A356820 -id:A356820 - cp's OEIS frontend

A356819 Expansion of e.g.f. exp(-x * exp(2*x)).

1, -1, -3, -1, 41, 239, 229, -8401, -87151, -324577, 3238541, 70271519, 601086265, 142860431, -81504662539, -1393683935281, -10777424809951, 63537986981183, 3552608426329117, 60283510555017023, 441644419610814281, -6191820436867600081

Offset: 0

Seiichi Manyama, Aug 29 2022

sign

Cf. A216689, A292952, A356812, A356820.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x*exp(2*x))))

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (-x)^k/(1-2*k*x)^(k+1)))

a(n) = sum(k=0, n, (-1)^k*(2*k)^(n-k)*binomial(n, k));

G.f.: Sum_{k>=0} (-x)^k / (1 - 2*k*x)^(k+1).

a(n) = Sum_{k=0..n} (-1)^k * (2*k)^(n-k) * binomial(n,k).

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

Seiichi Manyama, Aug 29 2022

nonn

Cf. A000248, A003725, A216689, A295552.

Cf. A277456, A336951, A351737, A355501, A356820.

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

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

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-3*k*x)^(k+1)))

a(n) = sum(k=0, n, (3*k)^(n-k)*binomial(n, k));

G.f.: Sum_{k>=0} x^k / (1 - 3*k*x)^(k+1).

a(n) = Sum_{k=0..n} (3*k)^(n-k) * binomial(n,k).

cp's OEIS Frontend

A356819 Expansion of e.g.f. exp(-x * exp(2*x)).

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A356827 Expansion of e.g.f. exp(x * exp(3*x)).

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Maple

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