A143631 -id:A143631 - 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

A121867 Let A(0) = 1, B(0) = 0; A(n+1) = Sum_{k=0..n} binomial(n,k)B(k), B(n+1) = Sum_{k=0..n} -binomial(n,k)A(k); entry gives A sequence (cf. A121868).

Original entry on oeis.org

1, 0, -1, -3, -6, -5, 33, 266, 1309, 4905, 11516, -22935, -556875, -4932512, -32889885, -174282151, -612400262, 907955295, 45283256165, 573855673458, 5397236838345, 41604258561397, 250231901787780, 756793798761989, -8425656230853383, -213091420659985440, -2990113204010882473

Offset: 0

N. J. A. Sloane, Sep 05 2006

Stirling transform of A056594.

			From _Peter Bala_, Aug 28 2008: (Start)
E_2(k) as linear combination of E_2(i), i = 0..1.
============================
..E_2(k)..|...E_2(0)..E_2(1)
============================
..E_2(2)..|....-1.......1...
..E_2(3)..|....-3.......0...
..E_2(4)..|....-6......-5...
..E_2(5)..|....-5.....-23...
..E_2(6)..|....33.....-74...
..E_2(7)..|...266....-161...
..E_2(8)..|..1309......57...
..E_2(9)..|..4905....3466...
...
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..220
A. Fekete and G. Martin, Problem 10791: Squared Series Yielding Integers, Amer. Math. Monthly, 108 (No. 2, 2001), 177-178.

Cf. A121868, A024430, A024429, A056594.

Cf. A000587, A143623, A143624, A143628, A143631. - Peter Bala, Aug 28 2008

List([0..30], n-> Sum([0..Int(n/2)], k-> (-1)^k*Stirling2(n,2*k)) ); # G. C. Greubel, Oct 09 2019

[(&+[(-1)^k*StirlingSecond(n,2*k): k in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Oct 09 2019

# Maple code for A024430, A024429, A121867, A121868.
M:=30; a:=array(0..100); b:=array(0..100); c:=array(0..100); d:=array(0..100); a[0]:=1; b[0]:=0; c[0]:=1; d[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)*a[k], k=0..n-1); c[n]:=add(binomial(n-1,k)*d[k], k=0..n-1); d[n]:=-add(binomial(n-1,k)*c[k], k=0..n-1); od: ta:=[seq(a[n],n=0..M)]; tb:=[seq(b[n],n=0..M)]; tc:=[seq(c[n],n=0..M)]; td:=[seq(d[n],n=0..M)];
# Code based on Stirling transform:
stirtr:= proc(p) proc(n) option remember;
            add(p(k) *Stirling2(n,k), k=0..n) end
         end:
a:= stirtr(n-> (I^n + (-I)^n)/2):
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 29 2011

a[n_] := (BellB[n, -I] + BellB[n, I])/2; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Mar 06 2013, after Alois P. Heinz *)

a(n) = sum(k=0,n\2, (-1)^k*stirling(n,2*k,2));
vector(30, n, a(n-1)) \\ G. C. Greubel, Oct 09 2019

[sum((-1)^k*stirling_number2(n,2*k) for k in (0..floor(n/2))) for n in (0..30)] # G. C. Greubel, Oct 09 2019

