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A143816
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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 B(n).
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11
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0, 1, 1, 1, 2, 11, 66, 352, 1730, 8233, 39987, 209793, 1240603, 8287281, 60473869, 463764484, 3647602117, 29165686541, 237499318823, 1984374301872, 17167462137733, 154885317758354, 1461156867801556, 14381004640256202
(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 A(n) = A143815 and C(n) = A143817 may be viewed as generalizations of the Bell numbers A000110. Define R(n) = sum {k = 0..inf} (3k)^n/(3k)! for n = 0,1,2,... . Then the real number R(n) is an integral linear combination of R(0) = 1 + 1/3! + 1/6! + ...., R(2) - R(1) = 1/1! + 1/4! + 1/7! + ... and R(1) = 1/2! + 1/5! + 1/8! + ... . Some examples are given below. The precise result is R(n) = A(n)*R(0) + B(n)*R(1) + C(n)*(R(2)-R(1)). This generalizes the Dobinski relation for the Bell numbers: sum {k = 0..inf} k^n/k! = A000110(n)*exp(1). See A143815 for more details. Compare with A143628 - A143631. The decimal expansions of R(0), R(2) - R(1) and R(1) may be found in A143819, A143820 and A143821 respectively.
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FORMULA
| a(n) = sum {k = 0..floor((n-1)/3)} Stirling2(n,3k+1). Let w = exp(2*Pi*i/3) and set F(x) = (exp(x) + w^2*exp(w*x) + w*exp(w^2*x))/3 = x + x^4/4! + x^7/7! + ... . Then the e.g.f. for the sequence is F(exp(x)-1). A143815(n) + A143816(n) + A143817(n) = Bell(n).
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EXAMPLE
| 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:
A143816:=[seq(b[n], n=0..M)];
(2)
with(combinat):
seq(sum(stirling2(n, 3*i+1), i = 0..floor((n-1)/3)), n = 0..24);
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CROSSREFS
| A000110, A024429, A024430, A143628, A143629, A143630, A143631, A143815, A143817, A143818, A143819, A143820, A143821.
Sequence in context: A199412 A074613 A039632 * A160852 A185627 A138792
Adjacent sequences: A143813 A143814 A143815 * A143817 A143818 A143819
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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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EXTENSIONS
| Spelling/notation corrections by Charles R Greathouse IV (charles.greathouse(AT)case.edu), Mar 18 2010
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