A143630 -id:A143630 - cp's OEIS frontend

A143815 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).

1, 0, 0, 1, 6, 25, 91, 322, 1232, 5672, 32202, 209143, 1432454, 9942517, 69363840, 490303335, 3565609732, 27118060170, 218183781871, 1861370544934, 16729411124821, 156706028787827, 1514442896327792, 14999698898942772, 151838974745743228, 1571513300578303070

Offset: 0

Peter Bala, Sep 03 2008

Compare with A024429 and A024430.
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.
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).
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.
For n > 0, the number of partitions of {1,2,...,n} into 3,6,9,... classes. - Geoffrey Critzer, Mar 05 2010

			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
  ...

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

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

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

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, 25}], x]*Table[n!, {n, 0, 25}] (* Geoffrey Critzer, Mar 05 2010 *)

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

a(n) = Sum_{k = 0..floor(n/3)} Stirling2(n, 3*k).
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)). - Geoffrey Critzer, Mar 05 2010
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
a(n) ~ n^n / (3 * (LambertW(n))^n * exp(n+1-n/LambertW(n)) * sqrt(1+LambertW(n))). - Vaclav Kotesovec, Jun 10 2025

A130151 Period 6: repeat [1, 1, 1, -1, -1, -1].

Original entry on oeis.org

1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1

Offset: 0

Paul Curtz, Aug 03 2007

			G.f. = 1 + x + x^2 - x^3 - x^4 - x^5 + x^6 + x^7 + x^8 - x^9 - x^10 - x^11 + ...
G.f. = q + q^3 + q^5 - q^7 - q^9 - q^11 + q^13 + q^15 + q^17 - q^19 - q^21 + ...

Michael Somos, Rational Function Multiplicative Coefficients
Index entries for linear recurrences with constant coefficients, signature (0,0,-1).

Cf. A131561, A131531, A257075.

Cf. A057077, A143621, A143622, A143628, A143629, A143630. - Peter Bala, Aug 28 2008

&cat [[1, 1, 1, -1, -1, -1]^^20]; // Wesley Ivan Hurt, Jul 05 2016

seq(op([1, 1, 1, -1, -1, -1]), n=0..30); # Wesley Ivan Hurt, Jul 05 2016

a[ n_] := (-1)^Quotient[n, 3]; (* Michael Somos, Apr 24 2014 *)
PadRight[{}, 100, {1, 1, 1, -1, -1, -1}] (* Wesley Ivan Hurt, Jul 05 2016 *)

{a(n) = (-1) ^ (n\3)}; /* Michael Somos, Feb 26 2011 */

a(n+6) = a(n), a(0)=a(1)=a(2)=-a(3)=-a(4)=-a(5)=1.
a(n) = ((-1)^n * (4 * (cos((2*n + 1)*Pi/3) + cos(n*Pi)) + 1) - 4) / 3. - Federico Acha Neckar (f0383864(AT)hotmail.com), Sep 01 2007
a(n) = (-1)^n * (4 * cos((2*n + 1) * Pi/3) + 1) / 3. - Federico Acha Neckar (f0383864(AT)hotmail.com), Sep 02 2007
G.f.: (1+x+x^2)/((1+x)*(x^2-x+1)). - R. J. Mathar, Nov 14 2007
a(n) = 3*a(n-1) - a(n-3) + 3*a(n-4) for n>3. - Paul Curtz, Nov 22 2007
a(n) = (-1)^floor(n/3). Compare with A057077, A143621 and A143622. Define E(k) = Sum_{n >= 0} a(n)*n^k/n! for k = 0,1,2,... . Then E(k) is an integral linear combination of E(0), E(1) and E(2) (a Dobinski-type relation). Precisely, E(k) = A143628(k)*E(0) + A143629(k)*E(1) + A143630(k)*E(2). - Peter Bala, Aug 28 2008
Euler transform of length 6 sequence [1, 0, -2, 0, 0, 1]. - Michael Somos, Feb 26 2011
a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(3^e) = -(-1)^e if e>0, b(p^e) = 1 if p == 1 (mod 4), b(p^e) = (-1)^e if p == 3 (mod 4) and p>3. - Michael Somos, Feb 26 2011
a(n + 3) = a(-1 - n) = -a(n) for all n in Z. - Michael Somos, Feb 26 2011
a(n) = (-1)^n * A257075(n) for all n in Z. - Michael Somos, Apr 15 2015
G.f.: 1 / (1 - x / (1 + 2*x^2 / (1 + x / (1 + x / (1 - x))))). - Michael Somos, Apr 15 2015
From Wesley Ivan Hurt, Jul 05 2016: (Start)
a(n) + a(n-3) = 0 for n>2.
a(n) = (cos(n*Pi) + 2*cos(n*Pi/3) + 2*sqrt(3)*sin(n*Pi/3)) / 3. (End)
a(n)*a(n-4) = a(n-1)*a(n-3) for all n in Z. - Michael Somos, Feb 25 2020

