A347948 -id:A347948 - cp's OEIS frontend

A347949 E.g.f.: 1 / (1 - Sum_{k>=1} x^prime(k) / prime(k)).

1, 0, 1, 2, 6, 64, 170, 2988, 14616, 180192, 1934712, 21673200, 300266736, 4220710272, 61785461712, 1003589762784, 17448621367680, 327598207658496, 6279739240655232, 134169095009652480, 2817563310900129024, 64570676279407718400, 1547773850801172960000, 38824156236466815920640

Offset: 0

Ilya Gutkovskiy, Sep 20 2021

nonn

Cf. A007840, A010051, A023360, A218002, A347948.

nmax = 23; CoefficientList[Series[1/(1 - Sum[x^Prime[k]/Prime[k], {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] (k - 1)! Boole[PrimeQ[k]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (k-1)! * A010051(k) * a(n-k).

A353165 Expansion of e.g.f. 1/(1 - Sum_{p prime} x^p / (p-1)!).

Original entry on oeis.org

1, 0, 2, 3, 24, 125, 900, 7987, 72240, 817488, 9536940, 127816931, 1830739680, 28597812685, 481145656992, 8650951895040, 166376275228800, 3392537585542577, 73342277086465296, 1672523286398099407, 40158685461798481560, 1012415090497566344400

Offset: 0

Seiichi Manyama, Apr 28 2022

nonn

Cf. A000040, A006153, A347948, A353163, A353164.

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

a(n) = if(n==0, 1, sum(k=1, n, isprime(k)*k*binomial(n, k)*a(n-k)));

a(0) = 1; a(n) = Sum_{p<=n, p prime} p * binomial(n,p) * a(n-p).

cp's OEIS Frontend

A347949 E.g.f.: 1 / (1 - Sum_{k>=1} x^prime(k) / prime(k)).

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