From Peter Bala, Aug 28 2008: (Start)
This sequence and its companion A121868 are related to the pair of constants cos(1) + sin(1) and cos(1) - sin(1) and may be viewed as generalizations of the Uppuluri-Carpenter numbers (complementary Bell numbers) A000587. Define E_2(k) = Sum_{n >= 0} (-1)^floor(n/2) * n^k/n! for k = 0,1,2,... . Then E_2(0) = cos(1) + sin(1) and E_2(1) = cos(1) - sin(1). It is easy to see that E_2(k+2) = E_2(k+1) - Sum_{i = 0..k} 2^i*binomial(k,i)*E_2(k-i) for k >= 0. Hence E_2(k) is an integral linear combination of E_2(0) and E_2(1) (a Dobinski-type relation). For example, E_2(2) = - E_2(0) + E_2(1), E_2(3) = -3*E_2(0) and E_2(4) = - 6*E_2(0) - 5*E_2(1). More examples are given below.
To find the precise result, show F(k) := Sum_{n >= 0} (-1)^floor((n+1)/2)*n^k/n! satisfies the above recurrence with F(0) = E_2(1) and F(1) = -E_2(0) and then use the identity Sum_{i = 0..k} binomial(k,i)*E_2(i) = -F(k+1) to obtain E_2(k) = A121867(k) * E_2(0) - A121868(k) * E_2(1). For similar results see A143628. The decimal expansions of E_2(0) and E_2(1) are given in A143623 and A143624 respectively. (End)
E.g.f.: A(x) = cos(exp(x)-1).
a(n) = Sum_{k=0..floor(n/2)} stirling2(n,2*k)*(-1)^(k). - Vladimir Kruchinin, Jan 29 2011

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)

A121868 Let A(0) = 1, B(0) = 0; A(n+1) = Sum_{k=0..n} binomial(n,k)B(k), B(n+1) = Sum_{k=0..n} -binomial(n,k)A(k); entry gives B sequence (cf. A121867).

Original entry on oeis.org

0, -1, -1, 0, 5, 23, 74, 161, -57, -3466, -27361, -155397, -687688, -1888525, 4974059, 134695952, 1400820897, 11055147275, 70658948426, 327448854237, 223871274083, -19116044475298, -314203665206509, -3562429698724513, -33024521386113840, -250403183401213513

Offset: 0

N. J. A. Sloane, Sep 05 2006

sign

Stirling transform of (I^(n+1)+(-I)^(n+1))/2 = (0,-1,0,1,..) repeated.

			From _Peter Bala_, Aug 28 2008: (Start)
E_2(k) as a linear combination of E_2(i), i = 0..1.
============================
..E_2(k)..|...E_2(0)..E_2(1)
============================
..E_2(2)..|....-1.......1...
..E_2(3)..|....-3.......0...
..E_2(4)..|....-6......-5...
..E_2(5)..|....-5.....-23...
..E_2(6)..|....33.....-74...
..E_2(7)..|...266....-161...
..E_2(8)..|..1309......57...
..E_2(9)..|..4905....3466...
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..250
A. Fekete and G. Martin, Problem 10791: Squared Series Yielding Integers, Amer. Math. Monthly, 108 (No. 2, 2001), 177-178.
V. V. Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

Cf. A121867, A024430, A024429.

Cf. A000587, A143623, A143624, A143628, A143631.

List([0..30], n-> Sum([0..Int(n/2)], k-> (-1)^(k+1)* Stirling2(n,2*k+1)) ); # G. C. Greubel, Oct 09 2019

[(&+[(-1)^(k+1)*StirlingSecond(n,2*k+1): k in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Oct 09 2019

# Maple code for A024430, A024429, A121867, A121868.
M:=30; a:=array(0..100); b:=array(0..100); c:=array(0..100); d:=array(0..100); a[0]:=1; b[0]:=0; c[0]:=1; d[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)*a[k], k=0..n-1); c[n]:=add(binomial(n-1,k)*d[k], k=0..n-1); d[n]:=-add(binomial(n-1,k)*c[k], k=0..n-1); od: ta:=[seq(a[n],n=0..M)]; tb:=[seq(b[n],n=0..M)]; tc:=[seq(c[n],n=0..M)]; td:=[seq(d[n],n=0..M)];
# Code based on Stirling transform:
stirtr:= proc(p) proc(n) option remember;
            add(p(k) *Stirling2(n, k), k=0..n) end
         end:
a:= stirtr(n-> (I^(n+1) + (-I)^(n+1))/2):
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 29 2011

stirtr[p_] := Module[{f}, f[n_] := f[n] = Sum[p[k]*StirlingS2[n, k], {k, 0, n}]; f]; a = stirtr[(I^(#+1)+(-I)^(#+1))/2&]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 11 2014, after Alois P. Heinz *)
Table[Im[BellB[n, -I]], {n, 0, 25}] (* Vladimir Reshetnikov, Oct 22 2015 *)

a(n) = sum(k=0,n\2, (-1)^(k+1)*stirling(n,2*k+1,2));
vector(30, n, a(n-1)) \\ G. C. Greubel, Oct 09 2019

