A143816 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).

0, 1, 1, 1, 2, 11, 66, 352, 1730, 8233, 39987, 209793, 1240603, 8287281, 60473869, 463764484, 3647602117, 29165686541, 237499318823, 1984374301872, 17167462137733, 154885317758354, 1461156867801556, 14381004640256202, 146852743814531169, 1546054541191452967

Offset: 0

Author

Peter Bala, Sep 03 2008

Keywords

• easy
• nonn

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} (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.

Examples

			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....
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..576
• Eric Weisstein's World of Mathematics, Bell Polynomial.

Crossrefs

Cf. A000110, A024429, A024430, A143628, A143629, A143630, A143631, A143815, A143817, A143818, A143819, A143820, A143821.

Programs

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)
seq(add(Stirling2(n,3*i+1),i = 0..floor((n-1)/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, 0):
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 20 2018

Mathematica

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 *)

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)+w^2*Bell_poly(n, w)+w*Bell_poly(n, w^2))/3; \\ Seiichi Manyama, Oct 13 2022

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).

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

Extensions

Spelling/notation corrections by Charles R Greathouse IV, Mar 18 2010