A380257 -id:A380257 - cp's OEIS frontend

A380214 Expansion of e.g.f. exp( 1/(1-3*x)^(2/3) - 1 ).

1, 2, 14, 148, 2076, 36152, 750344, 18055088, 493688976, 15108697632, 511379579104, 18959550197568, 763909806479296, 33227876172374912, 1551519044372535424, 77391560357497815808, 4106518327272819159296, 230931323981550384824832, 13718006864544800838290944

Offset: 0

Seiichi Manyama, Jan 16 2025

nonn

Cf. A049119, A380257.

CoefficientList[Series[Exp[ 1/(1-3*x)^(2/3) - 1],{x,0,18}],x]Range[0,18]! (* Stefano Spezia, Mar 31 2025 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(1/(1-3*x)^(2/3)-1)))

a(n) = Sum_{k=0..n} 2^k * 3^(n-k) * |Stirling1(n,k)| * Bell(k).
a(n) = (1/e) * (-3)^n * n! * Sum_{k>=0} binomial(-2*k/3,n)/k!.
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A380257.

A380258 Expansion of e.g.f. exp( (1/(1-5*x)^(2/5) - 1)/2 ).

Original entry on oeis.org

1, 1, 8, 106, 1954, 46082, 1323064, 44750644, 1741897340, 76672512316, 3764746706176, 203976645319448, 12086590557877144, 777464693554778776, 53948773488864143072, 4016672567726156437744, 319379204127841984947472, 27010128651142535536409360, 2420802590890201251989984128

Offset: 0

Seiichi Manyama, Jan 18 2025

nonn

Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A049376, A375173, A380257.

Cf. A004211, A025168.

CoefficientList[Series[Exp[ (1/(1-5*x)^(2/5) - 1)/2 ],{x,0,18}],x]Range[0,18]! (* Stefano Spezia, Mar 31 2025 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((1/(1-5*x)^(2/5)-1)/2)))

a(n) = Sum_{k=0..n} 5^(n-k) * |Stirling1(n,k)| * A004211(k) = Sum_{k=0..n} 2^k * 5^(n-k) * |Stirling1(n,k)| * Bell_k(1/2), where Bell_n(x) is n-th Bell polynomial.

a(n) = (1/exp(1/2)) * (-5)^n * n! * Sum_{k>=0} binomial(-2*k/5,n)/(2^k * k!).

cp's OEIS Frontend

A380214 Expansion of e.g.f. exp( 1/(1-3*x)^(2/3) - 1 ).

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