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A143815
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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).
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10
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1, 0, 0, 1, 6, 25, 91, 322, 1232, 5672, 32202, 209143, 1432454, 9942517, 69363840, 490303335, 3565609732, 27118060170, 218183781871, 1861370544934, 16729411124821, 156706028787827, 1514442896327792, 14999698898942772
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Compare with A024429 and A024430.
This sequence and its companion sequences B(n) = A143816 and C(n) = A143817 may be viewed as generalisations 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 generalises 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. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Mar 05 2010]
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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]) [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Mar 05 2010]
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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....
...
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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)
with(combinat):
seq(sum(stirling2(n, 3*i), i = 0..floor(n/3)), n = 0..24);
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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}] [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Mar 05 2010]
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CROSSREFS
| A000110, A024429, A024430, A143628, A143629, A143630, A143631, A143816, A143817, A143818, A143819, A143820, A143821.
Sequence in context: A001871 A000392 A099948 * A092491 A112308 A034336
Adjacent sequences: A143812 A143813 A143814 * A143816 A143817 A143818
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KEYWORD
| easy,nonn
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AUTHOR
| Peter Bala (pbala(AT)toucansurf.com), Sep 03 2008
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