0,5

Compare with A024429 and A024430.

This sequence and its companion sequences B(n) = A143816(n) and C(n) = A143817(n) may be viewed as generalizations of the Bell numbers A000110. Define a sequence R(n) of real numbers by R(n) = Sum_{k >= 0} (3*k)^n/(3*k)! for n = 0, 1, 2, .... It is easy to verify that this sequence satisfies the recurrence relation u(n+3) = 3*u(n+2) - 2*u(n+1) + Sum_{i = 0..n} binomial(n, i)*3^(n-i)*u(i). Hence R(n) is an integral linear combination of R(0), R(1) and R(2). Some examples are given below.

To find the precise form of the linear relation define two other sequences of real numbers by S(n) = Sum_{k >= 0} (3*k+1)^n/(3*k+1)! and T(n) = Sum_{k >= 0} (3*k+2)^n/(3*k+2)! for n = 0, 1, 2, .... Both S(n) and T(n) satisfy the above recurrence. Then by means of the identities S(n+1) = Sum_{i = 0..n} binomial(n, i)*R(i), T(n+1) = Sum_{i = 0..n} binomial(n, i)*S(i) and R(n+1) = Sum_{i = 0..n} binomial(n, i)*T(i) we obtain the result R(n) = A(n)*R(0) + (B(n) - C(n))*R(1) + C(n)*R(2) = A(n)*R(0) + B(n)*R(1) + C(n)*(R(2) - R(1)) (with corresponding expressions for S(n) and T(n)). This generalizes the Dobinski's relation for the Bell numbers Sum_{k >= 0} k^n/k! = A000110(n)*exp(1).

Some examples of R(n) as a linear combination of R(0), R(1) and R(2) - R(1) are given below. The decimal expansions of R(0) = 1 + 1/3! + 1/6! + 1/9! + ..., R(2) - R(1) = 1/1! + 1/4! + 1/7! + ... and R(1) = 1/2! + 1/5! + 1/8! + ... may be found in A143819, A143820 and A143821 respectively. Compare with A143628 through A143631.

For n > 0, the number of partitions of {1,2,...,n} into 3,6,9,... classes. - Geoffrey Critzer, Mar 05 2010

Alois P. Heinz, Table of n, a(n) for n = 0..576

Eric Weisstein's World of Mathematics, Bell Polynomial.

a(n) = Sum_{k = 0..floor(n/3)} Stirling2(n, 3*k).

Let w = exp(2*Pi*i/3) and set F(x) = (exp(x) + exp(w*x) + exp(w^2*x))/3 = 1 + x^3/3! + x^6/6! + ... . Then the e.g.f. for the sequence is F(exp(x) - 1).

A143815(n) + A143816(n) + A143817(n) = Bell(n).

E.g.f. is B(A(x)) where A(x) = exp(x) - 1 and B(x) = (1/3)*(exp(x) + 2*exp(-x/2)*cos(sqrt(3)*x/2)). - Geoffrey Critzer, Mar 05 2010

a(n) = ( Bell_n(1) + Bell_n(w) + Bell_n(w^2) )/3, where Bell_n(x) is n-th Bell polynomial and w = exp(2*Pi*i/3). - Seiichi Manyama, Oct 13 2022

a(n) ~ n^n / (3 * (LambertW(n))^n * exp(n+1-n/LambertW(n)) * sqrt(1+LambertW(n))). - Vaclav Kotesovec, Jun 10 2025

R(n) as a linear combination of R(i), i = 0..2.

==================================

R(n) | R(0) R(1) R(2)

==================================

R(3) | 1 -2 3

R(4) | 6 -5 7

R(5) | 25 -5 16

R(6) | 91 20 46

R(7) | 322 149 203

R(8) | 1232 552 1178

R(9) | 5672 991 7242

R(10) | 32202 -3799 43786

...

Column 2 of the above table is A143818.

R(n) as a linear combination of R(0),R(1) and R(2) - R(1).

=====================================

R(n) | R(0) R(1) R(2)-R(1)

=====================================

R(3) | 1 1 3

R(4) | 6 2 7

R(5) | 25 11 16

R(6) | 91 66 46

R(7) | 322 352 203

R(8) | 1232 1730 1178

R(9) | 5672 8233 7242

R(10) | 32202 39987 43786

...

# (1)

M:=24: a:=array(0..100): b:=array(0..100): c:=array(0..100):

a[0]:=1: b[0]:=0: c[0]:=0:

for n from 1 to M do

b[n]:=add(binomial(n-1, k)*a[k], k=0..n-1);

c[n]:=add(binomial(n-1, k)*b[k], k=0..n-1);

a[n]:=add(binomial(n-1, k)*c[k], k=0..n-1);

end do:

A143815:=[seq(a[n], n=0..M)];

# (2)

seq(add(Stirling2(n, 3*i), i = 0..floor(n/3)), n = 0..24);

# Alternative:

b:= proc(n, t) option remember; `if`(n=0, irem(t, 2),

add(b(n-j, irem(t+1, 3))*binomial(n-1, j-1), j=1..n))

end:

a:= n-> b(n, 1):

seq(a(n), n=0..25); # Alois P. Heinz, Feb 20 2018

a = Exp[x] - 1; f[x_] := 1/3 (E^x + 2 E^(-x/2) Cos[(Sqrt[3] x)/2]); CoefficientList[Series[f[a], {x, 0, 25}], x]*Table[n!, {n, 0, 25}] (* Geoffrey Critzer, Mar 05 2010 *)

(PARI) Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);

a(n) = my(w=(-1+sqrt(3)*I)/2); round(Bell_poly(n, 1)+Bell_poly(n, w)+Bell_poly(n, w^2))/3; \\ Seiichi Manyama, Oct 13 2022

Cf. A000110, A024429, A024430, A143628, A143629, A143630, A143631, A143816, A143817, A143818, A143819, A143820, A143821.

Sequence in context: A099948 A333017 A277973 * A209241 A369360 A092491

Adjacent sequences: A143812 A143813 A143814 * A143816 A143817 A143818

easy,nonn

Peter Bala, Sep 03 2008

approved