A143817 -id:A143817 - cp's OEIS frontend

A358498 a(n) = Sum_{k=0..floor(n/3)} (n-3*k)!.

1, 1, 2, 7, 25, 122, 727, 5065, 40442, 363607, 3633865, 39957242, 479365207, 6230654665, 87218248442, 1308153733207, 20929020542665, 355774646344442, 6403681859461207, 121666029429374665, 2433257782822984442, 51097345853568901207, 1124122393807037054665

Offset: 0

Seiichi Manyama, Nov 19 2022

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A136580, A358499, A358500.

PARI
```
a(n) = sum(k=0, n\3, (n-3*k)!);
```

a(n) = n * a(n-1) + a(n-3) - n * a(n-4) for n > 3.

a(n) ~ n! * (1 + 1/n^3 + 3/n^4 + 7/n^5 + 16/n^6 + 46/n^7 + 203/n^8 + 1178/n^9 + 7242/n^10 + ...), for coefficients see A143817. - Vaclav Kotesovec, Nov 24 2022

A357784 a(n) = Sum_{k=0..floor((n-2)/3)} 2^k * Stirling2(n,3*k+2).

Original entry on oeis.org

0, 0, 1, 3, 7, 17, 61, 343, 2231, 14301, 88561, 542011, 3397483, 22638993, 164336085, 1299899087, 10991061663, 97070035205, 881323166809, 8173386231395, 77489746906355, 754631383660729, 7590899551399869, 79174328607339767, 856889470005396071

Offset: 0

Seiichi Manyama, Oct 13 2022

nonn

Eric Weisstein's MathWorld, Bell Polynomial.

Cf. A357782, A357783.

Cf. A143817, A357833.

a(n) = sum(k=0, (n-2)\3, 2^k*stirling(n, 3*k+2, 2));

my(N=30, x='x+O('x^N)); concat([0, 0], Vec(serlaplace(sum(k=0, N\3, 2^k*(exp(x)-1)^(3*k+2)/(3*k+2)!))))

Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
a(n) = my(v=2^(1/3), w=(-1+sqrt(3)*I)/2); round((Bell_poly(n, v)+w*Bell_poly(n, v*w)+w^2*Bell_poly(n, v*w^2))/(3*v^2));

Let A(0)=1, B(0)=0 and C(0)=0. Let B(n+1) = Sum_{k=0..n} binomial(n,k)*A(k), C(n+1) = Sum_{k=0..n} binomial(n,k)*B(k) and A(n+1) = 2 * Sum_{k=0..n} binomial(n,k)*C(k). A357782(n) = A(n), A357783(n) = B(n) and a(n) = C(n).
Let w = exp(2*Pi*i/3) and set F(x) = (exp(x) + w*exp(w*x) + w^2*exp(w^2*x))/3 = x^2/2! + x^5/5! + x^8/8! + ... . Then the e.g.f. for the sequence is F(2^(1/3) * (exp(x)-1))/(2^(2/3)).
a(n) = ( Bell_n(2^(1/3)) + w * Bell_n(2^(1/3)*w) + w^2 * Bell_n(2^(1/3)*w^2) )/(3*2^(2/3)), where Bell_n(x) is n-th Bell polynomial.

cp's OEIS Frontend

A358498 a(n) = Sum_{k=0..floor(n/3)} (n-3*k)!.

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A357784 a(n) = Sum_{k=0..floor((n-2)/3)} 2^k * Stirling2(n,3*k+2).

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