A121868 -id:A121868 - cp's OEIS frontend

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).

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

A057077 Periodic sequence 1,1,-1,-1; expansion of (1+x)/(1+x^2).

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

Offset: 0

Wolfdieter Lang, Aug 04 2000

Abscissa of the image produced after n alternating reflections of (1,1) over the x and y axes respectively. Similarly, the ordinate of the image produced after n alternating reflections of (1,1) over the y and x axes respectively. - Wesley Ivan Hurt, Jul 06 2013

Alex Fink, Richard K. Guy, and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114. See Section 13.
T.-X. He and L. W. Shapiro, Fuss-Catalan matrices, their weighted sums, and stabilizer subgroups of the Riordan group, Lin. Alg. Applic. 532 (2017) 25-41, Theorem 2.5, k=2.
StackExchange, Finding sum(k=1..oo) (-1)^(k(k+1)/2), (2025) provides Dirichlet sums.
Index entries for linear recurrences with constant coefficients, signature (0,-1).
Index entries for sequences related to Chebyshev polynomials.

|a(n)|=A000012. Cf. A049310.

Cf. A000587, A121867, A121868, A130151, A143621, A143622, A143623, A143624, A143628.

&cat[[1, 1, -1, -1]^^20]; // Vincenzo Librandi, Feb 18 2016

seq((-1)^floor(k/2),k=0..70); # Wesley Ivan Hurt, Jul 06 2013
A057077 := proc(n)
    op(1+(n mod 4),[1,1,-1,-1]) ;
end proc:
seq(A057077(n),n=0..40) ; # R. J. Mathar, Feb 27 2025

a[n_] := {1, 1, -1, -1}[[Mod[n, 4] + 1]] (* Jean-François Alcover, Jul 05 2013 *)
PadRight[{},80,{1,1,-1,-1}] (* Harvey P. Dale, Jun 21 2015 *)

A057077(n) := block(
        [1,1,-1,-1][1+mod(n,4)]
)$ /* R. J. Mathar, Mar 19 2012 */

a(n)=(-1)^(n\2) \\ Charles R Greathouse IV, Nov 07 2016

def A057077(n): return -1 if n&2 else 1 # Chai Wah Wu, Jun 30 2024

G.f.: (1+x)/(1+x^2).
a(n) = S(n, 0) + S(n-1, 0) = S(2*n, sqrt(2)); S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind, A049310. S(n, 0)=A056594.
a(n) = (-1)^binomial(n,2) = (-1)^floor(n/2) = 1/2*((n+2) mod 4 - n mod 4). For fixed r = 0,1,2,..., it appears that (-1)^binomial(n,2^r) gives a periodic sequence of period 2^(r+1), the period consisting of a block of 2^r plus ones followed by a block of 2^r minus ones. See A033999 (r = 0), A143621 (r = 2) and A143622 (r = 3). Define E(k) = sum {n = 0..inf} a(n)*n^k/n! for k = 0,1,2,... . Then E(0) = cos(1) + sin(1), E(1) = cos(1) - sin(1) and E(k) is an integral linear combination of E(0) and E(1) (a Dobinski-type relation). Precisely, E(k) = A121867(k) * E(0) - A121868(k) * E(1). See A143623 and A143624 for the decimal expansions of E(0) and E(1) respectively. For a fixed value of r, similar relations hold between the values of the sums E_r(k) := Sum_{n>=0} (-1)^floor(n/r)*n^k/n!, k = 0,1,2,... . For particular cases see A000587 (r = 1) and A143628 (r = 3). - Peter Bala, Aug 28 2008
Sum_{k>=0} a(k)/(k+1) = Sum_{k>=0} 1/((a(k)*(k+1))) = log(2)/2 + Pi/4. - Jaume Oliver Lafont, Apr 30 2010
a(n) = (-1)^A180969(1,n), where the first index in A180969(.,.) is the row index. - Adriano Caroli, Nov 18 2010
a(n) = (-1)^((2*n+(-1)^n-1)/4) = i^((n-1)*n), with i=sqrt(-1). - Bruno Berselli, Dec 27 2010 - Aug 26 2011
Non-simple continued fraction expansion of (3+sqrt(5))/2 = A104457. - R. J. Mathar, Mar 08 2012
E.g.f.: cos(x)*(1 + tan(x)). - Arkadiusz Wesolowski, Aug 31 2012
From Ricardo Soares Vieira, Oct 15 2019: (Start)
E.g.f.: sin(x) + cos(x) = sqrt(2)*sin(x + Pi/4).
a(n) = sqrt(2)*(d^n/dx^n) sin(x)|_x=Pi/4, i.e., a(n) equals sqrt(2) times the n-th derivative of sin(x) evaluated at x=Pi/4. (End)
a(n) = 4*floor(n/4) - 2*floor(n/2) + 1. - Ridouane Oudra, Mar 23 2024