A143628 Define E(n) = Sum_{k >= 0} (-1)^floor(k/3)*k^n/k! for n = 0,1,2,... . Then E(n) is an integral linear combination of E(0), E(1) and E(2). This sequence lists the coefficients of E(0).

Original entry on oeis.org

1, 0, 0, -1, -6, -25, -89, -280, -700, -380, 13452, 149831, 1214852, 8700263, 57515640, 351296151, 1909757620, 8017484274, 5703377941, -428273438434, -7295220035921, -89868583754993, -970185398785810, -9657428906237364

Offset: 0

Peter Bala, Sep 05 2008

This sequence and its companion sequences A143629 and A143630 may be viewed as generalizations of the Uppuluri-Carpenter numbers (complementary Bell numbers) A000587. Define E(n) = Sum_{k >= 0} (-1)^floor(k/3)*k^n/k! = 0^n/0! + 1^n/1! + 2^n/2! - 3^n/3! - 4^n/4! - 5^n/5! + + + - - - ... for n = 0,1,2,... . It is easy to see that E(n+3) = 3*E(n+2) - 2*E(n+1) - Sum_{i = 0..n} 3^i* binomial(n,i)*E(n-i) for n >= 0. Thus E(n) is an integral linear combination of E(0), E(1) and E(2). Some examples are given below.

This sequence lists the coefficients of E(0). See A143629 and A143630 for the sequence of coefficients of E(1) and E(2) respectively. The functions F(n) := Sum_{k >= 0} (-1)^floor((k+1)/3)*k^n/k! and G(n) = Sum_{k >= 0} (-1)^floor((k+2)/3)*k^n/k! both satisfy the above recurrence as well as the identities E(n+1) = Sum_{i = 0..n} binomial(n,i)*F(i), F(n+1) = Sum_{i = 0..n} binomial(n,i)*G(i) and G(n+1) = - Sum_{i = 0..n} binomial(n,i)*E(i). This leads to the precise result for E(n) as a linear combination of E(0), E(1) and E(2), namely, E(n) = A143628(n)*E(0) + A143629(n)*E(1) + A143630(n)*E(2). Compare with A121867 and A143815.

			E(n) as linear combination of E(i),
i = 0..2.
====================================
..E(n)..|.....E(0).....E(1)....E(2).
====================================
..E(3)..|......-1......-2........3..
..E(4)..|......-6......-7........7..
..E(5)..|.....-25.....-23.......14..
..E(6)..|.....-89.....-80.......16..
..E(7)..|....-280....-271......-77..
..E(8)..|....-700....-750.....-922..
..E(9)..|....-380....-647....-6660..
..E(10).|...13452...13039...-41264..
...
a(5) = -25 because E(5) = -25*E(0) - 23*E(1) + 14*E(2).
a(6) = -89 because E(6) = -89*E(0) - 80*E(1) + 16*E(2).

