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A143815 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) = Sum_{k = 0..n} binomial(n,k)*C(k). This entry gives the sequence A(n). 10
1, 0, 0, 1, 6, 25, 91, 322, 1232, 5672, 32202, 209143, 1432454, 9942517, 69363840, 490303335, 3565609732, 27118060170, 218183781871, 1861370544934, 16729411124821, 156706028787827, 1514442896327792, 14999698898942772, 151838974745743228, 1571513300578303070 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Compare with A024429 and A024430.

This sequence and its companion sequences B(n) = A143816 and C(n) = A143817 may be viewed as generalizations of the Bell numbers A000110. Define a sequence R(n) of real numbers by R(n) := sum {k = 0..inf} (3k)^n/(3k)! 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 consider two other sequences of real numbers S(n) := sum {k = 0..inf} (3k+1)^n/(3k+1)! and T(n) := sum {k = 0..inf} (3k+2)^n/(3k+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 relation for the Bell numbers: sum {k = 0..inf} 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 - A143631.

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

LINKS

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

FORMULA

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

EXAMPLE

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....

...

MAPLE

# (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);

# third Maple program:

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

MATHEMATICA

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, 20}], x]*Table[n!, {n, 0, 20}] [Geoffrey Critzer, Mar 05 2010]

CROSSREFS

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

Sequence in context: A000392 A099948 A277973 * A209241 A092491 A112308

Adjacent sequences:  A143812 A143813 A143814 * A143816 A143817 A143818

KEYWORD

easy,nonn

AUTHOR

Peter Bala, Sep 03 2008

STATUS

approved

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Last modified July 20 16:34 EDT 2019. Contains 325185 sequences. (Running on oeis4.)