A024429 Expansion of e.g.f. sinh(exp(x)-1).

Original entry on oeis.org

0, 1, 1, 2, 7, 27, 106, 443, 2045, 10440, 57781, 340375, 2115664, 13847485, 95394573, 690495874, 5235101739, 41428115543, 341177640610, 2917641580783, 25866987547865, 237421321934176, 2252995117706961, 22073206655954547, 222971522853648704, 2319379362420267753

Offset: 0

Clark Kimberling

nonn

Number of partitions of an n-element set into an odd number of classes. - Peter Luschny, Apr 25 2011

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. A024430).

			G.f. = x + x^2 + 2*x^3 + 7*x^4 + 27*x^5 + 106*x^6 + 443*x^7 + 2045*x^8 + ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 226, 4th line of table.

G. C. Greubel, Table of n, a(n) for n = 0..400
A. Fekete and G. Martin, Problem 10791: Squared Series Yielding Integers, Amer. Math. Monthly, 108 (No. 2, 2001), 177-178.
S. K. Ghosal, J. K. Mandal, Stirling Transform Based Color Image Authentication, Procedia Technology, 2013 Volume 10, 2013, Pages 95-104.
Eric Weisstein's World of Mathematics, Stirling Transform.

Cf. A024430, A121867, A121868, A000110, A000587.

List([0..25], n-> Sum([0..Int(n/2)], k-> Stirling2(n,2*k+1)) ); # G. C. Greubel, Oct 09 2019

a:= func< n | (&+[StirlingSecond(n,2*k+1): k in [0..Floor(n/2)]]) >;
[a(n): n in [0..25]]; // G. C. Greubel, Oct 09 2019

b:= proc(n, t) option remember; `if`(n=0, t, add(
       b(n-j, 1-t)*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..28);  # Alois P. Heinz, Jan 15 2018
with(combinat); seq((bell(n) - BellB(n, -1))/2, n = 0..25); # G. C. Greubel, Oct 09 2019

CoefficientList[Series[Sinh[E^x-1], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Aug 04 2014 *)
Table[(BellB[n] - BellB[n, -1])/2, {n, 0, 20}] (* Vladimir Reshetnikov, Nov 01 2015 *)

x='x+O('x^50); concat([0], Vec(serlaplace(sinh(exp(x)-1)))) \\ G. C. Greubel, Nov 12 2017

def A024429(n) :
    return add(stirling_number2(n,i) for i in range(1,n+n%2,2))
# Peter Luschny, Feb 28 2012

S(n,1) + S(n,3) + ... + S(n,2k+1), where k = [ (n-1)/2 ] and S(i,j) are Stirling numbers of second kind.
E.g.f.: sinh(exp(x)-1). - N. J. A. Sloane, Jan 28 2001
a(n) = (A000110(n) - A000587(n)) / 2. - Peter Luschny, Apr 25 2011
G.f.: x*G(0) where G(k) = 1 - x*(2*k+1)/((2*x*k+x-1) - x*(2*x*k+x-1)/(x - (2*k+1)*(2*x*k+2*x-1)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 06 2013
G.f.: x*G(0)/(1+x) where G(k) = 1 - 2*x*(k+1)/((2*x*k+x-1) - x*(2*x*k+x-1)/(x - 2*(k+1)*(2*x*k+2*x-1)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 06 2013
G.f.: -x*(1+x)*Sum_{k>=0} x^(2*k)/((2*x*k+x-1)*Product_{p=0..k} (2*x*p-1)*(2*x*p-x-1)). - Sergei N. Gladkovskii, Jan 06 2013
G.f.: Sum_{k>=0} x^(2*k+1)/(Product_{i=0..2*k+1} 1-i*x). - Sergei N. Gladkovskii, Jan 06 2013
a(n) ~ n^n / (2 * (LambertW(n))^n * exp(n+1-n/LambertW(n)) * sqrt(1+LambertW(n))). - Vaclav Kotesovec, Aug 04 2014

Description changed by N. J. A. Sloane, Sep 05 2006

A024430 Expansion of e.g.f. cosh(exp(x)-1).