[sum((-1)^(k+1)*stirling_number2(n,2*k+1) for k in (0..floor(n/2))) for n in (0..30)] # G. C. Greubel, Oct 09 2019

From Peter Bala, Aug 28 2008: (Start)
This sequence and its companion A121867 are related to the pair of constants cos(1) + sin(1) and cos(1) - sin(1) and may be viewed as generalizations of the Uppuluri-Carpenter numbers (complementary Bell numbers) A000587.
Define E_2(k) = Sum_{n >= 0} (-1)^floor(n/2) *n^k/n! for k = 0,1,2,... . Then E_2(0) = cos(1) + sin(1) and E_2(1) = cos(1) - sin(1). Furthermore, E_2(k) is an integral linear combination of E_2(0) and E_2(1) (a Dobinski-type relation). For example, E_2(2) = - E_2(0) + E_2(1), E_2(3) = -3*E_2(0) and E_2(4) = - 6*E_2(0) - 5*E_2(1). More examples are given below. The precise result is E_2(k) = A121867(k) * E_2(0) - A121868(k) * E_2(1).
For similar results see A143628. The decimal expansions of E_2(0) and E_2(1) are given in A143623 and A143624 respectively. (End)
From Vladimir Kruchinin, Jan 26 2011: (Start)
E.g.f.: A(x) = -sin(exp(x)-1).
a(n) = Sum_{k = 0..floor(n/2)} Stirling2(n,2*k+1)*(-1)^(k+1). (End)

A143630 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(2).

Original entry on oeis.org

0, 0, 1, 3, 7, 14, 16, -77, -922, -6660, -41264, -233828, -1218392, -5607225, -19220589, 4397930, 1016675382, 14251497833, 151695504253, 1432992328055, 12527186450276, 102042171190168, 760272520469199, 4849866087637364

Offset: 0

Peter Bala, Sep 05 2008

This sequence and its companion sequences A143628 and A143629 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(2). The precise result 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) = 14 because E(5) = -25*E(0) - 23*E(1) + 14*E(2).
a(6) = 16 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, A143628, A143629, A143631, A143815, A143816, A143817, A143818.

# Compare with 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)*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:
A143630:=[seq(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}]]; A143630 = Table[c[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*Bell_poly(n, -w)+w^2*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) = C(n). The other sequences are A(n) = A143628 and B(n) = A143631. Compare with A143817.
From Seiichi Manyama, Oct 15 2022: (Start)
a(n) = Sum_{k = 0..floor((n-2)/3)} (-1)^k * Stirling2(n,3*k+2).
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). (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

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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A121867 Let A(0) = 1, B(0) = 0; A(n+1) = Sum_{k=0..n} binomial(n,k)*B(k), B(n+1) = Sum_{k=0..n} -binomial(n,k)*A(k); entry gives A sequence (cf. A121868).

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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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A121868 Let A(0) = 1, B(0) = 0; A(n+1) = Sum_{k=0..n} binomial(n,k)*B(k), B(n+1) = Sum_{k=0..n} -binomial(n,k)*A(k); entry gives B sequence (cf. A121867).

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A143630 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(2).

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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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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).

A121867 Let A(0) = 1, B(0) = 0; A(n+1) = Sum_{k=0..n} binomial(n,k)B(k), B(n+1) = Sum_{k=0..n} -binomial(n,k)A(k); entry gives A sequence (cf. A121868).

A121868 Let A(0) = 1, B(0) = 0; A(n+1) = Sum_{k=0..n} binomial(n,k)B(k), B(n+1) = Sum_{k=0..n} -binomial(n,k)A(k); entry gives B sequence (cf. A121867).

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).