 1)
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{a(n)=local(X=x+x*O(x^n),A=1+X);A=(exp(X*exp(X))+exp(X/exp(X)))/2;n!*polcoeff(A,n)}
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{a(n)=n!*polcoeff(sum(m=0,n,x^m*cosh(m*x+x*O(x^n))/m!),n)}
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a(n) = sum(k=0, n\2, (n-2*k)^(2*k)*binomial(n, 2*k)); \\ Seiichi Manyama, Feb 15 2023
Showing 1-5 of 5 results.
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