Seiichi Manyama, Table of n, a(n) for n = 0..578
Eric Weisstein's World of Mathematics, Bell Polynomial.

A000587, A121867, A143629, A143630, A143631, A143815, A143816, A143817, A143818.

# Compare with A143815
#
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
a[n]:= -add(binomial(n-1,k)*c[k], k=0..n-1);
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);
end do:
A143628:=[seq(a[n], n=0..M)];

m = 23; a[0] = 1; b[0] = 0; c[0] = 0; For[n = 1, n <= m, n++, a[n] = -Sum[ Binomial[n-1, k]*c[k], {k, 0, n-1}]; 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}]]; Table[a[n], {n, 0, m}] (* Jean-François Alcover, Mar 06 2013, after Maple *)

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)+Bell_poly(n, -w)+Bell_poly(n, -w^2))/3; \\ Seiichi Manyama, Oct 15 2022

Define three sequences A(n), B(n) and C(n) by the relations: A(n+1) = - Sum_{i = 0..n} binomial(n,i)*C(i), B(n+1) = Sum_{i = 0..n} binomial(n,i)*A(i), C(n+1) = Sum_{i = 0..n} binomial(n,i)*B(i), with initial conditions A(0) = 1, B(0) = C(0) = 0. Then a(n) = A(n). The other sequences are B(n) = A143630 and C(n) = A143629. Compare with A143815. Also a(n) = A143629(n) + A000587(n).
From Seiichi Manyama, Oct 15 2022: (Start)
a(n) = Sum_{k = 0..floor(n/3)} (-1)^k * Stirling2(n,3*k).
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). (End)

A143817 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 C(n).

Original entry on oeis.org

0, 0, 1, 3, 7, 16, 46, 203, 1178, 7242, 43786, 259634, 1540540, 9414639, 61061613, 428890726, 3266930298, 26581123093, 226393705465, 1986997358251, 17827284972818, 163278469610570, 1531115974317975, 14771302315885372, 147267150734530892, 1521022490460243316

Offset: 0

Peter Bala, Sep 03 2008

Compare with A024429 and A024430.

This sequence and its companion sequences A(n) = A143815 and B(n) = A143816 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.

			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....

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

Cf. A000110, A024429, A024430, A143628, A143629, A143630, A143631, A143815, A143816, A143818, A143819, A143820, A143821.

# (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:
A143817:=[seq(c[n], n=0..M)];
# (2)
seq(add(Stirling2(n,3*i+2),i = 0..floor((n-2)/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, 2):
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 20 2018

a[n_] := Sum[ StirlingS2[n, 3*i+2], {i, 0, (n-2)/3}]; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Mar 06 2013 *)

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

a(n) = Sum_{k = 0..floor((n-2)/3)} Stirling2(n,3k+2).
Let w = exp(2*Pi*i/3) and set F(x) = (exp(x) + w*exp(w*x) + w^2*exp(w^2*x))/3 = x^2/2! + x^5/5! + x^8/8! + ... . 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 * Bell_n(w) + w^2 * 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

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

A143629 Define E(n) = Sum_{k>=0} (-1)^floor(k/3)*k^n/k! for n = 0,1,2,... . Then E(n) is an integral linear combination of E(0), E(1) and E(2). This sequence lists the coefficients of E(1).

