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%I #28 Dec 29 2023 10:25:23
%S 0,1,1,1,2,11,66,352,1730,8233,39987,209793,1240603,8287281,60473869,
%T 463764484,3647602117,29165686541,237499318823,1984374301872,
%U 17167462137733,154885317758354,1461156867801556,14381004640256202,146852743814531169,1546054541191452967
%N 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).
%C Compare with A024429 and A024430.
%C 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} (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} k^n/k! = A000110(n)*exp(1). See A143815 for more details. Compare with A143628 through A143631. The decimal expansions of R(0), R(2) - R(1) and R(1) may be found in A143819, A143820 and A143821 respectively.
%H Alois P. Heinz, <a href="/A143816/b143816.txt">Table of n, a(n) for n = 0..576</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%F a(n) = Sum_{k = 0..floor((n-1)/3)} Stirling2(n,3k+1).
%F 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).
%F a(n) = ( Bell_n(1) + w^2 * Bell_n(w) + 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
%e R(n) as a linear combination of R(0),R(1)
%e and R(2) - R(1).
%e =======================================
%e ..R(n)..|.....R(0).....R(1)...R(2)-R(1)
%e =======================================
%e ..R(3)..|.......1........1........3....
%e ..R(4)..|.......6........2........7....
%e ..R(5)..|......25.......11.......16....
%e ..R(6)..|......91.......66.......46....
%e ..R(7)..|.....322......352......203....
%e ..R(8)..|....1232.....1730.....1178....
%e ..R(9)..|....5672.....8233.....7242....
%e ..R(10).|...32202....39987....43786....
%p # (1)
%p M:=24: a:=array(0..100): b:=array(0..100): c:=array(0..100):
%p a[0]:=1: b[0]:=0: c[0]:=0:
%p for n from 1 to M do
%p b[n]:=add(binomial(n-1,k)*a[k], k=0..n-1);
%p c[n]:=add(binomial(n-1,k)*b[k], k=0..n-1);
%p a[n]:=add(binomial(n-1,k)*c[k], k=0..n-1);
%p end do:
%p A143816:=[seq(b[n], n=0..M)];
%p # (2)
%p seq(add(Stirling2(n,3*i+1),i = 0..floor((n-1)/3)), n = 0..24);
%p # third Maple program:
%p b:= proc(n, t) option remember; `if`(n=0, irem(t, 2),
%p add(b(n-j, irem(t+1, 3))*binomial(n-1, j-1), j=1..n))
%p end:
%p a:= n-> b(n, 0):
%p seq(a(n), n=0..25); # _Alois P. Heinz_, Feb 20 2018
%t m = 23; a[0] = 1; b[0] = 0; c[0] = 0; For[n = 1, n <= m, n++, b[n] = Sum[Binomial[n - 1, k]*a[k], {k, 0, n - 1}]; c[n] = Sum[Binomial[n - 1, k]*b[k], {k, 0, n - 1}]; a[n] = Sum[Binomial[n - 1, k]*c[k], {k, 0, n - 1}]]; A143816 = Table[ b[n], {n, 0, m}] (* _Jean-François Alcover_, Mar 06 2013, after Maple *)
%o (PARI) Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
%o a(n) = my(w=(-1+sqrt(3)*I)/2); round(Bell_poly(n, 1)+w^2*Bell_poly(n, w)+w*Bell_poly(n, w^2))/3; \\ _Seiichi Manyama_, Oct 13 2022
%Y Cf. A000110, A024429, A024430, A143628, A143629, A143630, A143631, A143815, A143817, A143818, A143819, A143820, A143821.
%K easy,nonn
%O 0,5
%A _Peter Bala_, Sep 03 2008
%E Spelling/notation corrections by _Charles R Greathouse IV_, Mar 18 2010
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