A143815 - OEIS

%I #26 Feb 23 2024 04:22:29

%S 1,0,0,1,6,25,91,322,1232,5672,32202,209143,1432454,9942517,69363840,

%T 490303335,3565609732,27118060170,218183781871,1861370544934,

%U 16729411124821,156706028787827,1514442896327792,14999698898942772,151838974745743228,1571513300578303070

%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 A(n).

%C Compare with A024429 and A024430.

%C This sequence and its companion sequences B(n) = A143816(n) and C(n) = A143817(n) may be viewed as generalizations of the Bell numbers A000110. Define a sequence R(n) of real numbers by R(n) = Sum_{k >= 0} (3*k)^n/(3*k)! 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.

%C To find the precise form of the linear relation define two other sequences of real numbers by S(n) = Sum_{k >= 0} (3*k+1)^n/(3*k+1)! and T(n) = Sum_{k >= 0} (3*k+2)^n/(3*k+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's relation for the Bell numbers Sum_{k >= 0} k^n/k! = A000110(n)*exp(1).

%C 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 through A143631.

%C For n > 0, the number of partitions of {1,2,...,n} into 3,6,9,... classes. - _Geoffrey Critzer_, Mar 05 2010

%H Alois P. Heinz, <a href="/A143815/b143815.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/3)} Stirling2(n, 3*k).

%F 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).

%F A143815(n) + A143816(n) + A143817(n) = Bell(n).

%F 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

%F a(n) = ( Bell_n(1) + Bell_n(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(i),

%e i = 0..2.

%e ====================================

%e ..R(n)..|.....R(0)....R(1)....R(2)..

%e ====================================

%e ..R(3)..|.......1......-2.......3...

%e ..R(4)..|.......6......-5.......7...

%e ..R(5)..|......25......-5......16...

%e ..R(6)..|......91......20......46...

%e ..R(7)..|.....322.....149.....203...

%e ..R(8)..|....1232.....552....1178...

%e ..R(9)..|....5672.....991....7242...

%e ..R(10).|...32202...-3799...43786...

%e ...

%e Column 2 of the above table is A143818.

%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....

%e ...

%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 A143815:=[seq(a[n], n=0..M)];

%p # (2)

%p seq(add(Stirling2(n,3*i),i = 0..floor(n/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, 1):

%p seq(a(n), n=0..25);  # _Alois P. Heinz_, Feb 20 2018

%t 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]

%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)+Bell_poly(n, w)+Bell_poly(n, w^2))/3; \\ _Seiichi Manyama_, Oct 13 2022

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

%K easy,nonn

%O 0,5

%A _Peter Bala_, Sep 03 2008