Original entry on oeis.org

0, 1, 0, -2, -7, -23, -80, -271, -750, -647, 13039, 152011, 1232583, 8750796, 57405464, 349329354, 1899818951, 8008845556, 5981853002, -425732481925, -7285403175563, -89895756043392, -970910901819211, -9663021449412616

Offset: 0

Peter Bala, Sep 05 2008

This sequence and its companion sequences A143628 and A143630 may be viewed as generalizations of the Uppuluri-Carpenter numbers (complementary Bell numbers) A000587. Define E(n) = Sum_{k>=0} (-1)^floor(k/3)*k^n/k! = 0^n/0! + 1^n/1! + 2^n/2! - 3^n/3! - 4^n/4! - 5^n/5! + + + - - - ... for n = 0,1,2,... . It is easy to see that E(n+3) = 3*E(n+2) - 2*E(n+1) - Sum_{i = 0..n} 3^i*binomial(n,i)*E(n-i) for n >= 0. Thus E(n) is an integral linear combination of E(0), E(1) and E(2). This sequence lists the coefficients of E(1). Some examples are given below. The precise result for E(n) as a linear combination of E(0), E(1) and E(2) is E(n) = A143628(n)*E(0) + A143629(n)*E(1) + A143630(n)*E(2). Compare with A121867 and A143815.

			E(n) as linear combination of E(i),
i = 0..2.
====================================
..E(n)..|.....E(0).....E(1)....E(2).
====================================
..E(3)..|......-1......-2........3..
..E(4)..|......-6......-7........7..
..E(5)..|.....-25.....-23.......14..
..E(6)..|.....-89.....-80.......16..
..E(7)..|....-280....-271......-77..
..E(8)..|....-700....-750.....-922..
..E(9)..|....-380....-647....-6660..
..E(10).|...13452...13039...-41264..
...
a(5) = -23 because E(5) = -25*E(0) - 23*E(1) + 14*E(2).
a(6) = -80 because E(6) = -89*E(0) - 80*E(1) + 16*E(2).

A000587, A121867, A143628, A143630, A143631, A143815, A143816, A143817, A143818.

# Compare with A143818
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
a[n]:= -add(binomial(n-1,k)*c[k], k=0..n-1);
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);
end do:
A143629:=[seq(b[n]-c[n], n=0..M)];

m = 23; a[0] = 1; b[0] = 0; c[0] = 0; For[n = 1, n <= m, n++, a[n] = -Sum[ Binomial[n - 1, k]*c[k], {k, 0, n - 1}]; 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}] ]; A143629 = Table[b[n] - c[n], {n, 0, m}] (* Jean-François Alcover, Mar 06 2013, after Maple *)

Define three sequences A(n), B(n) and C(n) by the relations: A(n+1) = - Sum_{i = 0..n} binomial(n,i)*C(i), B(n+1) = Sum_{i = 0..n} binomial(n,i)*A(i), C(n+1) = Sum_{i = 0..n} binomial(n,i)*B(i), with initial conditions A(0) = 1, B(0) = C(0) = 0. Then a(n) = B(n) - C(n). The other sequences are A(n) = A143628(n) and C(n) = A143630(n). The values of B(n) are recorded in A143631. Compare with A143818. Also a(n) = A143628(n) - A000587(n).

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).

Original entry on oeis.org

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

Peter Bala, Sep 03 2008

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.

			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....

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

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

# (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

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 *)

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

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

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

A143818 Let 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), R(1) and R(2). This sequence gives the coefficients of R(1).

Original entry on oeis.org

0, 1, 0, -2, -5, -5, 20, 149, 552, 991, -3799, -49841, -299937, -1127358, -587744, 34873758, 380671819, 2584563448, 11105613358, -2623056379, -659822835085, -8393151852216, -69959106516419, -390297675629170, -414406919999723

Offset: 1

Peter Bala, Sep 03 2008

The coefficients of R(0) and R(2) are listed in A143815 and A143817 respectively.

			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...
...

A000110, A024429, A024430, A143628, A143629, A143630, A143631, A143815, A143816, A143817.