Original entry on oeis.org

1, 0, 1, 3, 8, 25, 97, 434, 2095, 10707, 58194, 338195, 2097933, 13796952, 95504749, 692462671, 5245040408, 41436754261, 340899165549, 2915100624274, 25857170687507, 237448494222575, 2253720620740362, 22078799199129799, 222987346441156585, 2319210969809731600

Offset: 0

Clark Kimberling

nonn

Number of partitions of an n-element set into an even number of classes.

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. A024429).

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 226, 5th line of table.
S. K. Ghosal, J. K. Mandal, Stirling Transform Based Color Image Authentication, Procedia Technology, 2013 Volume 10, 2013, Pages 95-104.
L. Lovasz, Combinatorial Problems and Exercises, North-Holland, 1993, pp. 15, 148.

Alois P. Heinz, Table of n, a(n) for n = 0..576
A. Fekete and G. Martin, Problem 10791: Squared Series Yielding Integers, Amer. Math. Monthly, 108 (No. 2, 2001), 177-178.
Eric Weisstein's World of Mathematics, Stirling Transform.

Cf. A024429, A121867, A121868, A000110, A000587.

List([0..25], n-> Sum([0..Int(n/2)], k-> Stirling2(n,2*k)) ); # G. C. Greubel, Oct 09 2019

a:= func< n | (&+[StirlingSecond(n,2*k): k in [0..Floor(n/2)]]) >;
[a(n): n in [0..25]]; // G. C. Greubel, Oct 09 2019

b:= proc(n, t) option remember; `if`(n=0, t, add(
       b(n-j, 1-t)*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..28);  # Alois P. Heinz, Jan 15 2018
with(combinat); seq((bell(n) + BellB(n, -1))/2, n = 0..20); # G. C. Greubel, Oct 09 2019

nn=20;a=Exp[Exp[x]-1];Range[0,nn]!CoefficientList[Series[(a+1/a)/2,{x,0,nn}],x]  (* Geoffrey Critzer, Nov 04 2012 *)
Table[(BellB[n] + BellB[n, -1])/2, {n, 0, 20}] (* Vladimir Reshetnikov, Nov 01 2015 *)

{a(n)=polcoeff(sum(m=0, n, x^(2*m)/prod(k=1, 2*m, 1-k*x +x*O(x^n))), n)} \\ Paul D. Hanna, Sep 05 2012

def A024430(n) :
    return add(stirling_number2(n,i) for i in range(0,n+(n+1)%2,2))
# Peter Luschny, Feb 28 2012

a(n) = S(n, 2) + S(n, 4) + ... + S(n, 2k), where k = [ n/2 ], S(i, j) are Stirling numbers of second kind.
E.g.f.: cosh(exp(x)-1). - N. J. A. Sloane, Jan 28 2001
a(n) = (A000110(n) + A000587(n)) / 2. - Peter Luschny, Apr 25 2011
O.g.f.: Sum_{n>=0} x^(2*n) / Product_{k=0..2*n} (1 - k*x). - Paul D. Hanna, Sep 05 2012
G.f.: G(0)/(1+x) where G(k) = 1 - x*(2*k+1)/((2*x*k-1) - x*(2*x*k-1)/(x - (2*k+1)*(2*x*k+x-1)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 05 2013
G.f.: G(0)/(1+2*x) where G(k) = 1 - 2*x*(k+1)/((2*x*k-1) - x*(2*x*k-1)/(x - 2*(k+1)*(2*x*k+x-1)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 05 2013
a(n) ~ n^n / (2 * (LambertW(n))^n * exp(n+1-n/LambertW(n)) * sqrt(1+LambertW(n))). - Vaclav Kotesovec, Aug 04 2014

Description changed by N. J. A. Sloane, Jun 14 2003 and again Sep 05 2006

A143623 Decimal expansion of the constant cos(1) + sin(1).