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
a[n]:=add(binomial(n-1,k)*b[k], k=0..n-1);
b[n]:=add(binomial(n-1,k)*c[k], k=0..n-1);
c[n]:=add(binomial(n-1,k)*a[k], k=0..n-1);
end do:
A143818:=[seq(b[n]-c[n], n=0..M)];

m = 24; a[0] = 1; b[0] = 0; c[0] = 0; For[n = 1, n <= m, n++, a[n] = Sum[Binomial[n - 1, k]*b[k], {k, 0, n - 1}]; b[n] = Sum[Binomial[n - 1, k]*c[k], {k, 0, n - 1}]; c[n] = Sum[Binomial[n - 1, k]*a[k], {k, 0, n - 1}] ]; A143818 = Table[c[n] - b[n], {n, 0, m}] (* Jean-François Alcover, Mar 06 2013, after Maple *)

a(n) = A143816(n) - A143817(n). a(n) = sum {k = 0..floor((n-1)/3)} (Stirling2(n,3k+1) - Stirling2(n,3k+2)). Let R(n) = sum {k = 0..inf} (3k)^n/(3k)! for n = 0,1,2,... . Then R(n) = A143815(n)*R(0) + A143818(n)*R(1) + A143817(n)*R(2). Some examples are given below. 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 A024429, A024430 and A143628--A143631

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

A143631 Let A(0) = 1, B(0) = 0 and C(0) = 0. Let A(n+1) = - Sum_{k = 0..n} binomial(n,k)C(k), B(n+1) = Sum_{k = 0..n} binomial(n,k)A(k) and C(n+1) = Sum_{k = 0..n} binomial(n,k)*B(k). This entry gives the sequence B(n).

Original entry on oeis.org

0, 1, 1, 1, 0, -9, -64, -348, -1672, -7307, -28225, -81817, 14191, 3143571, 38184875, 353727284, 2916494333, 22260343389, 157677357255, 1007259846130, 5241783274713, 12146415146776, -210638381350012, -4813155361775252

Offset: 0

Peter Bala, Sep 05 2008

The other sequences are A(n) = A143628(n) and C(n) = A143630(n). Compare with A121867 and A121868. See also A143816.

Seiichi Manyama, Table of n, a(n) for n = 0..578
Eric Weisstein's World of Mathematics, Bell Polynomial.

A121867, A121868, A143628, A143629, A143630, A143816.

# Compare with A143816
#
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
a[n]:= -add(binomial(n-1,k)*c[k], k=0..n-1);
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);
end do:
A143631:=[seq(b[n], n=0..M)];

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}]]; A143631 = Table[ -b[n], {n, 0, m}] (* Jean-François Alcover, Mar 06 2013, after Maple *)

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 15 2022

a(n) = A143629(n) + A143630(n).
From Seiichi Manyama, Oct 15 2022: (Start)
a(n) = Sum_{k = 0..floor((n-1)/3)} (-1)^k * Stirling2(n,3*k+1).
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). (End)

A143625 Decimal expansion of the constant E_3(0) := Sum_{n >= 0} (-1)^floor(n/3)/n! = 1 + 1/1! + 1/2! - 1/3! - 1/4! - 1/5! + + + - - - ... .

Original entry on oeis.org

2, 2, 8, 4, 9, 4, 2, 3, 8, 2, 4, 0, 9, 6, 3, 5, 2, 0, 8, 9, 9, 9, 0, 5, 0, 0, 1, 9, 2, 6, 3, 0, 8, 2, 7, 0, 2, 1, 6, 1, 5, 1, 3, 2, 6, 2, 9, 9, 4, 9, 5, 8, 9, 7, 8, 5, 9, 8, 2, 8, 8, 9, 8, 0, 0, 3, 7, 3, 7, 1, 0, 1, 5, 7, 5, 1, 9, 7, 3, 4, 5, 9, 4, 0, 3, 7, 4, 4, 9, 5, 1, 2, 5, 2, 4, 6, 3, 4, 4, 8, 8