Original entry on oeis.org

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

Offset: 1

Peter Bala, Aug 30 2008

cos(1) + sin(1) = Sum_{n >= 0} (-1)^floor(n/2)/n! = 1 + 1/1! - 1/2! - 1/3! + 1/4! + 1/5! - 1/6! - 1/7! + + - - ... .
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(1) - E_2(0), E_2(3) = -3*E_2(0) and E_2(4) = -5*E_2(1) - 6*E_2(0). The precise result is E_2(k) = A121867(k) * E_2(0) - A121868(k) * E_2(1).
The decimal expansion of the constant cos(1) - sin(1) = E_2(1) is recorded in A143624. Compare with A143625.

			1.38177329067603622405 ... .

Cf. A049469, A049470, A057077, A121867, A121868, A143624, A143625.

RealDigits[Cos[1]+Sin[1],10,120][[1]] (* Harvey P. Dale, Mar 01 2019 *)

sqrt(2)*sin(1+Pi/4) \\ Charles R Greathouse IV, Feb 03 2025

Equals sin(1+Pi/4)*sqrt(2). - Franklin T. Adams-Watters, Jun 27 2014

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

A143624 Decimal expansion of the negated constant cos(1) - sin(1) = -0.3011686789...

Original entry on oeis.org

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

Offset: 0

Peter Bala, Aug 30 2008

cos(1) - sin(1) = Sum_{n>=0} (-1)^floor(n/2)*n/n! = 1/1! - 2/2! - 3/3! + 4/4! + 5/5! - 6/6! - 7/7! + + - - ... . Define E_2(k) = Sum_{n>=0} (-1)^floor(n/2)*n^k/n! for k = 0,1,2,... . Then E_2(1) = cos(1) - sin(1) and E_2(0) = 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(1) - E_2(0), E_2(3) = -3*E_2(0) and E_2(4) = -5*E_2(1) - 6*E_2(0). The precise result is E_2(k) = A121867(k) * E_2(0) - A121868(k) * E_2(1). The decimal expansion of the constant cos(1) + sin(1) is recorded in A143623. Compare with A143625.

			 -0.30116867893975678925156571418732239589025264018...

Eric Weisstein's World of Mathematics, Spherical Bessel Function of the First Kind

Cf. A049469, A049470, A057077, A121867, A121868, A143623, A143625, A174549.

RealDigits[Cos[1] - Sin[1], 10, 100][[1]] (* Amiram Eldar, Aug 07 2020 *)

cos(1)-sin(1) \\ Charles R Greathouse IV, Feb 04 2025

Equals sin(1-Pi/4)*sqrt(2). - Franklin T. Adams-Watters, Jun 27 2014
Equals j_1(1), where j_1(z) is the spherical Bessel function of the first kind. - Stanislav Sykora, Jan 11 2017
From Amiram Eldar, Aug 07 2020: (Start)
Equals -Integral_{x=0..1} x*sin(x) dx.
Equals Sum_{k>=1} (-1)^k/((2*k-1)! * (2*k+1)) = Sum_{k>=1} (-1)^k/A174549(k). (End)

Added sign in definition. Offset corrected by R. J. Mathar, Feb 05 2009

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)

A358607 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * (n-2*k)!.

Original entry on oeis.org

1, 1, 1, 5, 23, 115, 697, 4925, 39623, 357955, 3589177, 39558845, 475412423, 6187461955, 86702878777, 1301486906045, 20836087009223, 354385941189955, 6381537618718777, 121290714467642045, 2426520470557921223, 50969651457241797955, 1121574207307049758777

Offset: 0

Seiichi Manyama, Nov 23 2022

nonn

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

Cf. A358608, A358609, A358611.

Cf. A121868, A136580.