Offset: 1

Peter Bala, Aug 30 2008

Define E_3(n) = Sum_{k >= 0} (-1)^floor(k/3)*k^n/k! = 0^n/0! + 1^n/1! + 2^n/2! - 3^n/3! - 4^n/4! - 5^n/5! + + + - - - ... for n = 0,1,2,... . It is easy to see that E_3(n+3) = 3*E_3(n+2) - 2*E_3(n+1) - Sum_{i = 0..n} 3^i*binomial(n,i) * E_3(n-i) for n >= 0. Thus E_3(n) is an integral linear combination of E_3(0), E_3(1) and E_3(2). See the examples below.
The decimal expansions of E_3(1) and E_3(2) are given in A143626 and A143627. Compare with A143623 and A143624.
E_3(n) as linear combination of E_3(i), i = 0..2.
=======================================
..E_3(n)..|....E_3(0)...E_3(1)...E_3(2)
=======================================
..E_3(3)..|.....-1.......-2........3...
..E_3(4)..|.....-6.......-7........7...
..E_3(5)..|....-25......-23.......14...
..E_3(6)..|....-89......-80.......16...
..E_3(7)..|...-280.....-271......-77...
..E_3(8)..|...-700.....-750.....-922...
..E_3(9)..|...-380.....-647....-6660...
..E_3(10).|..13452....13039...-41264...
...
The columns are A143628, A143629 and A143630.

			2.284942382409635208999050...

Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education Journal, Vol. 31, No. 1, pp. 24-28, Winter 1997.

Cf. A143623, A143624, A143626, A143627, A143628, A143629, A143630.

RealDigits[ N[ (2*E^(3/2)*(Cos[Sqrt[3]/2] + Sqrt[3]*Sin[Sqrt[3]/2]) + 1)/(3*E), 105]][[1]] (* Jean-François Alcover, Nov 08 2012 *)

Offset corrected by R. J. Mathar, Feb 05 2009

A358608 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * (n-3*k)!.

Original entry on oeis.org

1, 1, 2, 5, 23, 118, 715, 5017, 40202, 362165, 3623783, 39876598, 478639435, 6223397017, 87138414602, 1307195728565, 20916566490983, 355600289681398, 6401066509999435, 121624183842341017, 2432546407886958602, 51084541105199440565, 1123879103593765338983

Offset: 0

Seiichi Manyama, Nov 23 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A358607, A358609, A358611.

Cf. A143630, A358498.

PARI
```
a(n) = sum(k=0, n\3, (-1)^k*(n-3*k)!);
```

a(n) = n * a(n-1) - a(n-3) + n * a(n-4) for n > 3.

a(n) ~ n! * (1 - 1/n^3 - 3/n^4 - 7/n^5 - 14/n^6 - 16/n^7 + 77/n^8 + 922/n^9 + 6660/n^10 + ...), for coefficients see A143630. - Vaclav Kotesovec, Nov 25 2022

cp's OEIS Frontend

A143815 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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A130151 Period 6: repeat [1, 1, 1, -1, -1, -1].

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Magma

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A143628 Define E(n) = Sum_{k >= 0} (-1)^floor(k/3)*k^n/k! for n = 0,1,2,... . Then E(n) is an integral linear combination of E(0), E(1) and E(2). This sequence lists the coefficients of E(0).

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A143817 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 C(n).

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Extensions

A143629 Define E(n) = Sum_{k>=0} (-1)^floor(k/3)*k^n/k! for n = 0,1,2,... . Then E(n) is an integral linear combination of E(0), E(1) and E(2). This sequence lists the coefficients of E(1).

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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).

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A143818 Let 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), R(1) and R(2). This sequence gives the coefficients of R(1).

A143815 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).

A143817 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 C(n).

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).

A143631 Let A(0) = 1, B(0) = 0 and C(0) = 0. Let A(n+1) = - Sum_{k = 0..n} binomial(n,k)C(k), B(n+1) = Sum_{k = 0..n} binomial(n,k)A(k) and C(n+1) = Sum_{k = 0..n} binomial(n,k)*B(k). This entry gives the sequence B(n).