PARI
```
a(n) = sum(k=0, n\2, (-1)^k*(n-2*k)!);
```

a(n) = n * a(n-1) - a(n-2) + n * a(n-3) for n > 2.
a(n) ~ n! * (1 - 1/n^2 - 1/n^3 + 5/n^5 + 23/n^6 + 74/n^7 + 161/n^8 - 57/n^9 - 3466/n^10 - ...), for coefficients see A121868. - Vaclav Kotesovec, Nov 25 2022
a(2n) = 1+A215096(2n). a(2n+1) = A215096(2n+1). - R. J. Mathar, Jun 14 2024

A121869 Monthly Problem 10791, first expression.

Original entry on oeis.org

-1, 1, 0, -5, -15, 104, 1827, 7893, -207000, -5646249, -47897675, 1479282600, 74711288407, 1396956334921, -21032523700672, -2719998717430365, -104158663871982343, -715846242343471272, 189941380201812700699, 14820744271258596866013, 507768838531742620183176

Offset: 0

N. J. A. Sloane, Sep 05 2006

sign

G. C. Greubel, Table of n, a(n) for n = 0..330
A. Fekete and G. Martin, Problem 10791: Squared Series Yielding Integers, Amer. Math. Monthly, 108 (No. 2, 2001), 177-178.

Cf. A121870, A024429, A024430, A121867, A121868.

List([0..25], n-> (-1)*Sum([0..n], k-> Stirling2(n,k)) *Sum([0..n], k-> (-1)^k*Stirling2(n,k)) ); # G. C. Greubel, Oct 08 2019

a:= func< n | (-1)*(&+[StirlingSecond(n,k): k in [0..n]])*(&+[ (-1)^k*StirlingSecond(n,k): k in [0..n]]) >;
[a(n): n in [0..25]]; // G. C. Greubel, Oct 08 2019

with(combinat): seq(-bell(n)*BellB(n, -1), n = 0..25); # G. C. Greubel, Oct 08 2019

Table[-BellB[n]*BellB[n, -1], {n,0,25}] (* G. C. Greubel, Oct 08 2019 *)

a(n) = (-1)*sum(k=0,n, stirling(n,k,2))*sum(k=0,n, (-1)^k*stirling(n,k,2));
vector(25, n, a(n-1)) \\ G. C. Greubel, Oct 08 2019

[ -sum(stirling_number2(n, k) for k in (0..n))*sum((-1)^k* stirling_number2(n,k) for k in (0..n)) for n in (0..25)] # G. C. Greubel, Oct 08 2019

a(n) = A024429(n)^2 - A024430(n)^2.

A009061 Expansion of e.g.f. cos(sinh(x)*exp(x)).

Original entry on oeis.org

1, 0, -1, -6, -27, -100, -237, 742, 18025, 194904, 1689671, 12483570, 72272013, 155614004, -4305757029, -101460169442, -1561477983407, -20064006763728, -223375429298929, -2048612121431958, -11401251676320843, 95849085744834380

Offset: 0

R. H. Hardin

G. C. Greubel, Table of n, a(n) for n = 0..250
Eric Weisstein's MathWorld, Bell Polynomial.

Cf. A009496, A121868.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Cos(Sinh(x)*Exp(x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jul 23 2018

seq(coeff(series(factorial(n)*cos(sinh(x)*exp(x)), x,n+1),x,n),n=0..25); # Muniru A Asiru, Jul 24 2018

Table[SeriesCoefficient[Cos[Sinh[x] Exp[x]], {x, 0, n}] n!, {n, 0, 20}]
Table[2^n Re[BellB[n, I/2]], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 22 2015 *)

x='x+O('x^30); Vec(serlaplace(cos(sinh(x)*exp(x)))) \\ G. C. Greubel, Jul 23 2018

a(n) = 2^n*Re(B_n(i/2)), where B_n(x) is n-th Bell polynomial, i=sqrt(-1). Vladimir Reshetnikov, Oct 22 2015

Extended with signs by Olivier Gérard, Mar 15 1997

cp's OEIS Frontend

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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A057077 Periodic sequence 1,1,-1,-1; expansion of (1+x)/(1+x^2).

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A024429 Expansion of e.g.f. sinh(exp(x)-1).

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A024430 Expansion of e.g.f. cosh(exp(x)-1).

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A143623 Decimal expansion of the constant cos(1) + sin(1).

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A143624 Decimal expansion of the negated constant cos(1) - sin(1) = -0.3011686789...

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).

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).